New Multiobjective Optimization Approach to Rehabilitate and Maintain Sewer Networks Based on Whole Lifecycle Behavior Downloaded from ascelibrary.org by Tufts University on 10/28/17. Copyright ASCE. For personal use only; all rights reserved. Ahmad Altarabsheh 1; Amr Kandil 2; and Mario Ventresca 3 Abstract: This study proposes a new methodology for selecting renewal plans for sewer networks based on their impacts on the behavior of the networks over their whole lifecycle. The proposed approach combines a multiobjective genetic algorithm and Monte Carlo simulation to maximize network condition and serviceability while minimizing network risk of failure and total lifecycle cost for the entire planning period. The algorithm was applied to a sewer network in Sahab City, Jordan, in 16 different analysis scenarios that consider the uncertainty in the model variables. These different analysis scenarios varied the network age, deterioration rate, and available budget at each time step throughout the planning period. The model was then validated by statistically comparing its performance to an existing prioritization model that does not consider the long-term impact behavior of the wastewater system. The results show that the proposed algorithm outperforms the existing prioritization model because it results in statistically significant improvement in the network condition, risk of failure, serviceability, and the total lifecycle cost at the end of the planning period. DOI: 10.1061/(ASCE)CP.1943-5487.0000715. © 2017 American Society of Civil Engineers. Introduction The limited budgets that municipalities have, along with the rapidly dilapidating condition of the sewer network, increase the need to adopt optimized proactive renewal strategies. These strategies should aim to reduce the risk of failure and improve the structural and the operational conditions of the sewer networks with minimum lifecycle costs. Therefore, this study proposes a lifecycle cost analysis–multiobjective genetic algorithm (LCA–MOGA). The algorithm combines a multiobjective genetic algorithm [NSGA III by Deb and Jain (2014)] and Monte Carlo simulation to select near optimal renewal plans for sewer networks that meet the decision makers’ objectives and regulations. The algorithm bases its selection of renewal plans on their impacts on the long-term behavior of the sewer network over the whole lifecycle. The main premise of the proposed approach is that renewal plans may provide the best trade-off between the conflicting objective functions in a particular time step, but not the best trade-off between these objective functions over the lifecycle of the network. This may be due to the dynamic behavior of the wastewater system caused by the different dynamic socioeconomic factors that impact the behavior of the system (Marzouk and Omar 2013). An example of these dynamic factors is the available budget at each time step 1 Assistant Professor, Dept. of Construction Engineering and Management, School of Civil Engineering, Yarmouk Univ., Shafiq Irshidat St., 21163 Irbid, Jordan (corresponding author). ORCID: https://orcid.org /0000-0002-5358-1836. E-mail: Ahmad.gt@yu.edu.jo 2 Associate Professor, Dept. of Construction Engineering and Management, Lyles School of Civil Engineering, Purdue Univ., 550 W Stadium Ave., West Lafayette, IN 47907-2051. E-mail: akandil@purdue .edu 3 Assistant Professor, School of Industrial Engineering, Purdue Univ., 315 N. Grant St., West Lafayette, IN 47907-2023. E-mail: mventresca@ purdue.edu Note. This manuscript was submitted on June 3, 2016; approved on May 25, 2017; published online on October 25, 2017. Discussion period open until March 25, 2018; separate discussions must be submitted for individual papers. This paper is part of the Journal of Computing in Civil Engineering, © ASCE, ISSN 0887-3801. © ASCE of the planning period, which may change due to fluctuations in demand. Therefore, considering these factors would enable utilities to adjust their spending at the current time step according to the expected available budget in the future. For example, if a water utility forecasts a shortfall in funding in the future, it may reduce its spending at the current time step to allow an overall optimal performance. The proposed approach aims to provide a tool that enables an optimized proactive asset management that considers multiple criteria. A number of objective functions have been considered in the literature for optimizing renewal plans of sewer networks (Marzouk and Omar 2013; Ward and Savic 2012; Berardi et al. 2009) as discussed in the “Background” section. The objective functions considered in these studies focused on specific elements, such as maintenance or inspection costs that are considered in this study as part of the lifecycle cost. A number of other studies highlighted the importance of other objectives, such as network condition, risk of failure, and serviceability (Halfawy et al. 2008; Ward and Savic 2012). Therefore, the following four objective functions are considered in this study: (1) maximizing average network condition, (2) minimizing network risk of failure, (3) maximizing network serviceability, and (4) minimizing lifecycle cost. To simultaneously optimize the four aforementioned objectives, the developed model is integrated with the serviceability model developed in Altarabsheh (2016) to calculate the network serviceability objective function values (this model is briefly described in the “Methodology” section in this paper). Additionally, the developed model was integrated with a semi-Markov process to predict the sewer pipe condition at different time steps. In the future, this will allow the transition probability to change with time, thereby allowing for network condition to better be accounted for in the optimization. This capability is important because it overcomes the stationary property of discrete Markov chain processes where the transition probability is constant and does not change with time. Semi-Markov processes overcome such a drawback by allowing the value of the transition probability to change up or down as the pipes deteriorate or are renewed, respectively. Two null hypotheses were tested in this paper, which the authors aim to show are false. The first hypothesis is that the proposed 04017069-1 J. Comput. Civ. Eng., 2018, 32(1): 04017069 J. Comput. Civ. Eng. Downloaded from ascelibrary.org by Tufts University on 10/28/17. Copyright ASCE. For personal use only; all rights reserved. LCA–MOGA model results in fewer or equal diverse solutions over the entire planning period than a state-of-art model (Halfawy et al. 2008). The second hypothesis is that the proposed LCA– MOGA model results in worst or equal average network condition, average network risk of failure, network serviceability, and lifecycle cost over the entire planning period than the Halfawy et al. (2008) model. The two hypotheses are tested by implementing both the proposed LCA–MOGA model and Halfawy et al. (2008) to the same network. The solution diversity obtained from the two algorithms is then compared using the following metrics: the hypervolume indicator, the epsilon indicator, the spacing indicator, and the generational distance indicator. Also, the performance of the two algorithms is compared under different age and funding scenarios to highlight the strengths and the weaknesses of the two algorithms. The first null hypothesis would be rejected if the performance metrics indicate that the LCA–MOGA algorithm results in more diverse solutions than the Halfawy et al. (2008) model. On the other hand, the second null hypothesis would be rejected if the developed model results in better values for the objective functions at the end of the analysis period than the Halfawy et al. (2008) model under the same analysis scenarios. It is worth noting that the Halfawy et al. (2008) model was chosen for this comparison because it considers three of the four objective functions considered in this study (all except the serviceability objective function). Also, these objective functions achieve the concept of the proactive asset management approach. To the best of the authors’ knowledge, no previous study simultaneously considered the three objective functions considered by Halfawy et al. (2008) and the serviceability objective function considered in this study. This is important because not considering all of these objective functions will make the implementation of the proactive asset management approach incomplete. Also, another difference between the developed model and the one used in validation is that the latter selects renewal plans at the current time step without considering their impact on the long-term behavior of the sewer network. Background This section discusses the prioritization models developed in the literature to select renewal plans for sewer networks. One of these studies developed a multiobjective optimization approach to identify the rehabilitation strategies that maximize the structural condition, minimizing the construction cost, and minimizing the critical asset risk of failure (Ward and Savic 2012). The model used grading data from closed circuit television (CCTV) inspections based on the Water Research Center (WRc 2004) method to characterize sewer conditions. A commercial software package called InfoNet developed by Innovyze (2010) was used as a data preprocessing tool to manage and code the CCTV data. The data was further screened to remove the new pipes. A genetic algorithm was tested using a case study on a catchment provided by the South West Water Company in the United Kingdom. The results were compared to rehabilitation costs calculated manually using engineering best practices. The study identifies more cost-effective rehabilitation strategies than the manual rehabilitation calculations with savings ranging from £104,200 to £117,500. Finally, the study argued that, although the model cannot guarantee a global optimality of the solutions, it clearly results in better values for the objective functions with lower cost than solutions achieved with manual calculations. In another study, Marzouk and Omar (2013) developed a multiobjective genetic algorithm to prioritize the maintenance process for the sewer network in one time step. The study used three main objective functions: lifecycle maintenance cost, sewer service life, © ASCE and the sewer network condition. The study procedure starts by predicting the future condition of the sewer pipes using a Markov chain approach where the transition matrix was built based on the adopted maintenance and rehabilitation policies. A Monte Carlo simulation was used to account for the uncertainty in the maintenance and rehabilitation cost. Also, the net present value and the cost/benefit ratio was calculated by considering both the interest and the inflation rate as a stochastic variable with a normal and a beta probability distribution, respectively. Finally, a multiobjective genetic algorithm model is used to maximize both the