CIVIL ENGINEERING 365 ALL ABOUT CIVIL ENGINEERING



IntroductionThe USACE maintains and operates 239 lock chambers at 193 sites on the inland waterway system (USACE 2019b). Navigation locks are essential components of the inland waterway system because they enable barges and other vessels to cross over geographic features and dams that would otherwise make waterways unnavigable. Occasionally, locks must be dewatered to maintain, repair, or replace lock components. This disrupts navigation by preventing passage in single-lock systems and can increase waiting times at dual-lock systems. Dewatering events are often scheduled during times of the year when navigation traffic is at a minimum and at least 1 year in advance to minimize disruption by giving shippers an opportunity to plan. However, unscheduled outages can occur when lock components fail unexpectedly. Unscheduled outages tend to be more costly than scheduled outages because shippers incur “shipper carrier” costs (SCC) when vessels, cargo, and crews are stranded on the waterway and goods must be shipped by alternative means.At locks operated by the USACE, each hydraulic steel structure must be thoroughly inspected at least once every 25 years (USACE 2009). Several critical components are located under water, and dewatering is required to perform inspections. In many instances, where funds to maintain and repair infrastructure are limited, this has devolved into the practice of postponing dewatering events until they are required by regulation or the interim failure of a critical component causes an unscheduled outage. When locks are dewatered, the practice is to repair or replace only those components that have already failed; are clearly approaching imminent failure; or have deficiencies that severely affect performance, operations, or maintenance requirements. The condition of each component is assessed annually by engineers and lock operators through an operational condition assessment (OCA). The procedures for conducting an OCA are described in USACE (2019a). Lock managers use OCA ratings to evaluate and track the condition of lock components each year. For critical components located below the waterline, which can be difficult to assess visually, assessments of condition are based on operational performance.The OCA rating scale, which has evolved over time, currently consists of six levels, with six nominal grades of A, B, C, D, F, and CF, defined in Table 1. When a component meets the definition of OCA grades A, B, C, or D but shows initial signs of the next lower grade, a minus may be added to the grade. Lock operators and managers monitor the condition of lock components and assign OCA ratings annually (USACE 2019a). During an OCA rating, the operating condition of each lock component is evaluated based on the experience or knowledge of operators who are aware of performance issues. OCAs differ from periodic inspections, which are conducted by teams of a dozen or more experts in structural, electrical, mechanical, geotechnical, and hydraulic engineering using a combination of visual and objective measurement techniques to assess the condition of structures (McKay et al. 1999). OCA conditions may be assigned based on information collected during periodic inspections. During OCAs between periodic inspections, components may be downgraded when deficiencies affecting performance, operational procedures, or maintenance requirements are noted. Similarly, components may be upgraded following repair or replacement. The USACE Asset Management Program maintains a database to track the condition of each lock component.Table 1. OCA rating scaleTable 1. OCA rating scaleRatingDescriptorDefinitionAExcellentComponent was recently put into service and shows no signs of wear.BGoodComponent performs its intended function. Any deficiencies are normal wear and not actively progressing at a greater rate than normal wear.CFairComponent has a deficiency that is beginning to affect its performance, operational procedures, or maintenance requirements, or component is beginning to show a greater rate of change in degradation that has the potential to cause functional failure.DPoorComponent has a deficiency that increasingly or moderately affects its performance, operational procedures, or maintenance requirements, or component has a clear mode of failure due to an advanced state of degradation likely with an accelerating trend.FFailedComponent has a deficiency that substantially affects its performance, operational procedures, or maintenance requirements and is approaching complete failure, or component is clearly in the final stages of degradation trending toward complete failure (imminent failure).CFCompletely failedComponent is completely failed and does not perform its intended function, or component is red tagged.This research investigated whether alternatives to current practices for scheduling dewatering events and repairing or replacing lock components could reduce the costs of scheduled and unscheduled outages at navigation locks. Ten alternatives were considered, including the status quo. Each alternative was composed of two criteria, one that governed when dewatering events took place and one that governed which components were repaired or replaced during dewatering events. A decision model was developed to identify the alternative that minimized present value of total cost (PVC). Decision models are mathematical representations of decision problems that incorporate information about a decision-maker’s objectives, alternatives, and the sources of uncertainty that will influence the outcome realized under each alternative (Howard et al. 1972; Howard 1988; Edwards et al. 2007; Schultz et al. 2010). The decision model developed for this study is grounded in current USACE practice because it incorporates the methods and tools that are presently being used by USACE to manage the nation’s locks. These include the USACE’s OCA rating system, an empirical degradation model derived from histories of OCA ratings at USACE navigation locks, and the USACE’s SCC model, which was developed and is maintained by the USACE Planning Center of Expertise for Inland Navigation (PCXIN). The SCC model estimates the increase in transportation cost realized by waterway users as a result of unscheduled lock outages.Of particular interest in this study is the performance of alternatives that incorporate risk-based