AbstractInfrastructure rehabilitation comprises remedial and prevention measures; however, preventing failure is not always possible, and direct investigations to find evidence of failure are challenging. Urban buried pipes are among the infrastructure that needs recurrent remedial actions. At this point, it is important to raise the question of what the appropriate strategy to locate or rehabilitate leakage is. This paper aims to implement and evaluate a Bayesian decision model for the maintenance planning of a water network. This includes the treatment of uncertainties in the evaluation of the best decision in a short-term perspective. To this end, a utility-based optimization routine based on the Bayesian theory has been used. The proposed model, due to its simplicity, can facilitate the initial problem-structuring in the process of decision-making under uncertainty. The model has been demonstrated on a water distribution network in Sweden, optimizing the decisions for locating and rehabilitating leakages. The results show that the cost of interventions and probabilities of leakages has a significant influence on the most appropriate decision.IntroductionWhen a failure arises or is expected to emerge in infrastructure, it should be assessed not just for further trustworthy rehabilitation actions but also at the right time. The course of action depends greatly on the strategies delineated by the infrastructure owners.In water utilities, evaluating rehabilitation actions, e.g., water-loss management, is considered sensitive and subjective, varying from one stakeholder to another. Making the rehabilitation strategies as objective as possible, the chosen actions are normally weighted by a representative utility function (value) that can be translated to the cost of each strategy; this is the first step for evaluating/prioritizing decision plans. The cost can be a monetary function in most cases, which can be completed by a life cycle cost analysis for achieving a long-term result. The cost-based function can also be defined as nonmathematical values like social satisfaction or environmental impacts.Reviewing the decision-making process for rehabilitation of infrastructures, it helps to mention the following two main distinctions: •short-term versus long-term strategies; and•proactive versus reactive strategies.The long-term and proactive decision actions are based on failure prediction models, considered as risk-prone strategies. The basis is typically network characteristics and historical failure data, e.g., from Barton et al. (2019) and Giraldo-González and Rodríguez (2020). However, it should be mentioned that none of the decision options are risk-free. The need to cope with uncertainties in this respect makes risk analysis for assessing the consequences of decisions taken under uncertain circumstances an essential part of infrastructure rehabilitation strategies.Data is the other source of uncertainty in many cases, as a characteristic of an asset is a vital part of infrastructure condition assessment. However, the quality and quantity of data vary significantly between water sectors. Thereby, evaluating the reliability of the data is a challenge, which must be managed carefully for the most reliable result. The cost of more investigation needs to be assessed against the benefit of more knowledge, making it possible to update existing information, which is the basis of the Bayesian theory in decision-making analysis.The Bayesian theory has already been used for decision models by the water sector and in other engineering fields, such as construction and energy. Nielsen and Sørensen (2014) used Bayesian networks for the operation and maintenance of wind turbines. The focus was on theoretical development and critical evaluation of methods for risk-based decision-making. Moreu et al. (2017) proposed consequence-based management of bridges with the help of the Bayesian theory. The two cited references are relevant considering the basic decision theory; however, the fields of application impose differences in decision scenarios, uncertainties, and consequences.During the last 20 years, more than 50 stochastic and probabilistic models for pipeline failure prediction have been presented; however, these technologies for rehabilitation and replacement planning have rarely been implemented in the water sector (Jenkins et al. 2015). Prioritizing the buried pipes for maintenance, rehabilitation, and replacement, Kabir et al. (2018) proposed a Bayesian belief network (BBN) model. Earlier, Kabir et al. (2015) studied the use of BBN for assessing the risk of failure in metallic pipes; further, in another study, Kabir et al. (2016) developed a fuzzy Bayesian belief network (FBBN) for safety assessment of oil and gas pipelines. These works focused on failure prediction models and the asset deterioration process considering the likelihood of failure. The modeling relied on a complex representation of the systems, which required a comprehensive dataset. Poulakis et al. (2003) linked the Bayesian theory with hydraulic simulation. The proposed model is used for leakage identification in a water distribution network but does not involve the decision process. Francis et al. (2014) developed a knowledge-based Bayesian model for predicting pipe breaks in a water distribution system in the mid-Atlantic United States. Later, Elmasry et al. (2017) developed a Bayesian model for sewer pipelines, using probabilities of occurrences and conditional probabilities taken from observations. A risk assessment model for prioritizing sewer pipes inspection with the help of Bayesian networks was proposed by Anbari et al. (2017).In relation to the listed references, it can be noted that these studies have almost exclusively focused on failure prediction models and the asset deterioration process considering the likelihood of failure. To the authors’ knowledge, only a few works have demonstrated the evaluation of the required decisions in the case of inspection/rehabilitation using the Bayesian decision theory. The less sophisticated decision model proposed in this study is aimed as a helpful tool for the operative level of loss management