International Journal of Computer Information Systems and Industrial Management Applications.
(2013) pp. 308-316
ISSN 2150-7988 Volume 5 (2012)
© MIR Labs, www.mirlabs.net/ijcisim/index.html
Cross Entropy Optimization for Optimal Design of
Water Distribution Networks
A. Shibu1 and M. Janga Reddy2
1,2
Department of Civil Engineering,
Indian Institute of Technology Bombay,
Powai, Mumbai – 400 076, India.
Phone: +91 22 2576 7320; Fax : +91-22-2576 7302
E-mail: 1shibuiitb@yahoo.in (Corresponding author)
Abstract: This paper presents cross entropy (CE) based
methodology for optimal design of water distribution network
(WDN). Design of WDN involves selection of suitable diameter
for each pipe in the network from the list of commercially
available diameters. The CE methodology is applied to two bench
mark WDN design problems taken from literature for validation.
The first WDN problem deals with determining optimal pipe
sizes for planning a new system, while the second WDN deals
with rehabilitation of existing WDN by parallel piping. The
performance of CE is compared with the results of past studies
and it is found that the CE resulted in good optimal solutions.
Then, the model is applied to a case study in India. The results
suggest that CE method is very effective in optimal design of
water distribution networks and has the capability of rapid
convergence to optimum solutions.
Keywords: water distribution networks, metaheuristics, cross
entropy, optimization.
I. Introduction
Water distribution network (WDN) consists of a set of pipes
of different diameters and lengths connected with one another
at various junctions called nodes. The diameters and lengths of
pipes are designed in such a manner that they deliver the
required amount of water with sufficient pressure to the
demand nodes without failure. The optimal design of WDN
aims to find a combination of the diameters that are feasible
and results in minimum cost. Several researchers have
formulated different models for optimal design of WDNs. Few
studies modeled it as a nonlinear model and solved using
Nonlinear Programming (NLP) techniques by treating discrete
pipe sizes as continuous variables. The main disadvantage of
these NLP methods is the required rounding-off of continuous
decision variables to commercially available sizes, sometimes
which can lead to network infeasibilities as well as raise
questions on optimality of the adjusted solution. Some other
studies formulated linear models and solved it using Linear
Programming (LP) techniques. However, these methods are
capable of maintaining the constraint on discrete pipe sizes
(without the need for rounding off solutions), but it requires
approximation of non-linear functions, which may not
represent the reality as it is.
In spite of development of many conventional techniques
for optimization, each of these techniques has its own
limitations. To overcome those limitations, recently
metaheuristic techniques are being used for solving
combinatorial optimization problems. By using these
techniques, the given problem can be represented more
realistically. These also provide ease in handling the
non-linear relationships of the formulated model [1]. Genetic
algorithms, particle swarm optimization, ant colony
optimization algorithm, cross entropy algorithm etc. are some
of the techniques fall in this category. These evolutionary
algorithms search from a population of points, so there is a
greater possibility to cover the whole search space and
locating the global optimum.
The stochastic search approaches that were used for WDN
design include genetic algorithms ([2]; [3]; [4]; [5]; [1]; [6]),
Simulated annealing [7], shuffled leaping frog algorithms [8],
ant colony optimization algorithms [9], cross entropy
algorithms [10] etc. These techniques improve the quality of
the solution over the iterations by using heuristics.
The cross entropy (CE) method was motivated by an adaptive
algorithm for estimating probabilities of rare events in
complex stochastic networks, which involves variance
minimization. Later it was modified to a randomized
optimization technique, where the original variance
minimization was changed to cross entropy minimization
problem [11]. The CE method was successfully applied to
various optimization problems such as traveling salesman,
assignment problem etc.[11]. In the present study, CE method
is presented for optimization of the water distribution
networks.
II. Cross Entropy Method
The cross entropy method is an iterative technique based on
the concept of rare events, which involves two main stages: (i)
generation of random sample of initial population (i.e.,
solution vectors) with a set of parameters, and (ii) updating
this set of parameters which control the generation of random
data using the sample itself, with the aim of improving the
solution in the next iteration. The method derives its name
from the cross entropy or Kullback-Leibler distance- a well
MIR Labs, USA
Shibu and Reddy
309
known measure of ‘information’, which has been successfully
employed in various fields of engineering [11].