scheduling criteria and continuous health monitoring information. Under a risk-informed scheduling criteria, dewatering events are scheduled when the probability of unscheduled outage over the next 5 years exceeds a critical level pu*. The first of these strategies uses information about component condition gathered during the last inspection and the number of years since the last inspection to compute the probability of failure. The second risk-informed strategy uses perfect information obtained through a structural health monitoring (SHM) system to estimate the probability of outage over the next 5 years. SHM systems are of particular interest because they promise to greatly reduce costs by providing real-time information about the condition of infrastructure components (Farrar and Worden 2012). An estimate of the probability of unscheduled outage that is based on real-time information about component condition should be less uncertain than one based on data collected at the last dewatering.There are a few published accounts of efforts to model maintenance and repair decisions at navigation locks. Wang et al. (2009) described an optimization model to select projects in a network of locks to maximize net benefits in the waterway. These authors indicated that it is necessary to preserve the condition of lock components above a specific threshold in order to reduce the impacts to traffic over time and that raising this threshold increases lock availability and demand for inland waterway transportation. The results of the present study support the idea that raising the threshold condition increases lock availability, but the objective function in the present study minimizes cost because it is assumed that benefits to industry from actions at a single lock are limited to reducing SCC. Nevertheless, the idea that changes in lock performance influence the demand for inland transportation is important (Wang and Schonfeld 2007). More recently, Dang et al. (2019) studied the risk-based scheduling of welded joint inspections at navigation miter gates. Their results showed that heuristic decision rules based on periodic inspections are more costly than probabilistic thresholds. The present study confirms this. However, this paper also concludes that the benefits of a probabilistic threshold for dewatering are dwarfed by the benefits of raising the threshold condition for repair or replacement.Methods and MaterialsThe decision model is represented as an influence diagram in Fig. 1 to illustrate the overall structure of the model. Influence diagrams are graphical representations of decision problems that are useful for encoding, communicating, and visualizing the probabilistic dependencies among random variables (Howard and Matheson 1984). The rectangular decision node in the upper left hand corner defines the 10 alternatives, including the status quo policy. Each of the alternatives was composed of two criteria. The first was a criterion for scheduling dewatering operations (SCHEDULING CRITERION), and the second was a criterion for maintaining lock components (MAINTENANCE CRITERION). The model simulated the degradation and failure of lock components (OCA RATING), the occurrence of scheduled and unscheduled outages (OUTAGES), and the repair or replacement of lock components (REPAIR OR REPLACE) in each year of the planning horizon. The bidirectional edges between these nodes signify dynamic dependence, which is dependence of one random variable on the value of another variable in a preceding time interval. Oval nodes represent sources of uncertainty in estimating the net PVC over the planning horizon under each policy. Costs include fixed dewatering costs, repair and replacement costs, and SCCs.A navigation lock consists of four types of critical components, the pintle, the quoin contact blocks, the miter gate, and the miter gate contact blocks (Fig. 2), indexed in order of j∈{1,2,3,4}. To maintain consistency with the way that USACE assigns OCA ratings, riverside and landside components were treated as one single component, and upstream and downstream components were treated as separate components, indexed by k∈{Up,Down}. For the purpose of this study, the OCA rating scale was simplified by dropping minus signs and combining the F and CF OCA ratings. This reduced the OCA rating scale to five levels ordered from A to F and indexed by m={A,B,C,D,F}. The eight critical components degrade over time, and this process is governed by an empirical transition matrix, which was derived from a 7-year history of OCA ratings for these components at USACE navigation locks. The transition of any one critical component to an OCA rating of F causes an unscheduled outage. Dewatering events must be scheduled periodically to inspect, perform maintenance, and repair or replace critical components. The status quo criteria are to schedule dewatering operations once every 25 years and repair or replace only those components with an OCA rating of F.Alternative methods for scheduling and repairing or replacing lock components are summarized in Table 2. These alternatives have been constructed by the authors to illustrate different ways that decisions about scheduling and maintenance might be made. Scheduling criteria include operate to failure (OTF), dewatering on an interval (DWI), risk-informed scheduling (RISK), and structural health monitoring. Criteria for performing maintenance on lock components include fix as fail (FAF), repair or replace those components with OCA ratings of D or worse (RRD), and repair or replace those components with OCA ratings of C or worse (RRC). The status quo alternative is designated DWI-FAF. As shown in Table 3, the FAF maintenance criterion is considered only in conjunction with the OTF and DWI scheduling criteria because it would make little sense to implement a risk-informed criterion to schedule a dewatering event knowing in advance that no components would be repaired or replaced during the event. When components are repaired, their OCA rating in the next period is raised to a B. When components are replaced, their OCA rating in the next time period is raised to an A.Table 2. Descriptions of decision alternativesTable 2. Descriptions of decision alternativesnSymbolDescription1OTF-FAFOperate to failure and repair or replace components only when they transition to an OCA rating of F (e.g., fix as fail).2OTF-RRDOperate to failure and