overall network condition index value and the intended network service life while minimizing the net present value lifecycle maintenance costs. The proposed algorithm was then tested and validated using the city of Indianapolis’s combined sewer system. Six alternatives were considered for rehabilitation of the city’s large combined sewers, including: do nothing, routine cleaning, shotcrete, curedin-place pipes, slip lining with fiberglass reinforced pipes, open cut excavation, and replacement with reinforced concrete pipes. The study results indicate the advantages and the limitations of using probabilistic lifecycle cost. The main limitation of using a probabilistic lifecycle cost approach is that it requires higher computational cost than a deterministic approach. On the other hand, using a probabilistic lifecycle cost approach enables utilities to account for the uncertainty in interest and in inflation rates, which may lead to a better estimation for the lifecycle cost required to renew the sewer network. Finally, Halfawy et al. (2008) developed one of the most advanced models in this area (as explained in the “Introduction” section in this paper). Their work proposed a new integrated approach to maximize the average network condition, minimize the average network risk of failure, and minimize lifecycle cost for a single time step of the analysis. The developed algorithm first evaluates both the consequence and the likelihood of failure. The consequence of failure is estimated as the weighted average of the criticality level of a number of factors, including sewer type, sewer function, pipe diameter, pipe depth, surrounding soil, site seismicity, site land use, road classification, traffic volume, and proximity to critical assets. The likelihood of failure index, on the other hand, is calculated as the ratio between the current age of the sewer pipes and its remaining service life. The risk index is then calculated by multiplying the likelihood by consequence of failure. Based on the risk and condition indices, sewer pipes are ranked according to their urgency of intervention, then a multiobjective optimization algorithm is used to generate a set of feasible and optimal/near-optimal renewal plans based on their performance in the aforementioned objective functions for one time step in the analysis period. This procedure could be repeated for any number of time steps needed while updating network conditions based on the outcomes of the previous step. However, this may lead to the obtained renewal not being globally optimal (global optimal solution is a single solution that has the smallest network condition index, the smallest risk of failure index, the minimum lifecycle cost, and the largest serviceability over the entire planning horizon). Halfawy et al. (2008) also argued that obtaining such globally optimal renewal plans is a challenging task due to the large number of possible combinations of choices that need to be considered. From the review presented in this section, it is obvious that present prioritization models do not consider the long-term impacts of the decisions being made in one time step on the performance of the asset over its life. One possible reason for the present models adopting this approach is the large number of decision combinations and the uncertainty associated with their outcomes. Another possible reason is the high uncertainty in the model parameters, including the available budget at each time step and the 04017069-2 J. Comput. Civ. Eng., 2018, 32(1): 04017069 J. Comput. Civ. Eng. Si ðtÞ ¼ 1 − Fi ðtÞ ¼ exp½−ðλi tÞβi deterioration process of the sewer network, where these factors depend on many socioeconomic factors, such as population growth, water consumption, wastewater load concentration, and water utility financial status, such as utility revenue, expenditure, debt, and maximum acceptable user fee hike rate. Methodology Downloaded from ascelibrary.org by Tufts University on 10/28/17. Copyright ASCE. For personal use only; all rights reserved. The methodology of this study includes three main steps: (1) the formulation of the multiobjective optimization model, (2) the selection for the optimization technique, and (3) the implementation of the developed model. fi ðtÞ ¼ Average Condition Index Pn Minimize Ci ðAgei Þ i¼1 P n l i¼1 i li Si ðvÞ ¼ exp½−ðλi vÞβi → ln½Si ðvÞ ¼ −ðλi vÞβi ln½Si ðuÞ u → ln ¼ β i ln ln½Si ðvÞ v ð2Þ lnðln½Si ðuÞÞ − lnðln½Si ðvÞÞ lnðuÞ − lnðvÞ ð5Þ 1 →λi ¼ ð− ln½Si ðuÞÞ1=β i u ð6Þ Once these two parameters are established for every i ¼ f1; 2; : : : ; n − 1g, Eq. (7) is used to predict the transition probability matrix by substituting Eqs. (2)–(4) into it. However, since the cumulative PDF, CDF, and SF cannot be calculated analytically, a Monte Carlo simulation is used to estimate these functions numerically. After estimating the parameters for Weibull distributions that represent the sojourn time for each pipe in each condition state, the transition matrix for each pipe is then derived using these probability distributions. The derivation of the transition probabilities is illustrated in Eq. (7), which is used to compute all the transition probabilities pi;iþ1 ðtÞ and to populate the transition probability matrix for the semi-Markov process (Kleiner 2001) Pr½Xðt þ 1Þ ¼ i þ 1jXðtÞ ¼ i ¼ pi;iþ1 ðtÞ ¼ Fi ðtÞ ¼ Pr½T i ≤ t ¼ 1 − exp½−ðλi tÞβ i © ASCE → βi ¼ ð1Þ where n = number of pipes; Ci ðAgei Þ = condition of pipe (i) as a function of the Age determined using a semi-Markov model explained below; and li = length of pipe (i). The average condition index is evaluated based on the scale suggested by the Water Research Centre (2004). This scale is used to predict the condition of the sewer pipes at each time step using a semi-Markov deterioration model that was developed for this purpose. This model is created to model the transition probabilities of sewer pipes from one condition to the next as a function of time and to, therefore, overcome the stationary property of Markov chains. The semi-Markov deterioration process is implemented by first modeling the sojourn time (T i ) of the sewer pipe deterioration at every state (i) using a two-parameter Weibull distribution, which has been shown to be a good representative for sewer pipe deterioration within 95% confidence interval (Martin et al. 2007; Fujiu and Miyauchi 2007; Kumar et al. 2010; Duchesne et al. 2013). The probability density function (PDF) f i ðtÞ, cumulative distribution function (CDF) Fi ðtÞ, and survival function (SF) Si ðtÞ can be computed as follows: ð4Þ Both the shape (β i ) and the scale (λi ) parameters of Weibull distribution are estimated for every ith state by making two statements regarding the probability of the specific pipe to be in that condition for more than a specific number of years. The two statements are assumed to be made with u years and v years to produce two quantiles. Then the two Weibull distribution parameters are derived in the following manner: ( ) Si ðuÞ ¼ exp½−ðλi uÞβi → ln½Si ðuÞ ¼ −ðλi uÞβi Model Formulation Municipalities in most developed countries are facing three main challenges in managing their sewer networks (Younis and Knight 2014). These challenges include: (1) the rapidly deteriorating condition of the network, (2) the high cost of network renewal, and (3) the limited funds available for this renewal process. To overcome these limitations, water utilities have started to follow a proactive approach in managing their sewer network assets (Younis and Knight 2014; Halfawy et al. 2008). This approach makes use of the limited available funds to upgrade the network condition gradually with time and to reduce the risk of failures in the network. Therefore, this study aims to support this proactive asset management approach by simultaneously optimizing the following four objective functions, previously mentioned above: (1) maximize average network condition, (2) minimize risk of failure, (3) maximize network serviceability, and (4) minimize network lifecycle costs. These objective functions are explained in the following subsections in detail. ∂Fi ðtÞ ¼ λi β i ðλi tÞβ i −1 exp½−ðλi tÞβi ∂t ð3Þ f 1→i ðtÞ S1→i ðtÞ − S1→ði−1Þ ðtÞ i ¼ f1; 2; : : : ; n − 1g ð7Þ where number 1 = first condition state; and i ¼ ith condition state of the sewer pipe. Also, f 1→i ðtÞ and S1→i ðtÞ = probability density and survival functions for the sum of sojourn times in states (1; : : : ; i), respectively. It is clear from the above equations that the transition probabilities are time dependent, and the process is nonstationary. However, to apply these equations, this study assumes that the infrastructure asset can deteriorate only one state at a time up to failure (Baik 2003). It should be noted that this assumption is not always true, and sewer pipes can deteriorate more than one state at a time in cases of increased loading or natural disasters. However, both Madanat et al. (1995) and Baik et al. (2006) stated that in most cases, sewer pipes deteriorate only one state at a time and only in a few cases does it deteriorate more than one state at a time, which validates the assumption made above. Once the transition probability matrix is established, the state vector (QðtþkÞ ) at time (t þ k) can be expressed as QðtþkÞ ¼ Qt Pt;tþ1 Ptþ1;tþ2 : : : : : : Ptþk−1;tþk ð8Þ where Qt = vector that contains the probabilities that a pipe will be in certain state at time t. 04017069-3 J. Comput. Civ. Eng., 2018, 32(1): 04017069 J. Comput. Civ. Eng. Table 1. Determining Weibull Distribution Parameters for the Illustrated Example Condition state Downloaded from ascelibrary.org by Tufts University on 10/28/17. Copyright ASCE. For personal use only; all rights reserved. 