A. Entropy and Cross Entropy
Entropy can be termed as a measure of uncertainty
associated with a process (measure of expected information
gain from a random variable) [12]. The probability
distribution of events if known provides a certain amount of
information. Shannon defined a quantitative measure of the
distribution in terms of entropy, called Shannon entropy given
by (1).
n
H ( X ) = − K ∑ p r ln p r
(1)
r =1
where H(X) represents the Shannon entropy corresponding to
the random variable X, K is a constant, and pr represents the
discrete probability corresponding to the variable at xr. The
uncertainty can be quantified with entropy taking into account
all different kinds of available information. Thus entropy is a
measure of uncertainty represented by the probability
distribution and is a measure of the lack of information about a
system. If complete information is available, entropy is equal
to zero, otherwise it is greater than zero.
Cross entropy is a distance measure from one probability
distribution to another. One of the well known definitions of
Cross entropy is the Kullback–Leibler distance measure [13],
serving to assess the similarity between two probability
distributions: the assumed distribution q(x) and the actual
distribution p(x). Cross entropy [D(P,Q)] is formulated as in
(2).
n
pr
(2)
qr
r =1
The interpretation of (2) is that in order to estimate a
probability distribution, the cross entropy should be
minimized. The goal is to find a distribution p(x) for which the
Kullback – Leibler distance between p(x)* and q(x)* is
minimal.
D ( P, Q) = ∑ pr ln
B. Principle of Minimum Cross Entropy
According to Laplace’s principle of insufficient reason, all
outcomes of an experiment should be considered equally
likely unless there is information to the contrary [13]. Suppose
a probability distribution for a random variable X= X = {x1, x2,
x3,…,xn} is assumed as Q = {q1, q2, q3,…,qn} based on
intuition. This constitutes the prior information in terms of a
prior distribution. While estimating the actual distribution P
={p1, p2, p3,…,pn} of random variable X, using all the given
information and make the distribution as near as possible to the
assumed distribution. Thus, according to the principle of
minimum cross entropy (POMCE), the cross entropy (CE),
D(P,Q) is minimized as in (3).
n
p
(3)
M inim ize D ( P , Q ) = ∑ p r ln r
q
r =1
r
This is referred to as the principle of minimum cross
entropy, which minimizes the Bayesian entropy [13]. Here
minimizing D(P,Q) is equivalent to maximizing the Shannon
entropy.
C. Cross Entropy Algorithm
The main steps involved in the cross entropy algorithm for
solving combinatorial optimization problem is given below.
1. Conversion of the combinatorial optimization problem to a
stochastic node network (SNN) problem.
2. Set the trial counter t = 0 and assume equal probabilities
for all the options as p0,r , where ‘r’ takes values from 1 to m.
The number of stochastic nodes, m=np*nd, where np is the
number of variables and nd is the number of available options.
3. Generate Nc sample vectors Xv(x1, x2,…,xm) for v = 1 to Nc
using the probability pt,r (i.e., generate a set of Nc possible
vectors each of size m, and having zeros and ones, where one
corresponds to choosing a specific node, and zero otherwise).
The value of Nc is taken as Nc =β*nd, where β is an integer
value. The m dimensional vector Xv (x1, x2,…,xm) has the
discrete probability of P =(p1, p2,…,pm).
4. Find out the performance function S(Xv) and check for
constraints corresponding to each of the random vectors Xv,
generated.
5. Now arrange the random vectors Xv, in the ascending
order(if the problem is a minimization problem) or descending
order (if it is a maximization problem) of their performance
function S(Xv) values. Now the top most vectors will be having
the best performance value and it is denoted as γt .
6. Choose a set (say ρc) of the top best performing vectors for
updating the probability vector pt,r to the probability vector
pt+1,r. Here ρc corresponds to percentage of the vectors
selected and its value varies between 10% and 20% but may
change as a function of the sample size N. The rth component
of pt+1,r is obtained as given by (4).
pt +1,r =
Bt ,r
TBt
(4)
where pt+1,r is the probability of success in the (t+1)th iteration
of node r, Bt,r is the total number of times node r was chosen
(frequency) out of the best top performance vectors (i.e., TBt
the total number of vectors in the elite set) at iteration t.
In order to avoid early convergence (stopping criteria of
probabilities of potential options approaching ZERO or ONE)
to a local optimum solution, a smoothing parameter ( α c ) is
used. The probability is modified as given by (5)
pt +1,r ← α c pt +1,r + (1 − α c ) pt ,r
(5)
Using the above probability-updating scheme, the
probability of choosing a node at each subsequent iteration
increases as the frequency of occurrence of the node in the
elite set increases. Updating the entire probability components
using (4) in conjunction with the smoothing formula (5) yields
the new probability vector pt+1,r. The main reason why such a
smoothing updating procedure performs better is that it
prevents the incidents of zeroes and ones in the reference
vector, as in case such values are obtained they will remain
permanently, which is obviously not required.