repair or replace components with OCA ratings of D or F.3OTF-RRCOperate to failure and repair or replace components with OCA ratings of C, D, or F.4DWI-FAFaDewater on an interval of 25 years and repair or replace components with OCA ratings of F. The baseline policy is DWI-FAF with an interval of 25 years. Shorter dewatering intervals of 5, 10, 15, and 20 years are also investigated.5DWI-RRDDewater on an interval of 25 years and repair or replace components with OCA ratings of D or F. Shorter dewatering intervals of 5, 10, 15, and 20 years are also investigated. During dewatering, all components with an OCA rating of D or F are repaired or replaced.6DWI-RRCDewater on an interval of 25 years and repair or replace components with OCA ratings of C, D, or F. Shorter dewatering intervals of 5, 10, 15, and 20 years are also investigated. During dewatering, all components with an OCA rating of C, D, or F are repaired or replaced.7RISK-RRDDewatering operations are scheduled for 2 years after the probability of an unscheduled outage within 5 years exceeds a critical value, pu(tu≤5)≥pu*. The value of pu(tu≤5) is calculated using the OCA ratings from the most recent inspection during last dewatering. During dewatering, all components with OCA ratings of D or F are repaired or replaced.8RISK-RRCDewatering operations are scheduled for 2 years after the probability of an unscheduled outage within 5 years exceeds a critical value, pu(tu≤5)≥pu*. The value of pu(tu≤5) is calculated using OCA ratings from the most recent dewatering and inspection event. During dewatering, all components with OCA ratings of C, D, or F are repaired or replaced.9SHM-RRDDewatering operations are scheduled for 2 years after the probability of an unscheduled outage within 5 years exceeds a critical value, pu(tu≤5)≥pu*. The value of pu(tu≤5) is calculated using the OCA ratings from an SHM program that provides perfect information about the condition of each component at all times. During dewatering, all components with OCA ratings of D or F are repaired or replaced.10SHM-RRCDewatering operations are scheduled for 2 years after the probability of an unscheduled outage within 5 years exceeds a critical value, pu(tu≤5)≥pu*. The value of pu(tu≤5) is calculated using the OCA ratings from an SHM program that provides perfect information about the condition of each component at all times. During dewatering, all components with OCA ratings of C, D, or F are repaired or replaced.Table 3. Array of decision alternativesTable 3. Array of decision alternativesCriteria for repairing or replacing lock componentsOperate to failure (OTF)Dewater on an interval ≤25  years (DWI)Risk-informed scheduling (RISK)Structural health monitoring (SHM)Fix as fail (FAF)OTF-FAFDWI-FAFa——Repair or replace D or worse (RRD)OTF-RRDDWI-RRDRISK-RRDSHM-RRDRepair or replace C or worse (RRC)OTF-RRCDWI-RRCRISK-RRCSHM-RRCThe decision-maker’s objective, represented by the terminal node in the lower right-hand corner of Fig. 1, is to minimize the expected PVC over the 50-year planning horizon E[PVC](1) E[PVC]=1Ns∑n=1Ns∑t=050(CF+CR+ϕCC(d))t(1+r)twhere the numerator = the total cost of an outage in year t of the planning horizon and total cost is expressed as the sum of fixed costs CF and variable costs CR and CC(d). The term CR = the cost of component repair or replacement and CC(d) = the shipper carrier cost incurred during an unscheduled outage, which is a function of outage duration d. Outages may be scheduled or unscheduled and the coefficient ϕ is the ratio of scheduled to unscheduled outage cost at the lock. For shippers, the cost of scheduled outages should be less than the cost of unscheduled outages because, when outages are scheduled 1 year or more in advance, shippers have time to plan for the outage and can take action to mitigate their losses. Therefore, for unscheduled outages, ϕ=1, and for scheduled outages, 0≤ϕ≤1. The term ϕCC(d) is the shipper carrier cost associated with an outage in year t of the planning horizon depending upon whether that outage was scheduled or unscheduled.The denominator (1+r)t discounts costs incurred further in the future relative to those costs incurred closer to the present, and r is the discount rate, or rate of time preference. This discounted utility model was formulated by Samuelson (1937). It remains the standard way of representing intertemporal choice in economics when evaluating streams of benefits and costs over time (Frederick et al. 2002). A discount rate of 0.03 was used, which is typical for public projects. The planning horizon was discretized into 50 one-year intervals. This is consistent with the length of the planning horizon used within USACE for planning purposes. In addition, the practice of discounting means that costs incurred in later years have a diminishing effect on the decision. Beyond year 50, costs tend to carry much less weight in the decision. With regard to risk attitudes, we assumed that large organizations such as the USACE are risk neutral, so potential gains and losses weigh equally in the decision.Costs are incurred in those years when scheduled or unscheduled outages occur or SHM costs are incurred. The PVC associated with implementing a scheduling or maintenance criteria over the planning horizon is the sum of discounted costs incurred in each year of the planning horizon. An expected PVC was obtained by averaging the present value of total cost over NS=1,000 realizations of the Monte Carlo simulation because, in decision analysis, the conventional decision rule is to choose the alternative that maximizes expected net benefits (Edwards et al. 2007). However, in this case, the potential benefits were accounted for in terms of avoided shipper carrier costs and repair and replacement costs; hence, the decision rule was expressed here as to minimize the expected PVC.The fixed costs of an outage include mobilization ($350,000), closure and dewatering ($65,000), inspection ($25,000), and preventive maintenance ($400,000). For those alternatives that incorporate SHM, fixed costs also include an initial investment of $300,000 to purchase and install a SHM system in the first year of the simulation and a $30,000 per year maintenance cost in subsequent years. SHM cost estimates reflect USACE experience installing and operating SHM systems at navigation locks. Other fixed costs and repair or replacement costs were estimated from historical records of