under uncertainty even with limited input data.Considering the importance of rehabilitation strategies for making a utility-based decision plan, this study presents a critical evaluation of the decision strategy currently applied by the policymakers of the studied water sector in Uppsala, Sweden. This is presented with a case study described in a separate section. To this end, a theoretical model of the decision process through the application of the Bayesian decision theory has been used. The theory has been applied for decision-making under uncertainty. This risk-oriented approach focuses on the value of further information through more investigation and experimentation for handling uncertainty in the decision-making process (Parmigiani 2001).This study is based on the current manual signal monitoring technique applied by the selected water sector to the water supply system for leakage detection and associated remediation actions. The utility-based theoretical model developed considers the same approach of nonautomated monitoring of the same signal. The information stream from the monitoring system preceding the decision process, leading finally to the rehabilitation actions, is schematically visualized in Fig. 1. The decision tree shows decisions (e) and actions (a) with square nodes and random events (z and θ) with circles. The input to the decision tool is based on a manual evaluation of the measured flow signals. More advanced methods for data automation, real-time leakage detection, and other types of signal processing approaches are not explicitly treated in this paper.The aim of this study is to present Bayesian decision support in the case of water-loss management and rehabilitation in water distribution networks. This is demonstrated by considering a case study in Uppsala, Sweden. The decision strategy currently adopted by the water sector of the case study is translated in this paper to the Bayesian decision theory, considering the uncertainties for critical comparison.Utility-Based Decision AnalysisThe decision theory that is concerned with reaching the most rational outcomes is called utility-based decision-making (Briggs 2019). For evaluation of a utility-based decision plan in this study, a set of actions for rehabilitation of leakage is identified, and the optimal choice is evaluated by finding the lowest expected cost. Accordingly, a theoretical decision tree using Bayesian preposterior decision analysis is used.The decision-making process, like any other engineering activity, is characterized by different levels of uncertainties. The level of uncertainty may differ from simple deterministic decision models to probabilistic decision modeling. There is a trade-off between the accepted level of uncertainty and the knowledge-content in the process of decision-making; as such, it shall be evaluated whether the outcome will change considerably as a result of gathering more knowledge from the actual system by, for example, measurements, nondestructive tests (NDT), or destructive experiments. The focus of the Bayesian decision analysis (Benjamin and Cornell 1970) is on the uncertainties involved in the process of decision-making. This can be done by combining the information presently available with updated information from new evidence.Decision trees, which are commonly used in the decision-making process, can delicately visualize the decision alternatives, random outcomes, assigned probabilities, and utility or cost of each action. The random events are affected by chance due to uncertain outcomes and, as such, are considered uncontrollable circumstances (aleatoric events) (Magee 1964).In the decision tree, only those decisions, events, or results that are important and have consequences to compare will be involved in the decision process. Then, at each point, the course of uncertainty and the type or range of alternative outcomes will be identified. Assigning probabilities to different events or results of actions and, in a similar way, the costs and gains of various events and actions are essential parts of decision-making, which will be estimated in the process. Having established the mentioned structure and assigned values, the outcome of the defined alternatives is analyzed to find the best course of action (Magee 1964). Fig. 1 illustrates the schematic decision tree elaborated for the case study.The idealization of the decision scenario to a relatively simple decision tree is an advantage considering the computations, but it imposes limitations in representing real-world decision problems. If the decision-making follows a more complex protocol, simplifications have to be made, or a reformulation of the model is necessary. Another limitation is the modeling of consequences, which all have to be measured in comparable units. If social or environmental consequences are to be included in the model, they have to be recalculated to the same unit as the costs of the actions.Decisions and UncertaintiesIn order to build up a decision tree and minimize the cost/risk of choosing the best action, a list of possible inspection and remediation strategies adopted by Uppsala Water Company is assembled for the case study. These actions can be considered as corrective maintenance as the action takes place when anomalies in the monitored flow signal are detected. On the other hand, the monitoring strategy used as a starting point for this study could be considered as a preventive maintenance process. This facilitates the detection of early signs of leakage in the network before a failure that takes place in silence leads to catastrophic consequences. As such, the studied actions for inspection and remediation can be categorized as proactive-reactive strategies.Regardless of which type of strategy is chosen for the action, the decision-making process itself is an uncertain approach as it is based on uncertain inputs. The probability of failure/leakage, progressive deterioration, true states, mechanisms related to age or random events, access to the history of the failure, the variability of failure/action costs, taking into consideration the social cost, distinguishing noise from signals, and the