1 2 3 4 5 U (years) Xi,u V (years) Xi,v βi λi 26 27 34 21 5 0.500 0.500 0.500 0.500 0.500 40 40 50 32 10 0.100 0.100 0.100 0.100 0.100 2.787 3.054 3.113 2.850 1.732 0.034 0.033 0.026 0.042 0.162 To illustrate the use of the semi-Markov model derived above, the derivation is applied to a 20-year-old sewer pipe. First, the two parameters of the Weibull distribution need to be determined. The Baur and Hertz (2002) model was used for this purpose; the sojourn time values and their probabilities obtained from this model are listed in Table 1. Based on these values, Eqs. (5) and (6) were used to determine the Weibull distribution parameters as listed in Table 1. Next, parameters λi , β i and Eqs. (2)–(4) are used to produce PDF, fi(t), CDF, Fi(t), and SF, Si(t), for the waiting times Ti in every state i. Fig. 1 depicts the PDFs for this example. The next step is to find the sums of waiting times in the various states, T i 0 k , and the respective PDFs, CDFs, and SFs. These values are obtained numerically using Monte Carlo simulations to generate (n − 1) Weibull distributed random numbers with parameters λi and β i . Fig. 2 illustrates the resulting PDFs, CDFs, and SFs. It can be seen that, in this example, the mean time to failure is about 112 years. Recall that State 5 was defined as failure, thus the PDF of states 1 þ 2 þ 3 þ 4 defines the PDF of asset age at failure, given that it is as good as new at age zero. Further, it can be seen that the vast majority of buried assets of this type under similar sets of conditions are expected to last between 60 and 130 years. The survival function of the process demonstrates that the pipe was good as new at age zero. In this example, the asset at age 30 is about 40% likely to still be in Condition State 1, about 55% likely to be in State 2, about 5% likely to be in Condition State 3, and about 0% likely to be in State 4. The probability of failure at age 30 is virtually zero. The next step is to generate the age-dependent transition probabilities pi;iþ1 ðtÞ, using Eq. (7). The transition probability matrix for the 1-year-old sewer pipe analyzed in this example can be described as follows: 2 3 0.9999 0.0001 0 0 0 6 7 6 0 0.9994 0.0004 0 0 7 6 7 6 7 P1;2 ¼ 6 0 0 1 0 07 6 7 6 0 0 0 1 07 4 5 0 0 0 0 1 Because the pipe is 1 year old, the pipe is in State 1 with a probability of 100%. Therefore, the probability mass function can be written as Qt ¼ ½ 1 0 0 0 0 Finally, the probability mass function can be obtained using Eq. (8) as follows: 2 3 0.9999 0.0001 0 0 0 6 7 6 0 0.9994 0.0004 0 0 7 6 7 6 7 Qtþ1 ¼ Q1 P1;2 ¼ ½ 1 0 0 0 6 0 0 1 0 07 6 7 6 0 0 0 1 07 4 5 0 0 0 0 1 → Q2 ¼ ½ 0.99988 0.00012 0 0 0 Average Risk of Failure Index Pn Minimize i¼1 Ci ðAgei Þ COFi li Pn i¼1 li ð9Þ where COFi = consequence of failure for pipe (i). The COF is an index that helps to understand the cost and the consequences associated with sewer failure to the residents, the area, and the water company. As stated by Najafi and Gokhale (2005), there is no universal method or model to calculate the consequences and the social costs of sewer failure. In this study, a state-of-the-art model developed by Salman (2010) is used to determine the consequence of failure for each pipe by combining expert opinion with a geographical information system. This method is explained in detail in the “Results and Discussion” section in this paper. A multiplication of both the condition index and the consequence of failure was used to determine the risk of failure index for each sewer pipe (Halfawy et al. 2008; Salman 2010). This method provides a clear method to rank and prioritize sewer pipes and creates a quick method for the assessment of their overall risk. Lifecycle Cost Lifecycle cost analysis (LCA) of an asset can be defined as the process of evaluating the total costs (such as owning and operating costs) and benefits anticipated over the lifecycle of that asset. LCA considers all the significant decisions made in the assets life cycle, and thus can contribute to the reduction of asset construction, operation, and maintenance costs. Therefore, LCA is a more effective Fig. 1. PDF obtained for the illustrated example © ASCE 04017069-4 J. Comput. Civ. Eng., 2018, 32(1): 04017069 J. Comput. Civ. Eng. Downloaded from ascelibrary.org by Tufts University on 10/28/17. Copyright ASCE. For personal use only; all rights reserved. Fig. 2. Semi Markov results for the illustrated example method than investment analysis methods that are only based on initial construction costs when managing the sewer system (Marzouk and Omar 2013). LCA also considers the fact that costs are incurred at different points in time during the asset’s lifecycle and, therefore, have different time values. This makes these costs not directly addable. Therefore, FHWA (2014) recommends that future costs and benefits of a project be expressed in terms of constant dollars that are discounted to the present without any consideration to the inflation rate (since the benefits of the public sectors shouldn’t depend on the price changes but rather on the real gains). LCA also enables decision makers to evaluate different decision alternatives. For example, LCA can be used to evaluate the benefit of paying for maintenance now and postponing rehabilitation for later. This feature is used in the proposed approach to find the most cost-effective ways to manage a sewer network by evaluating different combinations of subsequent maintenance and rehabilitation (M&R) treatments. Based on the above discussion, the lifecycle cost is the third objective function of this proposed model Total lifecycle cost∶ Minimize TC n X ¼ RCi ðxÞ PðxÞi li þ LCpvm ð10Þ i¼1 where RCi ðxÞ = replacement cost for pipe I depends on the decision variable x determined using semi Markov process; Pi ðxÞ = transition probability of pipe (i) depends on the decision variable x determined using semiMarkov process LCpvm ¼ P X N X OCim li m¼1 i¼1 © ASCE ð1 þ iÞm ð11Þ where OCim = operation and maintenance cost for pipe (i) at year (m); i = discount rate; and LCpvm = lifecycle M&R cost used in this study. Network Serviceability Network serviceability has many different definitions (Savic et al. 2006; Arthur et al. 2009; Ashley et al. 2004). One of these definitions focuses on the reliability of the service provided by an infrastructure facility. In the case of a sewer network, this could be viewed as the network’s ability to provide uninterrupted service to customers. One of the main disruptors of sewer service is operational failures caused by blockages (Arthur et al. 2009). Blockages can cause loss of service and flooding, which can result in environmental pollution, health risks, property damage, and traffic disruption (Arthur et al. 2009). Blockages are considered to be the number one cause of losses in sewer serviceability across the world (Ashley et al. 2004). For example, Arthur et al. (2009) determined that 76% of sewerage-derived flooding incidents (>23,400 per year) in England and Wales were due to blockages. Also, in Australia, 70,000 properties across the country are affected by flooding almost every year (Marlow et al. 2011). This makes the prediction of blockages a key research challenge across the world. Therefore, in the present model, the objective function representing serviceability is evaluated by applying the Markov chain modulated Poisson process developed in Altarabsheh (2016) (the idea of this method is to determine the number of blockages in each sewer pipe as a stochastic process that depends on the pipe physical attributes and the pipe condition) as shown in Eq. (12) 04017069-5 J. Comput. Civ. Eng., 2018, 32(1): 04017069 J. Comput. Civ. Eng. Network serviceability∶ Minimize n X NOBi ðxÞ ð12Þ i¼1 where NOBi ðxÞ = number of blockages for pipe (i) as a function of the decision variables. discussed in detail in the “Case Study” section in this paper, but only a subset of these pipe groups could be selected due the availability of funds. Based on this, the number of possible solutions in the network can be specified as 43 ð14Þ x Budget Constraints Downloaded from ascelibrary.org by Tufts University on 10/28/17. Copyright ASCE. For personal use only; all rights reserved. The only constraint considered in the formulation of the proposed model is on the funds available for network maintenance and rehabilitation expressed by X RCi ðxÞ PðxÞi li ≤ Budget ð13Þ Optimization Technique Selection Decision Variables and Solution Space The proposed approach uses possible sewer pipe renewal strategies for each pipe in the network as its main decision variables. These possible renewal strategies depend on the current condition of the pipe, as shown in Table 2. For example, pipes in Condition 1, 2, or 3 could be renewed using routine cleaning methods. Pipes in Condition 4, on the other hand, could be renewed using shotcrete, cured-in-place pipes, or slip lining with fiberglass reinforced pipe. Finally, pipes in Condition 5 are mainly renewed using the open cut excavation and replacement method. The selection of the pipe renewal strategy to be applied also depends on the cost of the renewal strategies. Table 3 shows the estimated unit cost in U.S. dollars for each renewal plan listed in Table 2 (Marzouk and Omar 2013). Based on the decision variables discussed above, the size of the solution space is calculated using the concept of combinations where there are 43 groups of pipes that need to be renewed as Table 2. Decisions and Relevant States and Benefits (Data from Wirahadikusumah et al. 1999) Action Relevant states Benefit (year) Do nothing Routing cleaning Shotcrete Cured in place pipe Reinforced fiberglass slip-lining Dig and replace with concrete pipe 1,2,3 1,2,3 4 4 4 5 — 10 20 50 100 50 Decision (x) 1 2 3 4 5 6 Table 3. Estimated Unit Costs in ($=m) (Data from Wirahadikusumah et al. 1999) State 1 1 2 2 3 3 4 4 4 5 © ASCE Decision (x) Rehab cost Disruption cost Total cost 1 2 1 2 1 2 3 4 5 6 0 16 0 16 0 16 656 1,558 2,231 1,148 0 0 0 0 0 0 0 0 0 656 0 16 492 443 984 902 820 1,558 2,231 1,804 where x = possible number of the selected pipes to be renewed by the algorithm. From Eq. (14), for a range of values of x between 18 and 22 pipes (specified from the results in the next section where it was found that the algorithm selects the number of pipes in this range to be renewed), the number of possible combinations ranging from (6 × 1011 to 1 × 1012). The actual number of solutions in the solution space is even larger because there are many possible ways to renew each of the selected pipes. In addition to the large solution space that needs to be searched, the objective functions and constraints formulated above are: (1) discontinuous, (2) do not have a derivative, and (3) have discrete decision variables [as shown in Eqs. (1)–(5)]. This makes traditional mathematical optimization approaches not suitable for this problem, hence, alternative approaches need to be explored. Nominated Solutions To address the aforementioned characteristics of the objective functions, a multiobjective genetic algorithm (i.e., NSGA-III method by Deb and Jain 2014) is used to implement the proposed model. Several authors including Affenzeller et al. (2009), Al-Battaineh et al. (2005), and Berardi et al. (2009) demonstrated that evolutionary algorithms and, in particular, genetic algorithms (GA) are more suitable for handling of engineering optimization problems with similar objective functions than mathematical optimization methods. These authors showed that GAs can be used to solve problems similar to the one targeted by this research because: (1) they work with a coding of the parameter set not the parameters themselves, thus the algorithm doesn’t require the objective function to be continuous or have derivatives; and (2) they work with a population of strings simultaneously, climbing many peaks in parallel using probabilistic transition rules. Therefore, GAs are able to obtain more diverse solutions in the search space and have a lower probability of finding local minima than traditional optimization techniques, which move a single point in the decision space to the next using some deterministic transition rules. Observing Eqs. (1)–(4) shows that as the value of the total lifecycle cost objective function increases, the value of the three other objective functions decreases. Therefore, the total lifecycle