7. Check stopping conditions: If γt for subsequent iterations
remains unchanged and if pt converges to the degenerated
case (i.e. all the probabilities pt,r are close to zero and one)
then stop. Declare the last γt as the optimal solution γ* and its
associated vector X as the design vector X*, otherwise pt,r
pt+1,r and return to step 3.
Cross Entropy Optimization for Optimal Design of Water Distribution Networks
III. Model Formulation
The optimization problem is to determine the values of pipe
diameters that would minimize the cost of the system without
violating any of the constraints. Thus it is required to select
one diameter for each pipe from the list of commercially
available diameters. The optimization problem can be
expressed as,
np
Minimize Cost = ∑ C ( d i ) * li
(6)
i =1
subject to,
H j ≥ H min
, ∀j
j
q −q
in
j
out
j
− q j = 0 , ∀j
npu L
np L
HLi −
h p = 0 , L = 1,2,3,......., nL
p =1
i =1
L
∑
where,
∑
q inj =
310
a single reservoir which is located at elevation of 100 m. The
ground elevation for all nodes is 0. All pipes in the network are
of different lengths and the length of pipes is given in Table 1.
Data relevant to nodes is given in Table 2. The system
constraint on minimum pressure head requirement for all
nodes is defined as 30 m. No velocity constraint is taken into
account for this network. There are 6 commercially available
pipe diameters (nd=6) and unit cost of the pipes used in the
case study I are given in Table 3.
The study on Hanoi WDN was first carried out by [14].
Thereafter so many researchers [4], [7], [15] and [16] applied
various techniques to find optimal solution to Hanoi WDN.
The solution search space for the Hanoi WDN is 634.
(7)
(8)
(9)
n in
∑Q
i
(10)
∑Q
(11)
δ li Qi1.852
(12)
i =1
q out
j =
n out
i
i =1
HLi =
1.852 4.87
CHW
di
where C(di) corresponds to the cost per unit length of the pipe
having diameter di and li is the length of the ith pipe , Hj and
H min
are the available and minimum pressure heads at the jth
j
node; nd =number of demand nodes; q inj = flow entering the jth
node;
Figure 1. Layout of Hanoi WDN
q out
=flow leaving from the jth node ; qj= demand at the
j
jth node; HLi =head loss in ith pipe; npL=number of pipes in a
Table 1. Pipe length data for Hanoi WDN
loop; h p =head raised by the pump p, npuL=number of pumps
in a loop; nL=number of loops in the WDN. nin =number of
incoming pipes to the jth node; nout =number of outgoing
pipes from the jth node; and Qi = discharge or flow through the
ith pipe, δ =constant depending on the units of head loss,
length, diameter, and discharge; and CHW=Hazen William’s
roughness coefficient.
IV. Application of the Model
A. Case Study I: Hanoi WDN
The Hanoi water distribution network problem [14] as
shown in Figure 1, is an extensively studied WDN by many
researchers using a variety of optimization methods (such as
genetic algorithms, ant colony optimization, simulated
annealing etc.) is taken-up as case study I for testing the
performance of CE method. This network is a real WDN
constructed in Hanoi city at Vietnam, consists of 34 pipes and
32 nodes organized in three loops. The system is gravity fed by
1
Pipe
Length (m)
100
18
Pipe
Length (m)
800
2
1350
19
400
3
900
20
2200
4
1150
21
1500
5
1450
22
500
6
450
23
2650
7
850
24
1230
8
850
25
1300
9
800
26
850
10
950
27
300
11
1200
28
750
12
3500
29
1500
13
800
30
2000
14
500
31
1600
15
550
32
150
16
2730
33
860
17
1750
34
950
Pipe No.
Pipe No.
Shibu and Reddy
311
Table 2. Node demand data for Hanoi WDN
Node
No.
Nodal
Demand (m3/h)
Node No.
Nodal
Demand (m3/h)
1
-
17
865
2
890
18
1345
3
850
19
60
4
130
20
1275
5
725
21
930
6
1005
22
485
7
1350
23
1045
8
550
24
820
9
525
25
170
10
525
26
900
11
500
27
370
12
560
28
290
13
940
29
360
14
615
30
360
15
280
31
105
16
310
32
805
Table 3. Commercially available pipe diameters and unit cost
of pipes for Hanoi WDN
Sl.