shortage of data that affect the epistemic uncertainty are among the many uncertain properties that affect the accuracy of the decision. In this study, the effect of uncertainty on the decision-making process is simplified by performing a two-way sensitivity analysis considering the probability of leakage and the variability of some action costs.The true state of uncertainty is dealt with by considering the Bayes’ rule via updating the prior probabilities to give them greater accuracy. According to Bayes’ rule, considering new information and prior probabilities, the posterior probabilities are calculated (Benjamin and Cornell 1970) (1) P″[θi]=P[θi|zk]=P[zk|θi]P′[θi]∑jP[zk|θj]P′[θj]where P″[θi] = posterior probability of θ given z; P′[θi] = prior probability of θ; P[z|θ] = likelihood of z given θ; and the denominator at the right-hand side of the equation = total probability of z. See Fig. 1 for the connection between the decision modeling and the variables in Eq. (1). The posterior probability of θi is P″[θi|e,z], which will depend on the prior probability P′[θi], the decision e, and the stochastic outcome z.Utilities and CostsThe efforts to find the leakage and later maintain the failure come with a cost. However, taking no action when there is a water leak would not be free of charge either, not just because of the environmental/water cost but also because of the consequences that can follow from the leakage. In this study, the dissatisfaction of the water consumers could be added as a social cost of untreated leakages. By assigning a higher cost for leakage, the most favorable actions are those that aim to repair the failure. The present study maps the effect of the variability in political views by choice of the most effective decision.The posterior expected utilities are calculated (2) E[u|e,z,a]=∑iu(θi|a,e,z)p[θi|a,z]Then, the optimal expected utility for each decision outcome is gained (3) for which the expected utility for each decision alternative e becomes (4) E[u(e)]=∑ku*(e,z)P[zk|e]The experiment e with the lowest expected utility E[u(e)] represents the most rational decision.Case StudyThe case study describes the current rehabilitation decisions of a water network in Uppsala, Sweden, managed by the Uppsala Water Company using active leakage detection. The Uppsala Water Company is responsible for water supply, wastewater collection and treatment, and waste disposal in the Uppsala municipality.Two interviews were set up in October 2019 to get information about the current decision system of the rehabilitation strategies for the water mains in the study area. The information was collected by interviewing a few members of the Uppsala Water Company. As such, it is not expected to represent all the aspects of the company’s policy. The interviewees were the section manager and two water and sewage engineers, one of whom worked specifically with signal analysis and leakage detection.A signal-monitoring project based on flow and pressure measurements on the water and sewage system has helped the company to work actively with leakage detection and rehabilitation actions. The active leakage detection project started in 2013, using instruments such as a leak noise correlator. At that time, the leakage rate from the water pipes in Uppsala was about 25%. By equipping the networks with the Internet of Things (IoT) devices and combining this with active leakage detection/remediation, the leakage rate was reduced to 14%–16% in 2016 and remains more or less constant to date.The specific area selected, Länna, is located in east Uppsala and gets its drinking water from Lake Långsjön. The area was connected to Uppsala’s wastewater treatment plant, Kungsängsverket, in 2016. Länna is equipped with an IoT flow meter on the main pipe directly after the water treatment plant with a history of water consumption from May 2016 up to date. Länna has experienced several network failures, which caused significant leakages during the measurement period.The Current Decision-Making ProcessThe data presented in this section shows the current routine of Uppsala Water Company’s service team, who monitors the flow/pressure signals of the water mains to detect a failure in the water system.A brief but elucidating description of the water sector’s policy revealed that there was no structured rehabilitation strategy for the water system. The strategy in place can be categorized as undocumented and driven by subjective protocols. Moreover, no digital documented record of failures was available at the time.The Uppsala Water Company’s decision strategy is a data-driven (measurement-based) method achieved by monitoring. The manual analysis of flow/pressure signals, involving an operator visually examining graphs on a computer screen, gives an insight into the condition of the network whenever unexpected changes or extreme levels in the flow signal are found.Länna’s distribution network is monitored for flow data at the station located on the PE100 main water pipe with a diameter of 225 mm (inner 198 mm) at the outlet of the water treatment plant serving Länna. The IoT sensor transmits every 10 s via a platform named ADCON to the company. When anomalies in signals are observed as possible signs of defects in the network, identified as abnormal changes in the flow, the rehabilitation team can consider several decision alternatives: •M1 = taking no action;•M2 = more investigation;•M3 = full rehabilitation and replacement; and•M4 = partial rehabilitation and renovation.While M2 comes at the first level of decision-making in this study, M3 and M4 are represented in the second level of decision strategies. However, M1 is a passive strategy (do nothing) that exists in both levels of actions.The severity of the possible leakage is estimated from the change of the flow level and the duration of the anomaly. For minor anomalies in signals that remained for less than 24 h, rehabilitation would either not be undertaken or be postponed until further supporting evidence, M1, as this could be caused by abnormal consumption of water from either households or industry but not necessarily indicate any damage in the network.All the leakages that are not