cost objective function is competing with the other objective functions, which makes the problem multiobjective in nature. Hence, similar to other multiobjective optimization problems, this problem does not have a unique optimal solution (since the network condition, consequence of failure and serviceability will increase at a different rate as the lifecycle cost increases). Instead, it has a set of solutions Table 4. Ranges of Discrete Genes’ Values State D1 D2 D3 D4 D5 04017069-6 J. Comput. Civ. Eng., 2018, 32(1): 04017069 Range 0–1 0–1 0–1 0–3 0–1 J. Comput. Civ. Eng. Downloaded from ascelibrary.org by Tufts University on 10/28/17. Copyright ASCE. For personal use only; all rights reserved. Fig. 3. Lifecycle analysis multiobjective genetic algorithm © ASCE 04017069-7 J. Comput. Civ. Eng., 2018, 32(1): 04017069 J. Comput. Civ. Eng. Table 5. First Year Pareto Front for the Illustrated Example Average condition index 3.961168549 3.564807919 3.371404 3.21937 Average ROF index Serviceability LCA ($) 3.969381 3.054441 3.685577 3.763304 88.02596775 79.21795376 74.9201 71.54155 19,415.73 217,596 369,297.8 370,315.1 Downloaded from ascelibrary.org by Tufts University on 10/28/17. Copyright ASCE. For personal use only; all rights reserved. that form a trade-off between the objective functions. These solutions are called nondominated or Pareto-optimal and need to be considered by the decision maker to select the one that best fits their needs. The criteria for selecting nondominated solutions may vary from one decision maker to another depending on their interests and purpose. For example, one decision maker may prefer solutions with low cost regardless of the value of the other objective functions due to budget deficits. Others may, for example, be more interested in minimizing the risk of failure than the other objectives. In this study, two criteria are considered in selecting a solution from the Pareto set (also called the Pareto front) containing nondominate and feasible solutions that satisfy the problem constraints. The first criterion is to select solutions that give the best trade-off for the objective functions. The reference point approach suggested by Deb et al. (2006), in Eq. (15), is used for this purpose vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u M uX fi ðxÞ − z̄i 2 dij ¼ t ωi max fi − fmin i i¼1 ð15Þ and f min = population maximum and minimum funcwhere f max i i tion values of ith objective. The second criterion is to select the solution that focuses on the most critical pipes in the network. These most critical pipes are considered to be the ones in near failure conditions (in Condition 4 or 5) in this study. Within these critical pipes, those pipes with higher risk of failure and higher expected number of blockages are considered as the most critical among all the other pipes in the network. To satisfy these two solution selection criteria, all the nondominated solutions need to be examined. Based on the previous discussion of the decision variables (and the fact that represent possible renewal techniques for pipes), the range values of genetic algorithm genes are listed in Table 4. A value of zero in any of these genes represents the fact that the pipes were excluded from the renewal planning process. As can been seen in Table 4, the present pipe condition dictates the range values of the genes that represent them. These values are based on the renewal plans assigned to each pipe based on its condition according to Table 2. For example, pipes in condition state 1, 2, and 3, can either be cleaned (represented in Table 4 by number 1 in columns 1, 2, and 3 for pipes in condition 1, 2, and 3, respectively) or can be left without any renewal (represented in Table 4 by number 0 in columns 1, 2, and 3 for pipes in condition 1, 2, and 3, Table 6. Second Year Pareto Front from the First Solution in Table 5 for the Illustrated Example Average condition index Fig. 4. Flowchart of LCA-MOGA planning model implementation © ASCE 3.159080653 3.443895 3.316481843 3.91498568 04017069-8 J. Comput. Civ. Eng., 2018, 32(1): 04017069 Average ROF index Serviceability LCA ($) 3.615665 3.406894 3.93611 3.569332 70.20179229 76.531 73.69959652 86.99968178 401,043.9 258,636.8 322,343.4 23,091.43 J. Comput. Civ. Eng. respectively). Also, pipes in condition 4 can be renewed using shotcrete (represented in Table 4 by number 1 in column 4), cured in place (represented in Table 4 by number 2 in column 4) and reinforced fiberglass slip-lining (represented in Table 4 by number 3 in column 4) or can be left without any renewal (represented in Table 4 by number 0 in column 4). Finally, pipes in condition 5 can be renewed using dig and replace (represented in Table 4 by number 1 in column 5) or they can be left without renewal (represented in Table 4 by number 0 in column 5). Downloaded from ascelibrary.org by Tufts University on 10/28/17. Copyright ASCE. For personal use only; all rights reserved. Model Implementation In order to simultaneously optimize the aforementioned four objectives, the present model is implemented as multiobjective genetic algorithm (MOGA). The algorithm and the structure of the model is shown in Figs. 3 and 4, respectively. From Fig. 4 it can be seen that, the algorithm starts by an input from the user on the network data, optimization data, and cost data. Network data contains information about the number of pipes, their physical attributes, their age and their consequence of failure. Optimization data contains information about the objective functions and constraints used in the optimization. Also, it contains information about the optimization method used in the analysis along with the decision variables and the number of time the algorithm should be repeated. Finally, the cost data contains information about the renewal cost for each pipe. Fig. 4 and the first part of Fig. 3 show that after getting inputs from the user, the algorithm apply semi Markov process to determine the condition of each pipe. The algorithm starts by dividing the analysis period into smaller planning periods (increments of one or more years). At the beginning of each planning period, Table 7. Second Year Pareto Front from the Third Solution in Table 5 for the Illustrated Example Average condition index 3.343112 3.098567 2.988044 3.27406 Average ROF index Serviceability LCA ($) 2.162486 2.774476 3.307794 2.600404 74.29137141 68.85705376 66.40097 72.75689518 14,146.34 136,418.5 191,680.3 48,672.06 Table 8. Second Year Pareto Front from the Fourth Solution in Table 5 for the Illustrated Example Average condition index 2.779105 2.862264 2.767417 2.472507 Average ROF index Serviceability LCA ($) 3.751162 3.387967 3.181224 3.59985 61.75789257 63.60587 61.49815063 54.94459337 220132.3 178552.9 225976.5 373431.5 the objective functions, the constraints and the possible values of decision variables are calculated based on the semi-Markov deterioration model explained in the “Methodology” section and the serviceability model developed by Altarabsheh (2016). Then NSGA III is used to generate a set of feasible renewal plans. From this generated set, a feasible solution is selected randomly and used to update the decision variables for the following time step. This selection may not be the most desirable among the non-dominated solutions at the current time step but may lead to a better total lifecycle cost after considering the consequent decisions in following time steps. The selection of this solution randomly will allow a range of possible solutions to be evaluated based on their impact on the optimization objectives over the entire analysis period (not just in one-time step). The randomly selected solution is used to update a number of parameters before running the optimization algorithm for the following time step. These parameters include the age and the expected number of blockages for each pipe. The algorithm updates the age of each pipe based on the selected renewal strategy for the pipe according to Table 2. For example, if pipe in condition 1, 2, or 3 and it was selected by the algorithm to be renewed then routine cleaning method will be applied and the age of the pipe will be reduced by 10 years, and the same concept applied for pipes in condition 4 and 5. After that, the age of all pipes increase by 1 year and semi Markov process is run to update the condition of each pipe. Based on each pipe condition and physical characteristics, the serviceability model developed by Altarabsheh (2016) is applied to update the expected number of each pipe. After that, NSGA III algorithm is run again to obtain a set of nondominated renewal plans for the sewer network at the second time step, and the previous steps are repeated till the end of the analysis period. The algorithm is designed to select solutions from different regions of the Pareto-optimal set than those selected in previous time steps to enhance the diversity of the solutions. At the final time step, a single set of solutions is obtained from the analysis. Each solution set represents a possible renewal plan that could be applied to the sewer network at every time step of the analysis period. The above procedure is repeated 1,000 times (to allow for the selection of different random solutions at each time step), and each time the obtained set is stored in a matrix. The number of repetitions was determined by trial and error where it was found that the obtained solutions cease to improve after 1,000 repetitions. The purpose of repeating the process is explained in the second part of Fig. 3. Each branch in the second part of Fig. 3 represent a set of possible renewal strategies to be applied at each time step during the planning period. The purpose of the branching in the algorithm is to enable decision makers to select a renewal plan to be applied for the sewer network based on its long term impact on the value of the objective functions. This is done by selecting a solution randomly at each time step and update the network condition and the expected number of blockages based on this solution as explained early in this subsection, and by repeating the process 1,000 time, the long term impact of large number of Table 9. Differences between Halfawy et al. (2008) Algorithm and LCA-MOGA Algorithm Algorithm Objective functions Select renewal plan Halfawy et al. (2008) algorithm Three objective functions including (1) condition index, (2) risk of failure index, and (3) lifecycle cost LCA-MOGA algorithm In addition to the three objective functions considered by Halfawy et al. (2008) model, this model proposed fourth objective function which is the serviceability objective function Select renewal plan for the sewer network at the current time step based on its impact on the network behavior at the current time step Select renewal plan for the sewer network at the current time step based on its impact on the network behavior at the end of the planning period © ASCE 04017069-9 J. Comput. Civ. Eng., 2018, 32(1): 04017069 J. Comput. Civ. Eng. Downloaded from