repair and maintenance in USACE Tulsa District, where W.D. Mayo Lock and Dam is located. Component repair and replacement costs are summarized in Table 4. There is no cost associated with pintle repair because, in general, failed pintles cannot be repaired and must be replaced. SCCs are those costs incurred by shippers when, during unscheduled outages, vessels, crews, and cargo become stranded in the waterway and cargo must be shipped via the next least expensive mode of transportation. SCC were estimated as a function of outage duration using the 2015 update of USACE’s SCC model. That model simulates vessel and commodity flows through each lock in the USACE portfolio of locks and estimates the cost of increased waiting times and diversions of commodities to the next least expensive mode of transportation. Cumulative SCCs are shown in Fig. 3 for unscheduled outages lasting 1 to 100 days at W. D. Mayo Lock and Dam. After 10 days, cumulative SCCs are $0.67 million. These increase to $5.06 million after 30 days and $20.1 million after 90 days.Table 4. Repair and replacement costs and outage durations for lock componentsTable 4. Repair and replacement costs and outage durations for lock componentsComponentRepair cost ($)Replacement cost ($)Repair duration (days)Replacement duration (days)Quoin contact block250,0001,500,000U(7,14)U(30,45)Pintle—550,000—21Miter gate2,100,00012,500,00021U(30,45)Miter gate contact block250,000500,000U(7,14)U(30,45)Because lock managers at W.D. Mayo Lock and Dam have no specified pu* and, although the value of ϕ is believed to be low but is not known precisely, the decision model was simulated multiple times varying these parameters in a heuristic fashion to explore how optimal scheduling and maintenance criteria might differ in different operating environments. The value of ϕ was simulated at values ranging from 0 to 1.0. The lower values of ϕ were used to represent operating environments in which unscheduled outages are much more costly than scheduled outages. Higher values of ϕ represented operating environments in which there was little difference between the cost of scheduled and unscheduled outages, and ϕ=1 represented the extreme case of no difference in the costs of scheduled and unscheduled outages. The term pu* was varied from 0.1 to 0.9, with lower values representing a higher tolerance for unscheduled outage and higher values representing a lower tolerance for unscheduled outage. The expected PVC was calculated over Ns=1,000 realizations of a Monte Carlo simulation in which the degradation of components over time was simulated using a transition matrix, which is described subsequently. The number of simulation runs was sufficient to achieve convergence of the mean and variance of expected cost under each alternative (Ballio and Guadagnini 2004) (Fig. S1).For unscheduled outages, outage duration includes the time required to mobilize the fleet to dewater the lock and the time required to repair or replace the components (Table 4). Mobilization of the fleet requires 5 to 8 days and was modeled as a uniformly distributed random variable U(5,8), rounded to the nearest day. If multiple components would be repaired or replaced during an unscheduled outage, the maximum duration applied. For scheduled outages, the minimum length of an outage was 3 days, which is the minimum required to dewater, inspect, and refill the lock. If components were repaired or replaced during the scheduled outage, the repair and replacement durations in Table 4 also applied. As with unscheduled outages, if multiple components were repaired or replaced during an outage, the maximum repair or replacement duration was used. The maximum duration of an unscheduled outage during the simulation was 53 days, including mobilization, dewatering, and repair or replacement duration.At the beginning of the simulation, each component was assigned an OCA rating based on the most recent annual asset management review. In practice, OCA ratings are assigned to subcomponents of these components and to subcomponent failure modes. For the purpose of this simulation, subcomponents were lumped into components. At the beginning of each simulation, the initial OCA rating assigned to each component was that of the subcomponent and failure mode with the lowest approved OCA rating, with “minus” ratings treated as detailed previously. For W.D. Mayo, at the time of this study, the most recent OCA ratings for all components were B, except for the downstream miter gate (C) and downstream miter gate contact blocks (C).The degradation of components over time was simulated by a stochastic process in which the transitions from one OCA rating to lower OCA ratings were governed by an empirical transition matrix. The elements of the transition matrix T were conditional probabilities that described how likely it was that a component will transition to a lower OCA rating in the current period t, given that it was assigned a particular OCA rating in the preceding time period p(Mt|Mt−1)(2) T=[0.85710.142800000.96520.01820.01480.0016000.93400.04860.01730000.97910.020800001]For example, given that a component was rated A in the current period, the probability it would be rated A in the next time period was 0.857, and the probability it would be rated B in the next time period was 0.142. The transition matrix was derived from a database of 5,505 observations of OCA rating transitions for subcomponents of the four critical components at USACE locks during a 7-year period, from 2011 to 2018. Observations that showed improvements in ratings from one year to the next, implying corrective repair or replacement, were removed from the data before deriving the transition matrix. When multiple years separated OCA ratings, changes in OCA ratings were assumed to occur in the year they were observed, with the previous OCA rating assigned to intervening years. For this study, this transition matrix was sampled in each time period to determine if a component transitioned to a new OCA rating or remained in its current OCA rating.When a component transitions to an OCA rating of F, an outage occurs. During the outage, all components meeting the criteria for repair or replacement are repaired or replaced. Whether components are repaired or replaced is determined through a second stochastic process in which those components with higher OCA ratings immediately prior to the outage are more likely