among M1 are repaired directly or possibly within a couple of weeks. Leakage detection from the first sign to the remediation action is a daily routine with an assigned budget. For major irregularities in signals, the replacement or renovation (M3 and M4) of the broken pipe is the actual choice; however, taking these actions normally needs more investigation (M2) to establish that there actually is a failure and to find where it is located.Some leakages are visible at the ground surface, and large water retentions make them easy to detect. However, the problem lies in undetected leakage going on for a long time, which can cause seepage, damage infrastructure, and/or allow contamination to enter the pipe. The M2 action in the matter of further investigation consists of different alternatives, such as acoustic (listening and cross-correlation) and nonacoustic (tracer gas, ground-penetrating radar, thermography, and pigging) actions (Misiunas 2005). In this case, the acoustic action was the only practical choice.The process of inspection can be described as follows: 1.Surveying via hydrants, valves, service lines, and junctions (also known as spider-listening in Swedish) for leak detection.2.Partial area closure [network zoning and establishing water district metering (WDMs)] for leak detection.3.Using a listening stick to pinpoint the leak location.4.Using a leak noise correlator to pinpoint the leak location.5.Visual inspection via excavation.The inspection actions (1), (3), and (4) are considered conventional leak detection methods and can be seen either as a separate inspection method or in combination with (2), which is called WDM action. The excavation (5) that may be included in remediation actions (open versus trenchless technologies) is used separately as an inspection method.WDM can be performed in permanent or temporary ways on different scales based on the project’s specific objectives (such as network management, water balance, and pressure regulation). A small permanent district would give the best results in leakage monitoring and is called a district meter area (DMA). However, a temporary district metering (waste metering) divides the water network easily into defined areas; these are generally smaller than DMAs (Di Nardo et al. 2013).A listening stick is always used before the excavation action for finding the exact point of failure with accuracy in the range of centimeters. For major leakages, partial area closure and night consumption measures are performed. Other damage or defects found during the repairs could lead to new choices and a posterior decision; however, the operating team has focused mainly on the specific failure that has been raised each time. The condition of the buried pipes in the vicinity of the identified problem area is typically not assessed during the operations. The replacement strategy, M3, is the choice when the asset (such as a pipe and valve) is very prone to failure; the asset, in this case, does not deliver adequate service to customers, and its poor structural condition prevents renewal strategies to boost the asset’s condition. Depending on the problem a pipe is suffering, whether a structural or hydraulic issue, the replacement strategy might include replacement with a larger pipe, with the same or other materials, and/or building an auxiliary parallel pipe (Deb et al. 2002).The M4 enhances the asset’s condition by taking either remedial or preventive actions or a combination of the two. The rehabilitation action is chosen based on factors like the asset’s age, material, frequency of failure, hydraulic capacity along with water quality, soil condition, and other technical and strategic factors (Deb et al. 2002). This might include no-dig remediation actions, such as a cured-in-place pipe (CIPP), slip lining, structural liner, nonstructural liner, semistructural liner, pressure reduction, joint rehabilitation, and cathodic/anode protection.Based on the interviews conducted, leakage is a very probable event in Uppsala; however, for detecting and rehabilitating failures, the wrong decisions can be made occasionally. Blind excavations in the wrong places or wrong decisions as a result of signal misinterpretation are some examples.For documenting the history of the failures and rehabilitation actions, the Uppsala Water Company records excavations within an operational failure report (paper format); the report describes the rehabilitation action taken, the type of damage, and the location of the failure.Despite all these remedial actions, the Uppsala Water Company has no officially documented strategies for prioritizing the rehabilitation actions in areas that are more prone to failures.The Utility-Based Decision AnalysisIn this case study, only leakage from the water main is considered as a sign of system failure with the help of continuous monitoring of the flow signal for detection of anomalies. However, the data treatment, such as preprocessing, filtering, resampling, smoothing, finding peaks, and signal patterns, is out of the scope of this paper and will not be further investigated in this study. The outcome of such analyses with the possibility of leakage prediction is expected to be the a priori probability of leakage, considered as P′[θ] in the decision analysis.The utility or cost of each decision as a source of uncertainty in the decision-making process was set to a defined range to perform a two-sided sensitivity analysis. The expected costs were calculated for a combination of leakage detection methods and maintenance remedies. The lowest expected cost obtained represents the most rational strategy based on the expected utility theory.The decision alternatives studied in this case consist of several common water main rehabilitation actions. The first category of actions after evaluating the water flow signal in the case of anomalies includes the cases of do nothing or do more inspection to find the leakage. The latter consists of three inspection actions defined as e2 for conventional leakage detection, e3 for DMA and conventional leakage detection, and e4 for blind or trusted excavation.By taking a course of action from among the mentioned