ascelibrary.org by Tufts University on 10/28/17. Copyright ASCE. For personal use only; all rights reserved. renewal plans would be examined and the decision makers would select a renewal plan based on its long term impact according to their objectives as explained in the “Methodology” section. The algorithm is designed to select a solution set only once to ensure efficient computations. At the end of the analysis, all the dominated solutions are eliminated based on the total value of the life cycle cost over the entire analysis period (not just at single time step), the value of the condition index, the risk of failure index, and the total number of blockages in the network over the entire analysis period. Finally, a single solution is selected from the obtained non-dominated solutions using the two criteria discussed in the “Methodology” section. To illustrate the concept of the proposed model, assume a hypothetical sewer network with average condition index of 4, average risk of failure index of 4, and serviceability of 100. The proposed algorithm is applied to rehabilitate the network for 2 consecutive years. First the algorithm run to obtain the Pareto front for the first year, the results are shown in Table 5. A random solution is selected from the obtained Pareto front (in this case the first solution is selected by the algorithm), and based on the selected solution the objective function values are updated. Next the algorithm run again to rehabilitate the network for the second year, the resulting Pareto front is shown in Table 6. Eq. (15) is used to select the solution with best trade-off between the objective functions, in this case the second solution in Table 6 was selected. The selected solution for the 2 years planning period is stored, and the algorithm run again to select another solution from Table 5. In this case, the third solution in Table 5 was selected, and based on this solution the objective function values are updated. Next, the algorithm is run again to rehabilitate the network for the second year, the resulting Pareto front is shown in Table 7. Eq. (15) is used next to select the solution with the best trade-off between the objective functions in Table 7, in this case the third solution in Table 7 was selected and stored. Finally, the previous steps were repeated for the last time, and the algorithm select the last solution in Table 5. Based on the selected solution the objective functions are updated, and the algorithm select possible solutions to rehabilitate the network in the second year. The resulting Pareto front is shown in Table 8, and the second solution in the table was selected by Eq. (15) because it has the best trade-off between the objective functions. The three solutions selected from Tables 6–8 are compared based on the value of the condition index, risk of failure index, and serviceability at the end of the second year, and the total life cycle cost from the first and the second year. In this example, the three solutions are non-dominated, therefore, the decision makers can select a solution based on their interest. It is interesting to note that even with this small example, it is important to evaluate the impact of the selected solution in the first year on the value of the objective functions at the end of the planning period (2 years in this example). For instance, the third and the fourth solutions in Table 5 have close value of the objective function; however, at the second year selecting the fourth solution results in less life cycle cost and better average condition index. Finally, based on the discussion in this section and to better understand the proposed model, Table 9 summarize the main differences between the LCA-MOGA algorithm proposed in this paper and Halfawy et al. (2008) algorithm. Case Study The above described methodology was applied to a sewer network in Sahab city in the Hashemite Kingdom of Jordan. This city is © ASCE Fig. 5. GIS map for the sewer network in Sahab City (map data © Google 2016) located at the south east of the capital Amman and is considered to be the largest industrial city in Jordan. The reason for selecting this city is that an almost complete data set of 2,936 pipes was obtained, with only six pipes missing. All the pipes in the obtained network were recorded as “Existing and Active.” The wastewater system in Sahab city is managed by the Miyahona Company, and consists of 117.622 km of sanitary sewer lines. About (75%) of the pipes in the network have diameter of 200 mm and the diameter for the rest of the network pipes ranging between 150 and 1,000 mm diameter. The GIS map for this network is shown in Fig. 5, where the dots in the figure represent the sewer manhole. The sewer network is composed of about 2,936 pipes, many of which have similar attributes. Therefore, in this study sewer pipes were arranged into groups, each group contains pipes that have similar attributes including their length, diameter, slope, age, consequence of failure and material. Based on this criterion pipes, were classified into 43 groups as shown in Table 10. The assumption is that pipes in the same group will be rehabilitated using the same intervention option and at the same time. The consequence of failure for each pipe was calculated based on the procedure discussed by Salman (2010) using a GIS map. Six factors were used to calculate the consequence of failure, each factor was assigned a performance value within a range of 0–100 and a predetermined weight [both the performance values and the predetermined weight for each factor was adopted from Salman (2010)]. These factors are shown in Table 11. The final consequence of failure value was obtained by summing the multiplication of each factor performance value with its weight 04017069-10 J. Comput. Civ. Eng., 2018, 32(1): 04017069 J. Comput. Civ. Eng. Table 10. Pipes Classification Groups Downloaded from ascelibrary.org by Tufts University on 10/28/17. Copyright ASCE. For personal use only; all rights reserved. Group number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 Consequence of failure Diameter (mm2) Pipe material Age (years) Pipe slope Pipe length Number of pipes 5 5 5 5 5 4.5 3.75 3.75 3.75 3.75 3.25 3.25 3.25 4 4 4 4 4 2.75 2.75 2.75 2.75 2.75 2.25 2.25 2.25 2.25 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1,000 1,000 1,000 1,000 1,000 700 700 700 700 700 500 500 500 500 500 500 500 500 400 400 400 400 400 300 300 300 300 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 Concrete Concrete Concrete Concrete Concrete Concrete Concrete Concrete Concrete Concrete Concrete Concrete Concrete Concrete Concrete Concrete Concrete Concrete Concrete Concrete Concrete Concrete Concrete Concrete Concrete Concrete Concrete Ductile Ductile Ductile Ductile Ductile Ductile Ductile Concrete Concrete Concrete Concrete Concrete Concrete Concrete Concrete Concrete 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 6 7 7 7 6 7 7 7 7 7 6 6 7 7 7 7 7 6 6 6 6 0.030989 0.027409 0.078519 0.060338 0.042622 0.023777 0.029069 0.020816 0.044717 0.037419 0.030004 0.020988 0.053236 0.029423 0.035063 0.068258 0.037285 0.025854 0.04139 0.037011 0.08249 0.034226 0.034587 0.027302 0.070611 0.028557 0.037312 0.033556 0.086447 0.032972 0.013457 0.01018 0.049084 0.010703 0.044927 0.039526 0.025105 0.016089 0.011031 0.046623 0.02973 0.014171 0.010511 50.64346 54.5379 63.67037 65.72921 42.01262 23.23689 50.45008 46.48868 41.73077 48.26093 77.04085 75.84872 65.03962 51.97215 36.54481 39.85112 48.15462 50.7482 54.47313 42.80422 64.80058 47.14235 48.43013 48.17096 46.20398 51.2213 46.56812 34.01155 26.02838 19.23697 20.72254 24.58748 40.51824 26.87303 44.58581 45.7632 46.19918 41.49599 35.91789 42.22596 44.42968 41.17089 36.40783 22 17 17 7 9 4 5 11 13 9 26 16 11 18 16 10 10 9 28 15 12 11 12 30 24 16 18 45 67 14 32 14 45 9 659 225 368 200 68 159 130 95 35 Consequence of failure ¼ 6 X PV i × W i ð16Þ i¼1 where PV i = performance value with respect to factor i; and W i = weight of factor I; finally, the sum limit of 6 indicates that number of factors considered in this study to calculate the consequence of failure as listed in Table 11. Different values of pipe ages and deterioration rates were considered as shown in Tables 12 and 13, respectively. These values where used to evaluate the impact of the uncertainty in the different optimization variables on the proposed optimization model results and to understand the strength and the weaknesses of the proposed optimization models under different possible scenarios of the wastewater system. Table 12 shows the eight cases considered in this study regarding pipe ages. Case one assigns different deterioration levels to pipe groups that share the same risk of failure index. For example, the first five groups have a risk of failure index of 5, and therefore, were assigned different ages (and thus different deterioration levels). © ASCE The second case considers all the pipes with a risk of failure index above 3 to be over 90 years of age, and all the pipes with a risk of failure below 3 to have less than 50 years of age. The third case is the exact opposite of the second case where all the pipes with a risk of failure index above 3 would have ages below 50 years, and all the pipes with risk of failure above 3 would have ages over 90 years. The fourth, fifth and sixth cases considers only 10% of the 43 pipe groups to have an age above 90 years and the rest of the pipes to have ages below 50 years. These five pipe groups (10% of the 43 groups) were selected to have a risk of failure index of 5 in case four, a risk of failure index of 3.75 in case five, and a risk of failure index of 1.75 in case six. The seventh case considers all the pipes to be over 50 years old. Finally, the eighth case considers all the pipes to be over 80 years old. The planning period was chosen to be 20 years based on recommendations by Galán et al. (2009), who explained that beyond 20 years an update of the infrastructure information of the region would be required. This is particularly true for the present study because updated data would be needed on the expansion of the network, and the possible changes in the consequences of pipe 04017069-11 J. Comput. Civ. Eng., 2018, 32(1): 04017069 J. Comput. Civ. Eng. Table 11. Performance Values and Predetermined Weights for the COF Impact Factors Performance factor Type of roadway above the sewer pipe Distance of the sewer pipe from the nearest building or bridges Downloaded from ascelibrary.org by Tufts University on 10/28/17. Copyright ASCE. For personal use only; all rights reserved. Depth of the sewer pipe Number of building lateral connections Size of the pipe Building type Type of the impact factor Performance value Factor weight Arterial roads Collector roads Local streets Alley Not under roadway Located under a building Less than or equal to 3 m Greater than 3 m but less than or equal to 6 m Greater than 6 m but less than or equal to 9 m Greater than 9 m Less than 10 m Greater than 10 m More than 12 building lateral