to be repaired than replaced, and those with lower OCA ratings immediately prior to the outage are more likely to be replaced. The critical probabilities governing this process are summarized in Table 5. The elements of this table are the probabilities that the component is replaced rather than repaired given its most recent OCA rating.Table 5. Critical values for determining whether a component is repaired or replacedTable 5. Critical values for determining whether a component is repaired or replacedComponentABCDFMiter gate0.09250.24660.36230.49690.7888Contact blocks0.10210.27230.39990.54000.8102Critical values of the replacement probabilities in Table 4 were derived from Weibull distributions that USACE has used to model the time to failure of lock components and other infrastructure for asset management purposes. The parameters of these Weibull distributions were derived by an expert panel using engineering judgment for groups of navigation infrastructure components classified by the type of material used in fabrication. For miter gates, the parameters of the Weibull distribution were η=3.5 and β=84, and for miter gate and quoin contact blocks η=4.1 and β=60. Subsequently, these Weibull distributions were related to the OCA rating scale. The critical values in Table 5 represent the ratio of the implied age given the OCA rating in the preceding time period mt−1 and the implied age associated with CF. This assumed distribution of repairs and replacements is somewhat arbitrary because no data exist for validation. A plausible argument can be made for this model in that components failing from better conditions may be able to be repaired more frequently than other components that fail after being in poorer conditions. However, arguments could be made for other assumed distributions as well (e.g., uniform likelihood, bathtub curve, etc.). Without validation data, this research selected the USACE Weibull curves to serve as an indicator of repair or replacement likelihood. The sensitivity of this model choice has not been investigated at the time of this writing.This process of simulating component degradation and repair or replacement, sampled from distributions as described previously, produced a unique sequence of OCA ratings over the 50-year planning horizon for each realization of the Monte Carlo simulation. When one or more lock components transitioned to an OCA rating of F, the model recorded an outage Θ∈{N,U,S}, where N indicates no outage occurred in that year, U indicates an unscheduled outage occurred in that year, and S indicates a scheduled outage occurred in that year. Under the OTF criterion, all outages were unscheduled. Under the DWI, RISK, and SHM criteria, outages could be scheduled or unscheduled. An outage caused by the failure of a component was always unscheduled unless the criterion for scheduling an outage was met in the year that the component failed. If the scheduling criterion was met, the model used an additional stochastic process to determine whether at least one of the components failed before the scheduled outage could take place. Given that an outage is scheduled during the year, the probability that a component fails before the scheduled outage could take place is 0.5. If the component failed before the outage could take place, the outage was classified as unscheduled. Otherwise, the outage was classified as scheduled. The rationale for this process is that most of the cargo passes through W.D. Mayo during the first 6 months of the year, so outages tend to be scheduled in early summer, about halfway through the calendar year, to minimize impact on shippers.The criteria for scheduling outages have been previously described in the previous sections on decision alternatives. There are no criteria for scheduling outages for OTF alternatives. For DWI alternatives, outages were scheduled when the number of years since the last outage Y reached the maximum allowable years between outages. The model initialized each simulation of OCA rating transitions by sampling the number of years since the last outage at time t=0. For each alternative and each combination of pu* and ϕ, there are NS=1,000 simulations of OCA rating transitions, so this random variable was sampled 1,000 times, once for each realization of the Monte Carlo simulation. For OTF, RISK, and SHM alternatives, the number of years since the last outage was determined by sampling a uniform random variable between 0 and 25. For DWI alternatives, the model sampled a uniform random variable between 0 and the maximum dewatering interval. In subsequent years of the simulation, the model used a counter to track how many years had elapsed since the last outage.For RISK and SHM alternatives, outages were scheduled when the probability of an unscheduled outage occurring within 5 years exceeded a critical level pu*. For RISK alternatives, the probability of unscheduled outage was calculated given the OCA rating assigned to components at the last dewatering and inspection event. The probability that a component fails within 5 years given the OCA rating assigned at the last dewatering and inspection event can be calculated from the transition matrix T. The elements of T are one-step transition probabilities. The probability that a component transitions to another OCA rating in y years is a y-step transition probability. These y-step transition probabilities are the elements of a y-step transition matrix, which describes the probability that a component will have transitioned to any potential OCA rating by time t+y(mt+y), given its OCA rating at time t(mt). The y-step transition matrix can be found by multiplying the transition matrix by itself y times (3) The OCA rating is known in the year of dewatering and inspection but is uncertain in years between dewatering and inspection events. If it is known that the component did not fail in y years since the last dewatering and inspection event, a modified transition matrix is needed. The transition matrix can be modified to calculate the probability that the component will transition to another OCA rating given that it does not fail in y years. The matrix T is modified to T′ by setting the fifth-column elements in rows 1–4 to 0 and the fifth element in the fifth column representing the failed state to 1. The weights are then normalized so each row sums to 1 (4) T′=[T1,11−T1,5T1,21−T1,5T1,31−T1,5T1,41−T1,500T2,21−T2,5T2,31−T2,5T2,41−T2,5000T3,31−T3,5T3,41−T3,50000T4,41−T4,5000001]where the matrix T′ = modified 1-year transition matrix; and Xt = row vector with elements comprising a probability mass function that describes the probability that a component is in one of five OCA conditions Y years after the dewatering and inspection event, when the OCA state was known, given that it has not failed since then (5) where Xt−Y = row vector with 1 on the OCA state known in year t−Y and 0 elsewhere. The probability the component transitions to any particular OCA state over the next 5 years given its OCA rating at the last dewatering event Y years ago and the fact the component has not failed since is (6) The probability of an unscheduled outage over the next 5 years given that no outage has occurred in the Y years since the last dewatering and inspection event is calculated from the probability of each component failing over the next 5 years (7) p(Θt+5=U|Θt−Y+1≠U,Θt−Y+2≠U,…,Θt≠U)jk=1−∏jkp(Mt+5≠F′′|Mt−Y=mt−Y)jkUnder SHM alternatives, the simulated OCA ratings for each component were used to calculate the probability of unscheduled outage over 5 years. This simulates eliminating uncertainty about how each component has deteriorated since the last dewatering and inspection event but does not eliminate uncertainty about whether a component will fail. To calculate Xt+5 under SHM alternatives, Xt reduces to a row vector with 1 on the OCA state in time t and 0 elsewhere (8) The probability of unscheduled outage remains as described previously.The SHM system described in this study exists only as part of our computer simulation. It does not exist in practice, and it is not meant to be representative of all SHM systems. In practice, SHM systems can take many forms and provide a variety of different types of information to decision-makers with varying degrees of fidelity at varying periodicities, which may be regular or irregular. The SHM system in our model provides perfect information about component OCA ratings continuously during each period of the simulation. The information is described as continuous because, in the model, it is always available to the decision-maker. The information is described as perfect because the OCA ratings communicated to the decision-maker are exactly those assigned to each component in that period of the simulation. This information is not available to decision-makers operating under other scheduling criterion. In practice, it would be impossible to get perfect information about OCA ratings, and information transmitted once each year could not be described as continuous.ResultsRelative Frequency of Unscheduled OutagesThe relative frequency of unscheduled outages was estimated as the mean fraction of years with unscheduled outages during the planning horizon (Fig. 4). Under the status quo alternative, DWI-FAF with a 25-year dewatering interval, the relative frequency of unscheduled outages was 0.09, or nine outages per 100 years. This is similar to the relative frequency under the OTF-FAF alternative, 0.09. Reductions in the relative frequency of unscheduled outage can be achieved by changing either the scheduling criterion or the maintenance criterion. For OTF and DWI alternatives, pu* had no effect on the relative frequency of unscheduled outage. For RISK and SHM alternatives, changes in the critical probability of unscheduled outage pu* influenced the relative frequency of unscheduled outage in the range 0.1≤pu*≤0.6. However, these effects were small relative to the effect induced by changing the maintenance criterion. Under the status-quo scheduling alternative, DWI-FAF, changing the maintenance criterion from FAF to RRD reduced the relative frequency of unscheduled outage from 0.09 to 0.048, a decrease of 47%. Changing the maintenance criteria to RRC reduced the relative frequency of unscheduled outage to 0.03, a decrease of 67%. The largest potential reduction in the relative frequency of unscheduled outage, 82%, was achieved by implementing the RISK-RRC alternative with pu*=0.2.Relative Frequency of Scheduled and Unscheduled OutagesThe relative frequency of both scheduled and unscheduled outages under each alternative are summarized in Fig. 5. Under DWI-FAF, the combined relative frequency was 0.095. The combined relative frequency was at a minimum, 0.037, under the DWI-RRC alternative. This relative frequency is less than 0.04, which is the minimum relative frequency that one would expect under a 25-year dewatering interval. This lower-than-expected value for the DWI-RRC alternative is an artifact of having simulated outcomes over a 50-year planning horizon and uncertainty in the initialization of the years since the last outage variable Y. The RISK and SHM alternatives showed significant increases in the relative frequency of outages for pu* of 0.3 or less. The reason for this is that, in order to achieve the lower probability of unscheduled outage, lock managers must hold scheduled outages more frequently, increasing the relative frequency of all outages.Expected PVCThe sensitivity of the expected PVC to pu* is summarized in Fig. 6. Alternatives that incorporate RISK and SHM scheduling criteria were sensitive to pu*, but only in the range of 0.1 to 0.6. When ϕ=0, no SCC were incurred during scheduled outages, and SHM-RRC minimized the expected PVC when pu*=0.2. The RISK-RRC alternative minimized expected PVC otherwise. As ϕ increased, the SCC incurred during scheduled outages approached those of the unscheduled outages and the benefits of scheduling outages diminishes. This is reflected in the higher expected PVC when ϕ=0.25. The benefits of SHM-RRC were diminished, and the expected value of total cost reached a minimum with pu*=0.3. When ϕ=1, scheduled outage costs were the same as for unscheduled outages, and OTF-RRC minimized the expected PVC, insensitive to pu*. For values of ϕ=0 and 0.25, Fig. 6 reveals three groups of alternatives sharing the same maintenance criterion. The least expensive alternatives used RRC. The next least expensive alternatives used RRD. The most expensive alternatives used FAF.Sensitivity of the expected PVC to ϕ is summarized in Fig. 7 for selected values of pu*. Results are shown for the selected values of pu* over which changes in the critical probability of unscheduled outage influenced the expected PVC. In general, it is very difficult to manage for levels of pu*<0.2. For example, the lowest