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New Multiobjective Optimization Approach to
Rehabilitate and Maintain Sewer Networks Based
on Whole Lifecycle Behavior
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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
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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
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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
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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
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Table 1. Determining Weibull Distribution Parameters for the Illustrated
Example
Condition state
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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
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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)
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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
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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
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Range
0–1
0–1
0–1
0–3
0–1
J. Comput. Civ. Eng.
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Fig. 3. Lifecycle analysis multiobjective genetic algorithm
© ASCE
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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
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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).
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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
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J. Comput. Civ. Eng.
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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
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J. Comput. Civ. Eng.
Table 10. Pipes Classification Groups
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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
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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
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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
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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
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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
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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
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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
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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.
The results show that the proposed model produces significant savings in the life cycle cost, a better average condition, a lower risk of
failure and expected number of blockages, than the present model
used for validation under all the considered analysis scenarios.
These results also reject the null hypothesis, and confirm that
the developed model results in lower average condition, risk of failure and number of blockages with lower life cycle cost that the step
wise algorithm.
Finally, it was observed that both algorithms do not consider
the criticality of the selected pipes in selecting their solutions. This
issue will be covered in future work.
Supplemental Data
Figs. S1–S32 are available online in the ASCE Library (www
.ascelibrary.org).
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J. Comput. Civ. Eng.
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