connections Greater than or equal to eight but less than or equal to 12 Greater than or equal to five but less than or equal to seven With three or four building lateral connections With one or two building lateral connections With no building lateral connections Diameters larger than 900 mm Diameters larger than 600 mm but smaller than or equal to 900 mm Diameters larger than 300 mm but smaller than or equal to 600 mm Smaller than or equal to 300 mm Hospitals and schools Industrial buildings General business and governmental places Apartments and condos Miscellaneous buildings Residential and multifamily houses 80 60 30 10 0 100 75 50 10 0 Depth × 3 100 100 50 25 13 6.5 0 100 75 50 25 100 90 70 50 10 0 10 failures. These changes in consequences of pipe failures are particularly important as they dependent on land use. The 20-year analysis period was divided into 5-year time steps according to DeMonsabert et al. (1999), Burgi et al. (2008), and the 2012 wastewater master plan developed for city of Palm Bay, Florida (2012), that showed that plans for wastewater network renewal are typically made every 5 years. Finally, two cases for utility income at each time step were considered. These cases are shown in Table 14. Case 1 assumes a uniform utility income at each time step, while case 2 assumes a shortfall in funding at the third time step. The purpose of these cases is to evaluate and compare the behavior of the proposed model under funding deficits. Results and Discussion This section discusses the results of implementing the proposed model to the above described case study scenarios. First, the results obtained from implementing the developed semi-Markov deterioration model are presented. This is followed by a discussion of the results obtained from the proposed model in the 8 different cases and a comparison of the results to those of a model developed by Halfawy et al. (2008). Semi-Markov Model Results The first step in applying a semi-Markov model is to determine the coefficients (i.e., the scale and the shape coefficients) of the Weibull distribution. The values of the shape parameter are determined by applying Eq. (9), and are shown in Table 15. Next, these two parameters are used to determine the PDFs, CDFs and SFs for the sums of the waiting times in the various states T i→k using © ASCE 8 7 7 6 1 Monte Carlo simulations. Fig. 6 shows a sample of the results for pipe group 10. Fig. 5(c) illustrates the probability mass function of the process and how it changes over the life of the asset, given that it was new at age zero. In this example, the asset at age 26 was about 40% likely to still be in Condition 1, about 51% likely to be in Condition 2, about 9% to be in Condition 3, and about 0% likely to be in Conditions 4 or 5. This agrees with the values of sojourn times in Table 12 where the pipe is expected to transit from Condition 1 to 2 after 26 years. Optimization Model Results This section illustrates the result of the proposed model and compares them with those of a model proposed by Halfawy et al. (2008) for the analysis scenarios described in the “Results and Discussion” section. In order to evaluate the results produced in this section, criteria need to be set for the performance of the Pareto-optimal solutions selected by the algorithm. Therefore, this study uses the following three important criteria to evaluate this performance (Azevedo and Araujo 2011; Coello et al. 2007): (1) the coverage of the non-dominated solutions of the problem’s Pareto front, (2) the closeness of the non-dominated solutions to the problem’s Pareto front, and (3) the spread and spacing among of non-dominated solutions among each other. These criteria are evaluated using the following performance metrics: (1) the hypervolume indicator, (2) the generational distance indicator, (3) the spacing indication, and (4) the epsilon indicator. These metrics are explained in detail by Coello et al. (2007). One of the important determinants of the performance of MOGAs is the algorithm parameters. In this study, the impact of these parameters of solutions is evaluated by running the algorithm under a wide range parameters values, as shown in Table 16. 04017069-12 J. Comput. Civ. Eng., 2018, 32(1): 04017069 J. Comput. Civ. Eng. Table 12. Different Pipe Groups Age Scenarios Considered in This Study Downloaded from ascelibrary.org by Tufts University on 10/28/17. Copyright ASCE. For personal use only; all rights reserved. Pipe number Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 Pipe age (years) Scenario 1 Scenario 2 Scenario 3 Scenario 4 Scenario 5 Scenario 6 Scenario 7 Scenario 8 14 32 75 92 101 57 3 33 61 94 38 65 99 9 31 79 97 110 1 31 60 94 100 2 42 65 95 7 38 74 94 104 13 47 78 92 104 3 49 61 93 101 97 91 95 91 96 113 98 99 95 97 99 104 105 112 97 90 104 90 109 10 18 32 14 21 21 20 23 25 28 4 9 33 34 28 1 13 31 8 32 2 25 17 15 32 10 19 29 9 31 35 25 7 25 5 30 29 23 1 34 17 7 19 97 107 112 113 91 105 100 94 90 108 111 110 94 108 93 99 111 106 97 101 107 91 110 104 108 110 103 108 99 105 25 30 15 1 33 13 18 20 28 17 12 13 12 2 16 16 13 17 26 15 5 3 13 3 18 32 12 13 6 13 20 2 25 27 35 8 28 19 10 34 29 31 18 35 17 3 26 3 22 12 11 9 23 20 32 2 21 32 9 18 2 33 31 31 28 15 15 34 23 18 27 15 11 35 32 31 102 107 104 110 103 23 7 3 9 35 1 101 113 95 97 98 29 12 22 26 9 9 5 29 23 32 9 32 7 10 17 11 18 19 13 2 3 32 26 35 29 14 30 28 29 27 29 17 79 107 56 51 67 87 79 105 98 104 90 105 95 78 76 72 112 53 69 109 51 75 78 105 87 75 82 71 106 109 82 71 113 80 84 95 51 105 82 62 56 60 101 91 103 112 87 100 97 107 81 88 83 108 107 98 96 94 83 105 98 104 80 86 111 101 81 104 107 109 110 109 80 113 106 95 104 106 92 113 80 105 93 109 98 86 It is worth noting that different mutation and crossover operators were compared, and both simulated binary crossover (sbx) and polynomial mutation (pm) were chosen for the evaluation. The values the other main MOGA parameters are shown in Table 17. The Pareto fronts produced by the proposed model and that of Halfawy et al. (2008) are shown in Figs. S1–S16. A sample of these Pareto fronts for the first analysis scenario (explained in Tables 12 and 14) is shown in Figs. 8 and 9. It clear from these figures that the proposed LCA-MOGA approach results larger number of solutions with wider range of objective functions values than Halfawy et al. (2008) approach. For example, by comparing the results in Fig. 7 with those in Fig. 8. It can be seen that, the number of solutions presented in Fig. 7 are 850 non-dominated solutions, while only 50 non-dominated solutions are presented in Fig. 8. This variety of solutions obtained by the LCA-MOGA algorithm comparing with Halfawy et al. (2008) algorithm increase the opportunity of decision makers to achieve a renewal plan that satisfy their objectives. The difference in the results obtained from both LCA-MOGA and © ASCE Halfawy et al. (2008) are discussed in the remaining part of this section in more details. To investigate the proximity, diversity, and consistency of the obtained Pareto front, the aforementioned metrics were calculated. The values of these metrics are shown in Table 18 for the eight different analysis cases (explained in Table 12), assuming no funding deficit at the third time step. The results assuming a funding deficit at the third time step are shown in Table 19. It is clear from these Tables that the proposed algorithm outperforms the Halfawy et al. (2008), since the value of the Epsilon, Generational distance and spacing indicators for the proposed model is smaller than those for Halfawy et al. (2008) model. While the value of the hypervolume indicator for the proposed model is bigger than those for Halfawy et al. (2008) model. The values in Tables 18 and 19 are average values obtained from 865 non-dominated solutions obtained by the proposed algorithm and 50 obtained by the Halfawy et al. (2008) model as shown in Figs. 7 and 8, respectively, and as discussed early in this subsection. 04017069-13 J. Comput. Civ. Eng., 2018, 32(1): 04017069 J. Comput. Civ. Eng. Table 13. Assigned Sojourn Time for Each Pipe in Each Condition State Downloaded from ascelibrary.org by Tufts University on 10/28/17. Copyright ASCE. For personal use only; all rights reserved. Sojourn time Pipe number Condition 1 Condition 2 Condition 3 Condition 4 Condition 5 Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 32 21 22 23 32 22 21 30 24 26 32 41 22 34 22 44 27 3 26 32 43 18 19 35 21 30 19 30 22 41 27 27 32 23 30 18 19 19 33 30 28 20 26 33 22 23 24 33 23 21 32 25 27 33 42 22 35 22 45 28 3 27 33 44 18 19 36 21 31 19 31 22 42 28 28 33 23 31 18 19 19 34 31 29 20 27 41 27 29 30 42 30 27 40 32 34 41 53 28 44 28 57 35 3 34 41 56 23 24 45 27 39 24 39 28 53 35 35 41 30 39 23 24 24 43 39 36 26 34 25 17 17 19 26 18 16 24 19 21 25 33 17 27 17 35 21 2 21 25 34 14 15 28 16 24 15 24 17 33 21 21 25 18 24 14 15 15 26 24 22 16 21 6 4 4 4 6 4 4 5 4 5 6 7 4 6 4 8 5 0 5 6 8 3 3 6 4 5 3 5 4 7 5 5 6 4 5 3 3 3 6 5 5 3 5 Utility income ($) Scenario 1 Scenario 2 Time step 1 Time step 2 Time step 3 Time step 4 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 −1,500,000 1,000,000 1,000,000 To better judge the difference in the performance of the two algorithms, a one tailed z-test is performed to assess if the observed differences in the average values are indeed statistically significant. The use of this test was justified by the fact that only two populations were compared, and the sizes of both populations were large. Four different null hypotheses (one for each indicator) are tested for each scenario. The first three hypotheses state that the mean of the epsilon indicator, generation distance, and spacing obtained by the Halfawy et al. (2008) model are smaller than or equal © ASCE Shape parameter Pipe number Condition 1 Condition 2 Condition 3 Condition 4 Condition 5 Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Pipe Table 14. Utility Income at Different Time Step ($) Scenario Table 15. Shape Parameter Values for Each Group for Each Condition State 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 1.528026 2.32842 2.222583 2.125949 1.528026 2.222583 2.32842 1.629894 2.037368 1.880647 1.528026 1.192605 2.222583 1.438142 2.222583 1.111291 1.810993 1.629894 1.880647 1.528026 1.137135 2.71649 2.573517 1.397052 2.32842 1.629894 2.573517 1.629894 2.222583 1.192605 1.810993 1.810993 1.528026 2.125949 1.629894 2.71649 2.573517 2.573517 1.481722 1.629894 1.746315 2.444841 1.880647 1.521092 2.281637 2.182436 2.091501 1.521092 2.182436 2.390287 1.568626 2.007841 1.859112 1.521092 1.195143 2.281637 1.434172 2.281637 1.115467 1.792715 1.930616 1.859112 1.521092 1.140819 2.788668 2.641896 1.394334 2.390287 1.619227 2.641896 1.619227 2.281637 1.195143 1.792715 1.792715 1.521092 2.182436 1.619227 2.788668 2.641896 2.641896 1.476354 1.619227 1.730897 2.509801 1.859112 1.440919 2.188062 2.037162 1.969256 1.406612 1.969256 2.188062 1.476942 1.846178 1.737579 1.440919 1.114673 2.109917 1.342675 2.109917 1.036451 1.687934 1.641047 1.737579 1.440919 1.054959 2.568595 2.46157 1.312837 