alternatives, there will be uncertainties about the level of possible indications: z1 for no indication, z2 for vague indication, and z3 for distinct indication. The second level actions after inspections are defined by remediation strategies. These actions are defined as a1 for no action, a2 for full rehabilitation, and a3 for partial rehabilitation. For the Uppsala case, the true state of the leakage as evidence of pipe failure may have two discrete states: θ0 for no leakage and θ1 for leakage.Knowing that the leakage rate in the studied case is about 16% (current state), the a priori probability for the different states is considered as P′[θ0]=0.84 and P′[θ1]=0.16. However, P′[θ1] has been varied in the range from 0 to 1 in the sensitivity analysis.Case-Specific UncertaintiesGathering more knowledge, for instance, about the state of the failure, will help to cope with epistemic uncertainties. The processing of the flow signal could be an example for updating the decision-maker’s knowledge about the state of leakage in the network.Evidently, further inspection may help to find more robust evidence of leakage (true state); however, the risk of misleading information always exists. Considering the accuracy of each inspection, conditional probabilities (sample likelihoods) for each of the different states (z) due to the actions taken (e) are suggested, as shown in Table 1. Due to the lack of experimental data, the stated values for decision alternatives e2 and e3 are assumptions and, in themselves, affected by uncertainties. The numerical values were assigned after discussions with the personnel at the water company. The excavation alternative (e4) is expected to deliver a correct prognosis without uncertainty. More information about different leakage detection methods and their accuracies has been presented by Li et al. (2015).Table 1. Tentative sample likelihoods for the actions e2−e4 in the case studyTable 1. Tentative sample likelihoods for the actions e2−e4 in the case studyActionIndication zθ0θ1Conventional (e2)z1 no indicationP[z1|θ0]=0.90P[z1|θ1]=0.10z2 vague indicationP[z2|θ0]=0.07P[z2|θ1]=0.10z3 distinct indicationP[z3|θ0]=0.03P[z3|θ1]=0.80DMA (e3)z1 no indicationP[z1|θ0]=0.95P[z1|θ1]=0.05z2 vague indicationP[z2|θ0]=0.03P[z2|θ1]=0.05z3 distinct indicationP[z3|θ0]=0.02P[z3|θ1]=0.90Excavation (e4)z1 no indicationP[z1|θ0]=1P[z1|θ1]=0z2 distinct indicationP[z2|θ0]=0P[z2|θ1]=1The uncertainties about some values, such as the probability of leakage and the leakage cost, are evaluated by performing a two-way sensitivity analysis for these variables in a defined range. Performing the two-way sensitivity analysis facilitates understanding of when the best decision shifts from one alternative to another over the varied probability of leakage.Consequences and OutcomesThe outcome of every decision should be calculated by assigning a cost to the model inputs. These costs were adjusted to the current year’s price (2020) using an approximate inflation rate. Clearly, assigning a value or a range of prices to the outcomes of each decision is an uncertain approach; this affects the decision model outcomes. Moreover, some assumptions for the calculation of the outcome should be made for ease of the consequent analysis. As such, the role of uncertainty in this matter will be accentuated even further.The consequences included in this study are the water cost, water cost in the case of leakage and taking no action, several inspection strategies, and a few cases of rehabilitation actions, as mentioned previously.The social, environmental, and traffic-related costs, jointly with the role of political strategies, are converted to the consequences that a probable leakage would have if taking no remediation action. In this respect, the water cost of taking no remediation action in the case of leakage was varied from the same price as the water in the normal case to 500 times more expensive. This upper limit was shown to capture the region where the most optimal decision shifts between the possible options. Table 2 shows the activities included in cost estimation along with the important factors that affect the results.Table 2. Activities included in cost estimation and important variables that affect these outcomesTable 2. Activities included in cost estimation and important variables that affect these outcomesVariableAffecting factorsConventional leakage detectionCrew, work shift, leakage detection, and repair duration.WDM and conventional leakage detectionCrew, work shift, leakage detection and repair duration, number of flow meters, and price.ExcavationDifficulty of the work, soil structure, soil contamination, depth, need for sheet-pile, equipment, bedding material, slope, weather condition, traffic, groundwater level, duration of excavation remaining open, underground essential services, location, and so forth.Full rehab, pipe replacementPipe material and diameter, valves, bends and services, and choice of technology.Partial rehab, pipe renovationPipe material and diameter, valves, bends and services, and choice of technology.Water costPolitical strategies, water production and treatment cost, wastewater treatment cost, environmental and social costs, rate of leakage in each case, leakage detection, and repair duration.Water cost in case of leakage and taking no actionPolitical strategies, water production and treatment cost, wastewater treatment cost, environmental and social costs, rate of leakage in each case, and consequences of leakage by taking no action.The decision model inputs are shown in Table 3. Some of the parameter inputs were simultaneously varied to estimate the influence of the parameters. The goal was to observe the shift of the best decision resulting from the two-way sensitivity analysis performed for selected variables.Table 3. Considered prices for activities and updated costsTable 3. Considered prices for activities and updated costsVariableCost (SEK)Updated cost (SEK)Crew pricea (one person)400,000200 (per h)Flow-meter on networka (per set)10,00012,000Pipe replacementa (per meter)76,00090,000Water priceb (per m3)10.6015Leakage rate (l/s)—10,000Excavationc50,00052,000Table 3 presents the prices assigned