achievable probability of outage within 5 years is pu(tf≤t+5)=0.0896, when all eight components have an OCA rating of A. When all of the components have an OCA rating of B, pu(tf≤t+5)=0.1888. At pu*=0.2 or pu*=0.3, SHM-RRC was preferred at lower values of ϕ. As ϕ increased, the benefits of using a risk-informed scheduling criterion diminished, and DWI-RRC became the preferred alternative over a narrow range of ϕ. When ϕ≥0.7, OTF-RRC was preferred.Benefit of Structural Health MonitoringResults indicate that the RRC maintenance criterion dominated other maintenance criteria, meaning the RRC criterion always minimized the expected PVC. The potential benefit of implementing the SHM-RRC alternative over the status quo alternative, DWI-FAF, is the savings that would have been realized if the SHM-RRC alternative had been implemented instead. The expected benefit of SHM-RRC is the expected difference in the PVC under the DWI-FAF alternative and the PVC under the SHM-RRC alternative (9) E[BSHM-RRC]=1Ns∑n=1Ns[PVCDWI-FAF−PVCSHM-RRC]The expected benefit of SHM-RRC relative to the status quo alternative is summarized in Fig. 8. The maximum expected benefit, $15.4 million, was realized when ϕ=0 and pu*=0.2. Where the benefits are negative, DWI-FAF was preferred to SHM-RRC. Benefits were negative when lock managers managed for lower pu* and ϕ≥∼0.6. In these cases, the benefits were negative because the costs associated with the more frequent scheduled outages needed to achieve the desired levels of pu* exceeded the cost of unscheduled outages under the DWI-FAF alternative. When managing for higher levels of pu* (pu*≥0.4), the lines in Fig. 8 that show E[BSHM-RRC] for different shipper carrier cost ratios converge because fewer scheduled outages were needed to achieve the desired pu*, resulting in less sensitivity to ϕ.The benefits of SHM-RRC relative to the status quo alternative are significant. However, the decision model shows that much of that benefit can be attributed to the RRC maintenance criterion rather than the SHM scheduling criterion. This is illustrated in Fig. 9, which shows that benefits attributed to the SHM criteria diminished when managing for higher levels of pu* and operating in environments where ϕ is higher. These benefits are the expected differences between the PVC under the optimal alternative when SHM alternatives were excluded from the set of alternatives available to the decision-maker, MIN(PVC)SHM∉Ω and PVCSHM-RRC when SHM alternatives are included in the decision space (10) E[BSHM]=1Ns∑n=1Ns[MIN(PVC)SHM∉Ω−PVCSHM-RRC]where Ω = decision space, which is the set of alternatives available to the decision-maker; and E[BSHM] = portion of E[BSHM-RRC] that can be attributed to implementing the SHM scheduling criterion of the SHM-RRC alternative. Positive expected benefits were realized over only a narrow range of pu* and ϕ because the costs of implementing the SHM system outweighed the benefits of perfect information about the OCA rating of critical lock components. These results suggest that much of the benefit of SHM-RRC can be realized just by implementing one of the second best alternatives: RISK-RRC, DWI-RRC, or OTF-RRC. To see this, compare the E[BSHM] for ϕ=0.2 and pu*=0.2 in Fig. 9, $1.56 million, to the corresponding E[BSHM-RRC] in Fig. 8, $10.58 million. The difference, $9.02 million, can be attributed to maintenance component of the SHM-RRC alternative. Only 14.7% of the benefit of the SHM-RRC alternative can be attributed to having perfect information about the OCA rating. This pattern, in which the SHM component accounted for less than 15% of the benefits of SHM-RRC, held for all other combinations of pu* and ϕ when SHM-RRC was the optimal alternative.Opportunity Cost of DWI-FAFOpportunity costs are foregone benefits, the benefits that would have been realized had the optimal alternative been implemented instead of a suboptimal alternative. The expected opportunity costs E[OPC] of implementing the status quo alternative, DWI-FAF, rather than the optimal alternative are calculated by subtracting MIN(PVC)Ω,pu* the minimum PVC over all alternatives and potential values of pu* from PVCDWI-FAF(11) E[OPC]=1Ns∑n=1Ns[PVCDWI-FAF−MIN(PVC)Ω,pu*]This assumes that lock managers would adjust pu* to minimize PVC when operating a lock under a risk-informed scheduling criterion. The value of PVCDWI-FAF does not vary with pu*. Opportunity costs are summarized in Fig. 10, which show that, depending on the SCC ratio ϕ, opportunity costs range from $7.4 to $17.0 million over the 50-year planning horizon. A similar calculation can be used substituting PVCDWI-RRC for PVCDWI-FAF to demonstrate that much of that opportunity cost can be recovered just by switching from FAF to RRC.Shortening of the Dewatering IntervalIn the preceding analyses, the DWI alternatives each had a dewatering interval of 25 years. One strategy for reducing the frequency of unscheduled outage under DWI alternatives is to reduce the dewatering interval. The effect of reducing the dewatering interval on the relative frequency of unscheduled outages is summarized for the DWI alternatives in Table 6. Under the DWI-FAF alternative, shortening the dewatering interval had no effect on the relative frequency of unscheduled outage because components were repaired or replaced only after they failed and caused an unscheduled outage. Decreasing the dewatering interval under the FAF maintenance criteria leads only to higher costs. Under the RRD and RRC maintenance criteria, shortening of the dewatering interval to 5 years reduced the frequency of unscheduled outages by 27% and 40%, respectively. However, reducing the dewatering interval increased the frequency of scheduled outages and, depending on the SCC ratio, may result in higher PVC. Under the DWI-RRC alternative, reducing the dewatering interval from 25 years to 5 years reduced PVC as much as $4 million for SCC ratios of 0.25 and less and increased PVC as much as $15 million for SCC ratios greater than about 0.25 (Fig. 11).Table 6. Frequency of unscheduled outage realized by shortening the dewatering intervalTable 6. Frequency of unscheduled outage realized by shortening the dewatering intervalDWI alternative510152025DWI-FAF0.0900.0900.0900.0900.090DWI-RRD0.0350.0420.0450.0530.048DWI-RRC0.0180.0260.0260.0300.030Operational Costs to USACEIt is of interest to note the effect of each alternative on the operational costs to