2.188062 1.514812 2.46157 1.514812 2.109917 1.114673 1.687934 1.687934 1.440919 1.969256 1.514812 2.568595 2.46157 2.46157 1.3739 1.514812 1.641047 2.272219 1.737579 1.690563 2.486122 2.486122 2.224425 1.625541 2.348004 2.641505 1.761003 2.224425 2.012575 1.690563 1.28073 2.486122 1.565336 2.486122 1.207545 2.012575 2.348004 2.012575 1.690563 1.243061 3.018863 2.817605 1.509431 2.641505 1.761003 2.817605 1.761003 2.486122 1.28073 2.012575 2.012575 1.690563 2.348004 1.761003 3.018863 2.817605 2.817605 1.625541 1.761003 1.921094 2.641505 2.012575 3.172603 4.758904 4.758904 4.758904 3.172603 4.758904 4.758904 3.807123 4.758904 3.807123 3.172603 2.719374 4.758904 3.172603 4.758904 2.379452 3.807123 3.172603 3.807123 3.172603 2.379452 6.345205 6.345205 3.172603 4.758904 3.807123 6.345205 3.807123 4.758904 2.719374 3.807123 3.807123 3.172603 4.758904 3.807123 6.345205 6.345205 6.345205 3.172603 3.807123 3.807123 6.345205 3.807123 to the ones obtained by the proposed model. The fourth hypothesis states that the mean of the hyper volume obtained by the Halfawy et al. (2008) model is larger than that obtained by the proposed model. The results show that these four hypotheses were rejected within 95% confidence interval with p-value not more than 0.0128 in all cases. Therefore, the proposed model results in more diverse and consistent solutions than that Halfawy et al. (2008) model and the LCA-MOGA algorithm Pareto front are closer to the global Pareto front. The advantages of the proposed model over the one by Halfawy et al. (2008) could also be evaluated by examining the results of the eight analysis cases with and without funding deficits. For each scenario, a single solution is selected from the Pareto fronts obtained by both models. The single solution is selected using the two criteria (best tradeoff values, and more critical pipes), as explained in the “Methodology” section of this paper. In the Halfawy et al. (2008) model, the two criteria are applied at each time step. 04017069-14 J. Comput. Civ. Eng., 2018, 32(1): 04017069 J. Comput. Civ. Eng. Downloaded from ascelibrary.org by Tufts University on 10/28/17. Copyright ASCE. For personal use only; all rights reserved. Fig. 6. Semi Markov results: (a) PDFs of cumulative waiting time in various states; (b) CDFs of cumulative waiting time in various states; (c) SFs of cumulative waiting time in various states Table 16. Range of the Algorithm Parameters Parameter Range Maximum evaluations numbers Population size sbx rate sbx distribution index pm rate pm distribution index Monte Carlo steps 10,000–100,000 10–1000 0–1 0–500 0–1 0–500 100–10,000 Table 17. Chosen Values for the Algorithm Parameters Parameter Maximum evaluations numbers Population size sbx rate sbx distribution index pm rate pm distribution index Monte Carlo steps Range 50,000 100 0.9 15 0.1 20 1,000 For the proposed model, the best tradeoff is decided by using the best tradeoff values [Eq. (15)] at the end of the analysis period, while evaluating the pipe criticality criterion at each time step. The values of the objective functions at each time step for the selected solutions are shown in Table 20. These values show that the proposed algorithm results in saving in the life cycle cost, and at the © ASCE same time results in better network condition, risk of failure and serviceability at the end of the analysis period comparing with Halfawy et al. (2008) algorithm. Also, the results show that selecting the solution that gives the best tradeoff for the objective functions at one-time step does not necessarily lead to the best tradeoff for these objective functions at the end of the analysis period. For example, calculating the tradeoff criterion [Eq. (15)] for the results of the proposed algorithm for case 1 in Table 20 results in a dij value of 1.25. The dij value for the results of the Halfawy et al. (2008) model, on the other hand, is 0.72 which is lower than the 1.25 indicating that Halfawy et al. (2008) model results in better tradeoff of the objective functions than the proposed algorithm. However, the dij at the end of the analysis for the proposed algorithm was 0.488, which is lower than the dij of 0.694 obtain by the Halfawy et al. (2008) model at the end of the analysis indicating that the proposed algorithm results in better tradeoff of the objective functions at the end of the analysis period. Next, the two models were applied to the eight cases in Table 12, assuming a funding deficit at the third time step. The results shown in Table 21 indicate that the funding deficit at the third time step does not affect the selected renewal scenario for the networks in Cases 2, 4, 5, 6, and the results obtained from the proposed model in Case 3. Therefore, Cases 1, 7, 8, and the results obtained from Halfawy et al. (2008) in Case 3 were the only ones that are affected by this deficit. The results confirm that the proposed model results in saving in the life cycle cost, and at the same time results in better network condition, risk of failure and serviceability at the end of the analysis period comparing with Halfawy et al. (2008) algorithm. Therefore, the results presented in this section reject the null 04017069-15 J. Comput. Civ. Eng., 2018, 32(1): 04017069 J. Comput. Civ. Eng. Downloaded from ascelibrary.org by Tufts University on 10/28/17. Copyright ASCE. For personal use only; all rights reserved. Fig. 7. Renewal plans selected by LCA-MOGA algorithm for Scenario 1 for pipes age and utility income Fig. 8. Renewal plans selected by Halfawy et al. (2008) algorithm for Scenario 1 for pipes age and utility income hypothesis that the proposed model results in worst values for the four objective functions at the end of the analysis period than Halfawy et al. (2008) algorithm. One final aspect that need to be evaluated is the proposed model’s ability to select critical pipes in its selected renewal plans. A sample of the selection frequencies for scenario 2 by the proposed model is shown in Fig. 9. Figs. S17–S32 show all those frequencies. © ASCE These figures illustrate that both algorithms do not favor the most critical pipes in the network in their selection. For example, Fig. 9 shows that, the proposed model selects pipe group 6 which is 98 years old, and has consequence of failure of 4.5 in 1.7% of the solutions in the Pareto front. One the other hand, the model selects pipe group 41, which has an age of 17 years and a consequence of failure of 1.75 in 94% of the solutions in the Pareto front. It is important to consider which pipe groups are being selected by the 04017069-16 J. Comput. Civ. Eng., 2018, 32(1): 04017069 J. Comput. Civ. Eng. Downloaded from ascelibrary.org by Tufts University on 10/28/17. Copyright ASCE. For personal use only; all rights reserved. Fig. 9. Pipe group selection frequency applying LCA-MOGA algorithm on Scenario 2 Table 18. Solution Performance Metrics Assuming No Funding Deficit (All Points Are Statistically Significant) Table 19. Solution Performance Metrics Assuming a Funding Deficit (All Points Are Statistically Significant) LCA-MOGA LCA-MOGA Parameter Average Standard deviation Epsilon indicator Generational distance Hyper volume Spacing Stepwise model Average Standard deviation Scenario 1 1.028400 0.004114 0.000380 0.000015 1.372400 0.020568 0.000838 0.000017 1.186200 0.002300 1.164120 0.006420 Epsilon indicator Generational distance Hyper volume Spacing Scenario 2 1.029200 0.041168 0.000600 0.000012 1.373100 0.051460 0.001800 0.000054 Epsilon indicator Generational distance Hyper volume Spacing Stepwise model Parameter Average Standard deviation Average Standard deviation 0.003559 0.000069 0.023724 0.000321 Epsilon indicator Generational distance Hyper volume Spacing Scenario 1 1.073200 0.053660 0.000750 0.000023 1.242600 0.042928 0.001038 0.000052 1.206700 0.006400 1.194500 0.008200 0.048268 0.000128 0.024134 0.000246 1.186500 0.002400 1.164200 0.006900 0.059325 0.000072 0.047460 0.000138 Epsilon indicator Generational distance Hyper volume Spacing Scenario 2 1.074100 0.032223 0.001300 0.000026 1.243100 0.053705 0.001100 0.000044 1.207100 0.006700 1.194900 0.008400 0.024142 0.000268 0.060355 0.000168 Scenario 3 1.028800 0.041152 0.000700 0.000035 1.373100 0.051440 0.001300 0.000065 1.186500 0.002600 1.164700 0.007300 0.047460 0.000052 0.035595 0.000219 Epsilon indicator Generational distance Hyper volume Spacing Scenario 3 1.073300 0.032199 0.001300 0.000052 1.243500 0.032199 0.001900 0.000038 1.207600 0.006800 1.195300 0.008500 0.060380 0.000340 0.024152 0.000340 Epsilon indicator Generational distance Hyper volume Spacing Scenario 4 1.028900 0.030867 0.000800 0.000040 1.372600 0.051445 0.001000 0.000040 1.187000 0.003000 1.164600 0.006800 0.059350 0.000090 0.075210 0.000272 Epsilon indicator Generational distance Hyper volume Spacing Scenario 4 1.074100 0.042964 0.001500 0.000060 1.242800 0.021482 0.001300 0.000026 1.207600 0.006500 1.195000 0.008900 0.048304 0.000130 0.036228 0.000445 Epsilon indicator Generational distance Hyper volume Spacing Scenario 5 1.029100 0.030873 0.000800 0.000032 1.373200 0.054928 0.001600 0.000064 1.186900 0.002400 1.164200 0.007000 0.047476 0.000072 0.034926 0.000350 Epsilon indicator Generational distance Hyper volume Spacing Scenario 5 1.073900 0.032217 0.000800 0.000032 1.373300 0.032217 0.001300 0.000052 1.207500 0.006800 1.195000 0.008400 0.036225 0.000136 0.024150 0.000336 Epsilon indicator Generational distance Hyper volume Spacing Scenario 6 1.029200 0.030876 0.000800 0.000024 1.373300 0.041168 0.000800 0.000040 1.186800 0.002500 1.164500 0.006800 0.035604 0.000050 0.023736 0.000272 Epsilon indicator Generational distance Hyper volume Spacing Scenario 6 1.073700 0.032211 0.001400 0.000042 1.243500 0.042948 0.001100 0.000055 1.207600 0.007300 1.194600 0.008700 0.048304 0.000292 0.048304 0.000174 Epsilon indicator Generational distance Hyper volume Spacing Scenario 7 1.029400 0.051470 0.000400 0.000012 1.372500 0.041176 0.000900 0.000018 1.186700 0.002300 1.164900 0.007000 0.023734 0.000069 0.047468 0.000280 Epsilon indicator Generational distance Hyper volume Spacing Scenario 7 1.073500 0.053675 0.000900 0.000045 1.373300 0.053675 0.001800 0.000036 1.206700 0.006400 1.195200 0.009100 0.060335 0.000192 0.060335 0.000364 Epsilon indicator Generational distance Hyper volume Spacing Scenario 8 1.028500 0.041140 0.000500 0.000015 1.373100 0.030855 0.001300 0.000026 1.187000 0.002900 1.164300 0.006700 0.023740 0.000116 0.047480 0.000134 Epsilon indicator Generational distance Hyper volume Spacing Scenario 8 1.073500 0.042940 0.000900 0.000036 1.243200 0.032205 0.001100 0.000033 1.207200 0.006900 1.194900 0.009200 0.048288 0.000276 0.036216 0.000276 © ASCE 04017069-17 J. Comput. Civ. Eng., 2018, 32(1): 04017069 J. Comput. Civ. Eng. Table 20. Objective Function Values of the Selected Solutions from the Two Algorithms Scenario number LCC ($) Downloaded from ascelibrary.org by Tufts University on 10/28/17. Copyright ASCE. For personal use only; all rights reserved. 