to the activities considering the recommendations from the literature. The prices were adjusted using an approximate inflation rate to be representative of today’s price levels. Additionally, some of the costs were rounded to the nearest Swedish Krona (SEK) 5, 50, and 1,000. It should be noted that SEK 10 corresponds roughly to EUR 1.Considering the prices listed in Table 3, the following assumptions have been made to simplify the consequence analysis in the study. Finding the suspected leakage and fixing the failure would take place, ordinarily, within 7 working days. This assumption has been made based on the Uppsala Water Company’s statements through the interviews performed. This statement reveals that detection and repairing of the leakage are performed directly after the occurrence or at most within a couple of weeks. A shorter period is assumed (5 days) for the WDM method, as area closure might facilitate leak detection. However, the leakage cost (for the first and second scenario) is the same as for conventional leakage detection in 7 days because the water price is rounded up. The extra price for a night work shift in this respect was considered as well. The average working hours per day were assumed as 6 h as the personnel might spend some of their working hours on other projects. Two persons actively working on the leakage detection and remediation were expected to participate in the project according to the interviews performed.In Sweden, according to Stockholm Water and Waste Company (SVOA), the average price of water per cubic meter includes the cost for water treatment SEK 3.4/m3 and the network cost SEK 7.2/m3 (prices are from 2017). Nevertheless, no-cost estimation was considered in this study for the treatment of any extra amount of water that might infiltrate the sewage system from the leaking water pipes.The excavation was included in the two stages of the decision model: inspections and rehabilitation actions. As the Uppsala Water Company did not consider the trenchless technology for rehabilitation actions at the time of this study, this option was excluded from the possible choices. According to the Uppsala Water Company and SVOA, the average cost for digging the ground is about SEK 50,000 for a normal trench with no specific difficulties in excavation. However, this study considers the choice of excavation for inspection with a penalty factor of 1.6 to include the possibility of blind pits. There is no basis for this assumption, but the authors consider this factor to show the role of penalty policies in the proposed model. The excavation base cost (without penalty) is included as a part of a full rehabilitation action (M3, replacement) and partial rehabilitation (M4, renovation).Based on the history of old leakages via flow signals, the leakage rate can be calculated from the amplitude difference of the normal and leakage trends. However, the calculated leakage rate differs from one leakage case to another; therefore, an assumption has been made in this study considering the leakage rate and water cost calculation. Based on hydraulic principles, the leakage from a pipe can be considered as flow from orifices. Thus, the orifice equation seems suitable for calculating the flow rate through a leak as described by van Zyl and Malde (2017) (5) where Cd = leakage coefficient, considered as 0.6 in most cases; A = area of the hole in the pipe, assumed in this study as 60  mm2, corresponding to a hole with a diameter of 9 mm; h = pressure head differential over the hole; and g = acceleration due to gravity. According to the Uppsala Water Company, the maximum head for the studied area, Länna, is 5.2 bar, and the minimum head is 2.4 bar; for ease of calculation, the average head was considered. With all these assumptions, the leakage rate per failure was calculated to be 1  l/s. Table 4 shows the costs for the rest of the activities (one unit per stated dimension) related to the mentioned assumptions.Table 4. Calculated costs for the activities considering assumptionsTable 4. Calculated costs for the activities considering assumptionsVariableBase cost (SEK)Rounded cost (SEK)Full rehabilitation action, replacement (per meter)141,000145,000Partial rehabilitation action, renovation (per meter)169,200170,000Conventional leakage detection (per meter)16,80020,000WDM and conventional leakage detection (per meter)24,00030,000Excavation for inspection (per case)85,00085,000ResultsThree different scenarios were defined for performing the two-way sensitivity analysis by variation of selected variables: the water cost in the case of no remediation action, the excavation cost, and the leakage rate. The purpose was to present the effect of selected variables on shifting the best decision choice resulting from the utility-based model. The scenarios and resulting diagrams are presented as follows.Reference ScenarioThis scenario, which was taken as the base scenario, considers the costs listed in the last column of Tables 3 and 4; the leakage rate was assumed as 1  l/s. By assuming this rate for the probable leakage and SEK 15 as the cost of 1  m3 of water (Table 3), the leakage in the case of inspection/remediation in 7 days (604,800 s) costs SEK 9,072, which is rounded up to SEK 10,000.Fig. 2 presents the result of the two-way sensitivity analysis performed by variation of the leakage probability [0,1] and the ratio of the water cost when taking no remediation action to the water cost in the case of fixed failures [1,500]; the latter can be translated to the consequences of probable leakage events that have been left untreated. The shift of the best decision (first level and inspection) is displayed in Fig. 2, with different patterns for the lowest expected utility (cost) evaluated by Eq. (4). The figure shows that all four options may be optimal depending on the conditions.For different combinations of leakage probabilities and consequences of untreated leakage, Fig. 2 recommends an inspection decision. In this study, either excavation, WDM in combination with conventional leakage detection, conventional leakage detection by itself, or taking no action could be the most cost-effective alternative. For