USACE. The agency costs for each alternative are shown in Fig. 12 varying over the SCC ratio. The variable pu* was optimized for RISK and SHM alternatives. The value was pu*=0.2 for RISK-RRD and SHM-RRD and pu*=0.3 for RISK-RRC and SHM-RRC. For DWI-FAF, the present value of agency costs exceeded $10M, whereas those costs were minimized by the DWI-RRC at ϕ = 0–0.1 and RISK-RRC otherwise. Where ϕ≥0.2, RISK-RRC resulted in a present value of agency costs as low as $5.6M, depending on ϕ, and DWI-RRC resulted in a present value of agency costs less than $5.8M regardless of ϕ. As in previous results, a trend can be seen that RRD costs were lower than FAF costs, and RRC costs were the lowest of all alternatives. The data used to explore agency costs were generated using the same decision model as the data presented elsewhere in the paper, but on a different computer.ConclusionsThe decision model investigating whether alternatives to current practice could reduce cost or improve lock performance revealed interesting results with possible significant implications for USACE lock operations and maintenance practice. First, the status quo alternative, DWI-FAF, and the OTF-FAF alternative had the highest PVC and the highest frequencies of unscheduled outage. Collectively, these were the worst alternatives considered and suggest USACE could realize benefits from implementing one of the other alternatives in practice. Second, the most important factor affecting cost is the choice of maintenance criteria. The RRC alternatives consistently provided greater expected benefits than the FAF or RRD alternatives. Indeed, this cost savings from RRC outweighed the cost savings of risk-informed scheduling criteria. Much of the benefit of risk-informed scheduling alternatives can be achieved simply by dewatering on an interval (DWI) and repairing or replacing those components that are in an OCA rating of C or worse (RRC).Outages should only be scheduled if, by scheduling the outage, it would be possible to avoid SCC. Results indicate that, at W.D. Mayo Lock and Dam, outages should not be scheduled if the ratio of scheduled to unscheduled SCCs exceeds 0.7. In this case, OTF-RRC is preferred. However, there is only a relatively small (<$2  M) expected opportunity cost of implementing DWI-RRC even in this case.Risk-informed and SHM alternatives can reduce expected costs beyond the savings from DWI-RRC under certain cases. Risk-informed scheduling alternatives may be preferred when the ratio of scheduled to unscheduled shipper-carrier costs is low, ϕ<0.6, and the lock is being managed for a relatively low probability of unscheduled outage within 5 years, 0.2≤pu*≤0.4. SHM provides perfect information about the OCA rating of lock components, enabling a less uncertain estimate of the probability of unscheduled outage within 5 years. However, the SHM alternatives in this study may not always be preferred to the RISK alternatives because there are costs associated with installing and operating SHM systems that may outweigh the benefit of having less uncertain estimates of unscheduled outage probability.With respect to DWI alternatives, the analysis shows that shortening the status quo dewatering interval from 25 years had no effect on the frequency of unscheduled outage under a FAF maintenance criterion but reduced the relative frequency of unscheduled outages under the RRD and RRC maintenance criteria. However, when ϕ≥∼0.25, total cost increased because the savings associated with a reduced frequency of unscheduled outage was more than offset by the cost of scheduled outages.The RRC criteria was also found to reduce the USACE agency costs significantly compared with FAF and RRD criteria. For most SCC ratios, RISK-RRC resulted in the lowest agency costs, which was only slightly better than DWI-RRC. The administration of RISK-RRC is likely to have higher agency costs than that of DWI-RRC, and thus, DWI-RRC may be a better alternative if the agency wishes to minimize its costs. If the USACE were to adjust its operations to minimize its own cost without regard to the larger costs imposed on society as a whole, this would constitute what is known in economics as a principal agent problem. An agent with an objective function that differs from that of its principal will act in ways that are contrary to the interests of the principal, resulting in net costs. However, these particular results indicate that those alternatives that reduce PVC also result in significant agency cost savings.The methods introduced in this study enable data that are actively being collected by USACE to yield specific insights into how the cost of lock operations and maintenance could be reduced. Although these results provide insight for optimizing USACE lock operations and maintenance practice, they should be interpreted in light of the assumptions made in modeling the decision and the realities of business processes beyond those within the model framework. The existing USACE budget process may not be well aligned with implementing some of the alternatives in this study. For instance, in order to request maintenance funds 2 years in advance, lock managers must predict which components are in OCA C or worse condition without dewatering. If an incorrect amount of funding is requested due to prediction errors, it may not be possible to repair or replace components as assumed by the decision model. Also, this model assumed that SHM information can be represented as continuous information about OCA ratings. Although the OCA process is an example of continuous condition information, actual SHM systems providing specific information must be compared against other alternatives for more accurate insights. Accounting for the benefits to performance provided by preventative maintenance activities is difficult and not accounted for in these results. These results depend on the quality of data in the OCA database, which is relatively new and is still maturing. The results of this study were generated only for W.D. Mayo Lock and Dam and may not well represent other USACE locks. Finally, the costs of administrating the OCA process were not considered in the decision model. If USACE wishes to consider a change in its policies for dewatering locks, it is recommended that a larger study be conducted that considers a variety of locks, costs, and modeling assumptions.


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