Available Pipe Diameter
Unit Cost of Pipe
No.
inch
mm
($/m length)
1
12
304.8
45.73
2
16
406.4
70.40
3
20
508
98.38
4
24
609.6
129.30
5
30
762.0
180.75
6
40
1016.0
278.28
1) Model Run and Output for Case Study I
At the start of the algorithm, it is assumed that all the
options have equal probability of selection (i.e., P0,r = 1/6).
The performance function used for solving the model is
np
nn
i =1
j =1
S ( X v ) = ∑ C ( d i ) × li + ∑ PN × MAX (0, H min
− H j)
j
(13)
where S(Xv) is the performance function for the solution
vector, and PN is the penalty function rate for violating the
nodal pressure constraint.
At the end of the final iteration, only 34 (i.e., total number
of pipes) options will be having probability equal to one which
forms the optimal solution set, and rest of them will be having
a probability equal to zero. A program in MATLAB is
developed, which is linked to EPANET toolkit for simulation
of the WDN and to check the hydraulic feasibility. The
stopping criteria is arrived in 31,500 function evaluations with
smoothing parameter α = 0.35 and PN =100000000. The
output of the model run for Hanoi WDN is given in Tables 4 &
5, and also compared with the past studies.
Table 4. Nodal pressure corresponding to the optimal design
by Cross Entropy method for Hanoi WDN
Node
No.
Available
Nodal
Pressure
(m)
1(R)
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
100
97.1407
61.6704
57.1713
51.5992
45.7571
44.4013
42.8160
41.5661
40.6585
39.0991
35.6707
31.4625
33.3626
30.5197
30.4795
Remarks
Reservoir
Avail.
pressure
is more
than the
min.
pressure
required
Node
No.
Available
Nodal
Pressure
(m)
Remarks
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
32.9603
49.8247
55.0349
50.0175
40.6683
39.3963
43.4291
37.5807
33.7794
31.7037
30.9604
35.1562
30.7902
30.1112
30.6475
32.0296
Avail.
pressure
is more
than the
min.
pressure
required
Table 5. Comparison of Cross Entropy model result of Hanoi
WDN with past studies
Pipe No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
Total cost ($)
Pipe Diameter (inch) as per:
Cross Entropy
[15]
Method
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
30
40
40
30
30
30
30
24
24
24
24
24
16
24
16
16
12
12
12
12
16
16
12
12
20
16
16
24
24
24
24
24
20
40
40
40
20
20
20
12
12
16
40
40
40
30
30
30
30
30
30
20
20
24
12
12
16
12
12
12
16
16
16
16
12
16
12
12
12
12
20
12
16
16
16
20
24
24
6.18×106
6.11×106
6.15×106
[4]
Cross Entropy Optimization for Optimal Design of Water Distribution Networks
On comparing the results of the Cross Entropy Model for
Hanoi WDN problem [14] with results of earlier studies, it is
found that the optimum diameters obtained from the present
study is coming nearly same for all pipes in the network except
for few pipes. Also the optimal cost obtained is closer to
optimal costs of previous studies. Thus, the results obtained
from present study shows that the CE method is effective and
is well suited for optimal design of medium sized WDN like
Hanoi WDN.
B. Case Study II: Newyork City Tunnel WDN
Newyork City Tunnel WDN [15] is taken-up as a case
study II, for testing the performance of CE method. The layout
of WDN is shown in Figure 2. The network consists of 20
nodes, 21 pipes and 1 loop, and is fed by gravity from a
reservoir at a fixed head of 300 ft (91.44 m). The ground
elevation for all nodes is 0. This system is in place and requires
expansion. The pipe lengths, existing pipe diameters, and
nodal demands are given in Table 6, and a Hazen-Williams
constant of 100 is assumed for both the old tunnels and new
pipes [15]. The system constraint is the minimum pressure
head requirement for all nodes which is also given in Table 6.
Fifteen commercially available pipe diameters and their unit
cost are listed in Table 7. No velocity constraint is taken into
account for this network. The objective is to determine
whether a new pipe is to be laid parallel to an existing pipe or
not, and if needed what will be the diameter of a parallel pipe,
while the system is required to provide minimum hydraulic
gradients. This network is firstly studied in [17] and thereafter
studied by a number of other researchers ([4]; [5]; [6]). Due to
pipe aging, the existing gravity flow tunnels are inadequate to
meet the pressure requirements at nodes 16, 17, 18, 19, and 20
for the projected demands. Therefore new pipes can be added
in parallel to the existing pipes to meet the minimum pressure
head requirements. For this problem, 16 possible candidate
diameters are available including 15 commercially available
diameters and the ‘zero diameter-zero unit cost’ option.