instance, if the probability of leakage is assumed as 40%, the boundary for no action is 23, which multiplied with the water cost of SEK 10,000 becomes SEK 230,000, corresponding to the maximum consequences acceptable for making no action the most rational decision alternative. If the consequences are assumed higher, the decision should change sequentially from conventional leakage detection to excavation along a vertical line in Fig. 2. A consequence of SEK 2,700,000 for the same probability of leakage makes excavation the most rational inspection alternative.For a lower probability of leakage, the boundary of no action lies at higher consequences. An overlap between the best decision options can sometimes be detected, which is obvious in the numerical results but requires manual treatment for the graphical visualization. The competition between conventional leakage detection and WDM in combination with the conventional method dominates the decision space.Influence of Changes in Excavation CostThis has been performed to show the effect of the excavation (inspection) cost on the choice of the best decision. Fig. 3 presents three values for the excavation cost: SEK 50,000, SEK 95,000, and SEK 120,000, over the variation of leakage probability [0,1] and the ratio of the water cost when taking no remediation action to the water cost in the case of fixed failures [1,500].The results in Fig. 3 illustrate the impact of monetary penalty policies on less convenient actions, such as blind excavations in the case of inspection. This has been done to show the role of actor preferences, social and environmental factors, and so forth to demonstrate the flexibility of the proposed model in the case of the best decision. Changes in the excavation cost starting from SEK 50,000 make excavation the most cost-effective inspection alternative, which is dominant among the other alternatives. Raising the excavation cost to SEK 95,000, the threshold for leakage probability decreases significantly to about 25%–50%, and the consequences of untreated leakages shift up to SEK 2,750,000. The minimum expected utility for the excavation decision is achieved in this new range. The excavation alternative for inspection ceases to be an option when the excavation cost reaches SEK 120,000. A sensitivity analysis on the effect of other alternatives can be illustrated similarly for different thresholds.Influence of the Leakage RateThis scenario considers the real rate of leakage from an actual failure that happened in Länna in September 2017. The leakage rate was calculated from the differences of the normal trend and anomalies in the measured flow as a sign of leakage. The anomalies were observable from the flow signals on September 19, 2017, to September 25, 2017. By calculating the differences from the normal trend (October 24, 2017, to October 30, 2017), the volume of leaked water was 550  m3. This volume calculated from the observed leakage (in 7 days) is very close to the estimated assumed volume of leakage of 1  l/s for a leakage event (600  m3). The result of the two-way sensitivity analysis for this scenario is compared with the reference scenario that considers 1  l/s for leakage events (7 days); however, the results were close to identical, making differences difficult to detect. Considering this, five different values for the water cost as evidence for several leakage rates have been modeled (SEK 5,000, 8,000, 10,000, 50,000, and 100,000). The purpose was to compare the changes in the boundary shift of the no-action alternative for the assumed leakage rates. Fig. 4 shows the shift of the boundary for the no-action alternative (first level and inspection) over the varied probability of leakage [0,1] and the water cost in the case of no action and the presence of leakage.As the result shows, the boundary of the no-action alternative overlaps to some extent the lowest leakage rates, which cost either SEK 5,000 or 10,000. Considering these two values, for a 40% probability of leakage, the boundary of taking no action depends in some measure on the consequence of untreated leakage, which corresponds to SEK 225,000. However, with a higher rate of leakage, the boundary of the no-action alternative extends (SEK 50,000 and 100,000). As an example, for the leakage rate corresponding to SEK 100,000 (double dashed line in Fig. 4) and a 40% probability of leakage, the boundary of taking no action extends to about SEK 320,000 for the consequences of untreated leakage. This is because the utility-based model considers even the treated leakage with a higher water cost as an expensive action. In contrast, without doubt, as the leakage rate becomes more severe, it must be located and treated within the necessary time; so, the area dominated by the no-action alternative is expected to be reduced for higher leakage rates. Hence, it might seem impractical to consider a less advanced utility function in decision models by considering the cheapest alternative, as different criteria, not just monetary values, can be combined in the utility functions. A well-scheduled action plan and a budget threshold for the chosen alternatives could be an option to constrain the decision process. This would be conducive, especially when altering the utility functions to penalize no action.DiscussionIn the opinions of decision-makers, no specific decision model can be claimed to give a better result than any other approach (Mutikanga et al. 2011); there is a trade-off between simple models and multicriteria approaches that may lead to better accuracy. The proposed model with its limited input data seems to be a helpful tool for the operative level of loss management under uncertainty. However, the proposed model has been tested only for one case, and consequently, the outcomes are limited to that. Furthermore, it might seem impractical to consider a utility-based decision model for comprehensive water loss management as there are multiple criteria affecting this strategy. For example, as the results from the sensitivity analysis reveal, a decision based on the minimum expected utility action is not necessarily the desired choice as the preferred choice should