1 Average condition index Total LCC ($) 787,711.70 110,435.10 47,192.43 371,805.20 LCA-MOGA model 2.94 301.11 6.44 2.88 335.3 6.29 2.96 311.59 6.43 2.66 262.34 5.8 664,000.10 218,814.70 309,610.50 337,66.04 Stepwise optimization 2.99 359.79 6.52 2.89 413.41 6.33 2.72 369.2 5.99 2.79 313.13 6.1 188,541.90 210,619.90 34,88.81 767,86.78 LCA-MOGA model 1.47 267.08 3.89 1.36 314.25 3.41 1.37 669.7 3.45 1.3 359.34 3.17 199,678.40 155,134.20 107,64.32 17,313.40 Stepwise optimization 1.45 416.03 3.79 1.36 624.93 3.41 1.53 550.75 3.77 1.33 530.2 3.28 — — — 502,123.70 301,561.70 152,595.50 257,576.10 159,800.50 2.94 2.88 2.96 2.63 LCA-MOGA model 301.11 6.44 335.3 6.29 311.59 6.43 180.62 5.79 — — — 871,533.80 290,774.70 285,987.60 225,773.70 91,892.51 Stepwise optimization 2.99 159.79 6.52 2.89 413.41 6.33 2.72 269.2 5.99 2.71 561.21 5.98 — — — 894,428.50 28,945.15 28,963.42 19,402.60 13,110.68 LCA-MOGA model 2.94 301.11 6.44 2.88 335.3 6.29 2.96 311.59 6.43 1.26 370.84 2.92 2 3 4 Scenario number — — — 1,317,144.00 — — — 1,648,833.00 LCC ($) 644,642.10 540,568.60 452,550.80 54,075.56 3.02 2.76 2.23 2.20 LCA-MOGA model 350.61 7.03 320.40 6.42 259.19 5.27 255.41 5.13 — — — 1,691,837.00 867,132.40 149,036.90 18,602.22 870,937.70 Stepwise optimization 2.86 336.61 6.68 2.73 1,216.46 6.38 2.71 733.19 6.27 2.39 288.13 5.55 — — — 1,905,709.00 608,749.10 597,423.90 275,814.60 209,849.40 3.25 2.77 2.49 2.20 LCA-MOGA model 104.89 7.52 273.19 6.32 97.00 5.74 255.41 5.13 — — — 1,691,837.00 274,958.20 63,0157.20 632,561.10 315,302.90 Stepwise optimization 3.61 111.01 8.28 3.05 199.53 6.81 2.52 219.39 5.74 2.29 288.13 5.24 — — — 1,852,979.00 7 8 — — — 472,704.60 Total number of blockages Average risk of failure index Average condition index Total LCC ($) Table 21. Objective Function Values of the Selected Solutions from the Two Algorithms Assuming Fund Deficit at the Third Time Step Scenario number LCC ($) 1 — — — 90,421.85 Average Total condition number of index blockages Average risk of failure index Total LCC ($) 105,674.30 210,198.60 11,495.90 513,716.50 LCA-MOGA model 3.26 150.42 7.16 3.16 136.39 6.93 3.44 210.85 7.13 2.49 179.83 6.35 — — — 841,085.30 369,473.00 24,145.18 13,757.60 650,667.10 Stepwise optimization 3.09 110.69 6.82 3.60 207.25 7.17 4.08 291.09 7.37 3.05 225.39 6.61 — — — 1,058,042.88 LCA-MOGA model 1.47 267.08 3.89 1.36 314.25 3.41 1.37 469.70 3.45 1.30 359.34 3.17 — — — 479,437.39 433,772.30 38,138.35 97,529.23 13,361.77 Stepwise optimization 2.99 159.79 6.52 2.89 413.41 6.33 2.72 269.2 5.99 1.32 442.32 3.08 LCA-MOGA model 1.96 463.81 4.26 2.08 490.38 4.48 1.75 413.85 3.87 1.61 380.81 3.57 — — — 582,801.60 188,541.90 210,619.90 3,488.81 76,786.78 465,378.00 11,998.58 10,742.54 114,452.40 Stepwise optimization 1.95 439.58 4.24 2.04 894.76 4.44 1.99 350.43 4.35 1.95 855.06 4.26 — — — 602,571.50 199,678.40 155,134.20 10,764.32 17,313.40 Stepwise optimization 1.45 416.03 3.79 1.36 624.93 3.41 1.53 550.75 3.77 1.33 530.20 3.28 — — — 382,890.32 69,281.72 27,724.24 25,108.86 43,343.47 LCA-MOGA model 1.81 915.91 4.00 1.32 668.61 3.08 1.45 734.77 3.31 1.30 658.32 3.00 — — — 165,458.30 301,561.70 152,595.50 257,576.10 159,800.50 LCA-MOGA model 2.94 301.11 6.44 2.88 335.30 6.29 2.76 311.59 6.13 2.63 180.62 5.79 — — — 871,533.80 97,931.50 24,573.54 9,938.13 56,347.05 Stepwise optimization 1.78 1,001.91 3.92 1.50 618.25 3.37 1.80 523.77 3.95 1.88 456.84 4.11 — — — 188,790.20 55,818.03 138,844.80 288,279.10 998,641.00 Stepwise optimization 4.54 494.87 9.59 4.44 609.40 9.36 4.29 1,056.92 9.11 3.06 735.76 7.72 — — — 1,481,582.93 26,856.50 9,393.10 45,043.31 15,420.57 5 6 © ASCE Total number of blockages Average risk of failure index Table 20. (Continued.) — — — 96,713.49 2 3 04017069-18 J. Comput. Civ. Eng., 2018, 32(1): 04017069 J. Comput. Civ. Eng. Table 21. (Continued.) Scenario number LCC ($) Downloaded from ascelibrary.org by Tufts University on 10/28/17. Copyright ASCE. For personal use only; all rights reserved. 4 Average Total condition number of index blockages Average risk of failure index Total LCC ($) 28,945.15 28,963.42 19,402.60 13,110.68 LCA-MOGA model 2.94 301.11 6.44 2.88 335.30 6.29 2.96 311.59 6.43 1.26 370.84 2.92 — — — 90,421.85 26,856.50 9,393.10 45,043.31 15,420.57 Stepwise optimization 2.99 159.79 6.52 2.89 413.41 6.33 2.72 269.20 5.99 1.32 442.32 3.08 — — — 96,713.48 433,772.30 38,138.35 97,529.23 13,361.77 LCA-MOGA model 1.96 463.81 4.26 2.08 490.38 4.48 1.75 413.85 3.87 1.61 380.81 3.57 — — — 582,801.65 465,378.00 11,998.58 10,742.54 114,452.40 Stepwise optimization 1.95 439.58 4.24 2.04 894.76 4.44 1.99 350.43 4.35 1.95 855.06 4.26 — — — 602,571.52 69,281.72 27,724.24 25,108.86 43,343.47 LCA-MOGA model 1.81 915.91 4.00 1.32 668.61 3.08 1.45 734.77 3.31 1.30 658.32 3.00 — — — 165,458.29 97,931.50 24,573.54 9,938.13 56,347.05 Stepwise optimization 1.78 1,001.91 3.92 1.50 618.25 3.37 1.80 523.77 3.95 1.88 456.84 4.11 — — — 188,790.22 151,925.80 167,085.10 76,501.72 683,432.60 LCA-MOGA model 2.90 648.46 7.16 2.99 240.03 7.96 3.20 418.66 8.31 2.59 771.23 7.10 — — — 1,078,945.22 867,132.40 149,036.90 −516,169.00 870,937.70 Stepwise optimization 2.86 336.61 7.68 2.73 1,216.46 7.38 3.31 1,616.46 8.27 2.99 1,171.40 7.55 — — — 1,370,938.00 212,415.10 111,047.40 15,560.03 1,433,117.00 LCA-MOGA model 3.79 230.22 8.91 3.71 471.28 8.59 3.70 798.46 8.54 3.06 775.49 7.09 — — — 1,772,139.53 149,464.70 184,133.10 91,809.39 1,476,565.00 Stepwise optimization 3.81 260.41 8.83 3.74 503.50 8.67 3.67 793.05 8.48 3.01 768.45 6.93 — — — 1,901,972.19 5 6 7 8 model because the serviceability objective function depends on both pipe conditions and physical attributes. It is important for the proposed model to consider pipe criticality and not just age because pipes with similar ages may have different expected numbers of blockages because they may have different physical characteristics. For example, a pipe with 1.0 m diameter may have less expected blockages than a pipe with 0.2 m diameter although the two pipes have the same age and are in the same condition. The same may apply to the risk of failure index. © ASCE Although risk of failure depends on the condition of the pipe, it also depends on the consequence of failure for the pipe which is not related to the pipe age. The model favors critical pipes (however this criticality is defined), which is an important feature since water utilities usually set certain criteria for renewing pipes in the network. For example, a water utility may be interested in renewing pipes over 100 years old before they fail regardless of their risk of failure. They also may be interested in keeping certain pipes that have a high consequence of failure within a specific Condition (Condition 2 or 3 for example), not allowing them to approach failure. Utilities, may also be more interested in renewing pipes that serves vital area than other pipes regardless of their age or consequence of failure. The proposed model and the one by Halfawy et al. (2008) do not address this problem. Therefore, the proposed model is modified in the next chapter to solve this problem. Summary and Conclusion Present optimization approaches developed in the literature focus on selection renewal strategy for the sewer pipes in a particular time step rather than over the entire planning period. As a result, these models suffer from a number of drawbacks. First, these models make their decision for a specific time step without considering the impact of those decisions on the performance of the network in the future, and without considering the uncertainty in utility resources in the future. Second, present models focus on reducing the risk of the structural failure in the network and ignore the operational failure although they are the more prevalent threat to public health and the environment. This is apparent in the literature, since there is limited work on how to evaluate and improve the network serviceability. To overcome these limitations, this paper proposed and implemented an LCA-MOGA model. As the name indicates this algorithm uses the concept of the life cycle analysis along with multiobjective genetic algorithm to select optimal/near-optimal renewal strategies for sewer networks. The model optimizes the following four objective functions: (1) average condition of the sewer network, (2) average network risk of failure, (3) total expected number of blockages, and (4) total life cycle cost. The model integrates a semi-Markov deterioration model and a serviceability model developed in the literature to update the network condition and the expected number of blockages at each time step. The proposed algorithm selects a set of feasible and optimal/ near-optimal renewal plans to be applied at each time step, based on the value of the objective functions at the end of the analysis period. This allows the uncertainty in the performance of the wastewater system in the future to be considered while selecting the renewal plan to be applied at each time step. This results in the selected renewal plan at each time step to satisfy the imposed budget constraints, and also allow the wastewater system to achieve the desired performance of at the end of the analysis period. The developed model was applied to a sewer network in Sahab city in Jordan and different scenarios were used for the analysis. These scenarios varied pipes ages, sojourn times in each condition, and the utility income level at each time step. The developed model was validated by comparing its performance to that of a present model developed in the literature. The performance of the two algorithms was compared based on the proximity, diversity, and consistency of the solutions they obtained in their Pareto-fronts. The aforementioned properties were tested using four metrics including; hypervolume, generational distance, spacing, and epsilon indicators. The values of these indicators showed that the developed model statistically outperforms the present one in all four properties discussed 04017069-19 J. Comput. Civ. Eng., 2018, 32(1): 04017069 J. Comput. Civ. Eng. Downloaded from ascelibrary.org by Tufts University on 10/28/17. Copyright ASCE. For personal use only; all rights reserved. above. These results are further validated statistically using a z-test to make sure that the obtained results are not due to random chance. As a result, the null hypothesis that the proposed model results in higher average network condition, average network risk of failure, network serviceability, and life-cycle cost over the entire planning period than the step wise prioritization model, is rejected within a 95% confidence interval for all analysis scenarios. To further evaluate the amount of the difference in the value of the objective functions obtained from the two models, two criteria were adopted to choose solutions from the Pareto-front obtained by each model. The first criterion was to select solutions with best tradeoff between the objective functions, and the second was to select solutions that include the most critical pipes in the network. 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