Considering all 21 pipes for possible duplication, it results in
1621 possible designs.
312
Table 6. Data for Newyork city tunnel WDN
Nodal
Demand
(m3/h)
Minimum
Required
Nodal
Pressure
(m)
1
-205665
91.44
2
9419.317
77.72
180
3
9419.317
77.72
2530.49
180
4
8991.166
77.72
2621.95
180
5
8991.166
77.72
6
5823.17
180
6
8991.166
77.72
7
2926.83
132
7
8991.166
77.72
8
3810.98
132
8
8991.166
77.72
9
2926.83
180
9
17329.91
77.72
10
3414.63
204
10
101.941
77.72
11
4420.73
204
11
17329.91
77.72
12
3719.51
204
12
11937.25
77.72
13
7347.56
204
13
11937.25
77.72
14
6432.93
204
14
9419.317
77.72
15
4725.61
204
15
9419.317
77.72
16
8048.78
72
16
17329.91
79.25
Pipe
No.
Pipe
Length
(m)
Existing
Pipe
Diameter
(inch)
Node
No.
1
3536.59
180
2
6036.59
180
3
2225.61
4
5
17
9512.2
72
17
5861.588
83.15
18
7317.07
60
18
11937.25
77.72
19
4390.24
60
19
11937.25
77.72
20
11707.32
60
20
17329.91
77.72
21
8048.78
72
Table 7. Commercially available pipe diameters and unit cost
of pipe for Newyork city tunnel WDN
Sl.No.
Pipe Diameter
Unit Cost of Pipe
1
2
3
4
5
6
7
8
9
(inch)
36
48
60
72
84
96
108
120
132
(mm)
914.4
1219.2
1524
1828.8
2133.6
2438.4
2743.2
3048
3352.8
($/foot)
93.5
134
176
221
267
316
365
417
469
($/metre)
306.7
439.6
577.4
725
875.9
1036.7
1197.5
1368.1
1538.7
10
144
3657.6
522
1712.6
11
12
13
14
15
156
168
180
192
204
3962.4
4267.2
4572
4876.8
5181.6
577
632
689
746
804
1893
2073.4
2260.5
2447.5
2637.7
1) Model Run and Output for Case Study II
Figure 2. Layout of Newyork City Tunnel WDN
The 21 existing pipes are considered as such and 21 parallel
pipes for all the 21 pipes with 16 candidate diameters. At the
start of the iteration, it is assumed that all the potential
alternatives have equal probability of
selection
(i.e.,P0,r=1/16), since there are 16 candidate diameters
including ‘zero diameter- zero unit cost’ option. While using
Shibu and Reddy
313
EPANET, to avoid problems with consideration of zero
diameter pipes, negligibly small diameter (i.e., 0.0001 mm)
with zero unit cost is considered. The performance function
used for solving the model is given by (13).
As the iteration begins, some of the candidate diameters
becomes superior to the others based on the performance
values and their probability increases while for others gets
reduced. This step by step iterative procedure for the
modification to the probability of candidate diameters will
continue until they reach the stopping criteria of
approximately ones and zeros in the final iteration. At the end
of the final iteration, only 21(i.e., total number of pipes)
candidate diameters will be having probability equal to one
which forms the optimal solution set, and all the rest will be
having a probability equal to zero. The stopping criteria is
arrived in 36,000 function evaluations with smoothing
parameter α = 0.35 and PN =10000000. The output of the
model run for Newyork City Tunnel WDN is given in Table 8
and also compared with the past studies, which is given in
Table 9.
Table 8. Cross Entropy Model output for Newyork city tunnel
WDN
Pipe
No.
Pipe
Length
(m)
Existing
Pipe
Diamete
r
(inch)
Parallel
Pipe
Diameter
(inch)
Node
No.
Available
Nodal
Pressure
(m)
Minimum
nodal
Pressure
required
(m)
1
3536.59
180
0
1
91.44
91.44
2
6036.59
180
0
2
89.6743
77.72
Table 9. Comparison of optimal outputs obtained by various
approaches for Newyork city tunnel WDN
Diameter of parallel pipe (inch)
Pipe
No.