take an action to remediate the larger leakages in real-time. Thus, it seems crucial to reformulate the mathematical model to revise the decision scenario and utility functions in the case of no action.Despite applying less advanced utility functions, the studied model was able to provide a general rational guideline for the responsible actor at the operating level of an organization, enabling an early leakage detection and repair strategy as an essential tactic for water loss management that can minimize the loss volume. The proposed model seems to be useful in this respect by accelerating the awareness and location of leakages. This may even save some time for focusing on repair solutions.The decision model used in this study can facilitate the initial problem structuring of the process of decision-making. Moreover, the results give an insight into less costly decisions through variation of the leakage probability and the consequences that may be caused by untreated leakage.Knowledge about the structural condition of pipes seems essential as it enables a proactive approach, indicating the extent of the failure (Sægrov et al. 1999). The suggested model has the potential to be updated with information taken from deterioration models; this can be achieved by updating the a priori true states and probability of failures. Considering the actual leakage rate from the applicable failure prediction models makes real-time decision-making analysis possible for the suggested model.Nonetheless, the model’s limitations do suggest some issues for further research. As the results demonstrate, the cheapest action is not necessarily the desired choice; hence, a comparison of the current model with a traditional cost-benefit analysis by considering the life cycle assessment is a valuable topic for further research.The importance of strategic planning in water loss management and its effect on other areas of decision-making in an organization is another issue for further study. This can help to include different levels of the organization in the process of decision-making, which may be affected by diverse points of view in ranking the best decision choices. Moreover, including more actions considering the stakeholders’ preferences in the decision model is an issue that can be considered in further studies to make the model more comprehensive.ConclusionsThis study presents a utility-based decision model supported by the Bayesian theory in the case of water loss management (operative level); the model is applied to a case study from Uppsala, Sweden.The proposed model is a valuable tool for minimizing the time for leakage detection and localization besides giving a general insight about the least costly inspection decision in the context of uncertainty. Considering expert knowledge together with theoretical facts for defining the prior evidence guarantees awareness of the uncertainty, which leads to a more reliable decision-making process. This is the main strength of the Bayesian theory. The following conclusions can be drawn from the suggested decision model and the presented case study: •The water loss management run by the Uppsala Water Company lacks strategic planning (SP); however, even in the absence of a long-term failure prediction strategy, their experience was a successful example of leakage reduction. The proposed model developed a more systematic framework for supporting the problem-structuring part in the decision-making of the studied sector.•There are different levels of data scarcity in water utilities that could cause difficulties in estimating some inputs/relations in the process of decision-making; faced with this incomplete data, the proposed model nevertheless shows its ability even where the existing data is limited.•Determination of the probability and the consequences of leakage is the most challenging part of probabilistic decision modeling. Luckily, Bayesian theory can provide decision-makers with awareness, for example, of the uncertainties in (prior) failure probability, the accuracy of remedial actions, and the consequences of the recommended decisions. By considering these uncertainties discretely in the modeling, the proposed model shows briefly how these challenges can affect the selection of the best (most economical) decision.•By applying monetary penalty policies to a less convenient action (such as actor preference as well as social and environmental factors), the proposed model shows flexibility in the case of the best decision, e.g., Scenario 2 involving the influence of changes in the excavation cost.•Clearly, the cheapest action is not necessarily the most desirable choice as the results demonstrate in some cases, like the case of blind pits for inspection with different costs in the sensitivity analysis and influence of the leakage rate on the extension of the no action case.Data Availability StatementSome or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.AcknowledgmentsThis paper was produced as part of the Mistra InfraMaint research program with funding from Mistra, the Swedish Foundation for Strategic Environmental Research, and Stockholms Stadshus AB. The authors acknowledge the support from the staff at the division of Urban Pipes Planning, Uppsala Water Company, who provided this study with monitoring signals and information about their experience over the past years. Special thanks to Mr. Ronnie Branteström whose great job in water loss management in Uppsala County made this study possible for external entities from academia and industry.NotationThe following symbols are used in this paper:A=area of hole in pipe;a=decision level 2 action;Cd=leakage coefficient;E[·]=expectation;e=decision level 1 action (experiment);g=acceleration due to gravity;h=pressure head differential;P[·]=probability;P′[·]=a priori probability;P″[·]=posterior probability;Q=flow rate;u=utility;u*=optimal expected utility;z=random outcome of actions; andθ=random outcome of state.References Anbari, M. J., M. Tabesh, and A. Roozbahani. 2017. “Risk assessment model to prioritize sewer pipes inspection in wastewater collection networks.” J. Environ. 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