Existing
Pipe
Diameter
(inch)
Improved GA
[5]
Messy GA
[6]
1
180
0
0
Cross
Entropy
Method
0
2
180
0
0
0
3
180
0
0
0
4
180
0
0
0
5
180
0
0
0
6
180
0
0
0
7
132
0
144
144
8
132
0
0
0
9
180
0
0
0
10
204
0
0
0
11
204
0
0
0
12
204
0
0
0
13
204
0
0
0
14
204
0
0
0
15
204
120
0
0
16
72
84
96
96
17
72
96
96
96
18
60
84
84
84
19
60
72
72
72
20
60
0
0
0
21
72
72
72
72
Cost (in million $)
38.8
38.64
38.64
3
2225.61
180
0
3
87.2179
77.72
4
2530.49
180
0
4
86.4983
77.72
5
2621.95
180
0
5
85.861
77.72
6
5823.17
180
0
6
85.3664
77.72
No. of function
evaluations
96,750
37,186
36,000
7
2926.83
132
144
7
84.5863
77.72
Feasibility
Feasible
Feasible
Feasible
8
3810.98
132
0
8
84.328
77.72
9
2926.83
180
0
9
83.4469
77.72
10
3414.63
204
0
10
83.4373
77.72
11
4420.73
204
0
11
83.4745
77.72
12
3719.51
204
0
12
83.8627
77.72
13
7347.56
204
0
13
84.7651
77.72
14
6432.93
204
0
14
87.04
77.72
15
4725.61
204
0
15
89.4058
77.72
16
8048.78
72
96
16
79.2747
79.25
17
9512.2
72
96
17
83.1702
83.15
18
7317.07
60
84
18
79.6084
77.72
19
4390.24
60
72
19
77.7403
77.72
20
11707.3
60
0
20
79.4684
77.72
21
8048.78
72
72
The optimal solution obtained in the present study by using
the Cross entropy method is satisfying the minimum pressure
head requirement at all the nodes, and is resulting in minimum
cost.
On comparing the results of the CE method for Newyork
City Tunnel WDN with the results of past studies on the same
WDN, it is found that the number of parallel pipes to be added
is nearly same with only slight difference in one of the parallel
pipe diameters with approximately same cost for providing
parallel pipes. The number of function evaluations taken for
producing the optimum solution is less than the other
approaches. Thus, the results obtained from present study
shows that the cross entropy method is effective and is well
suited for the optimal design of large network like Newyork
City Tunnel WDN, which involves capacity expansion in
terms of adding parallel pipes without disturbing the existing
pipes.
C. Case Study III: Bengali Camp Zone WDN
The Bengali Camp Zone WDN of Chandrapur city in
Maharashtra State, India is taken as a case study III. This is a
real WDN of Chandrapur water supply system, and whose
network details are shown in Figure 3.
This WDN is built to serve a new residential area in the city.
The network was designed as an extension to the original
WDN of Chandrapur city. The projected population for the
year 2040 of Bengali Camp zone and the peak factor adopted
for the design of WDN are 47126 and 3 respectively. The
Cross Entropy Optimization for Optimal Design of Water Distribution Networks
Bengali Camp zone WDN consists of 34 nodes, 38 pipes, and
is fed by gravity from a tank at a fixed head of 206 m. The
existing pipe diameters, and nodal pressure for the Bengali
Camp zone WDN are given in Table 10. The pipe details and
lengths are given in Table 11; nodal elevations and nodal
demands are given in Table 12. A Hazen-Williams constant of
140 is assumed for all the pipes. The system constraint is
minimum pressure requirement for all nodes is 11 m. Twelve
commercially available pipe diameters and their unit cost are
listed in Table 13. No velocity constraint is taken into account
for this network. The solution search space of Bengali Camp
zone WDN is 1238.
314
Table 11. Pipe details for Bengali camp zone WDN
Pipe
No.
Start
Node
End
Node
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
107
1
3
4
5
6
7
7
4
17
18
19
22
23
24
25
26
27
28
1
3
4
5
6
7
8
9
17
18
19
22
23
24
25
26
27
28
29
Pipe
Length
(m)
61
413
83
165
715
193
413
220
72
77
165
660
330
715
330
248
468
138
715
Pipe
No.
Start
Node
End
Node
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
29
24
28
6
11
13
33
13
6
10
22
15
18
20
2
16
17
5
6
30
31
32
11
13
33
14
14
10
15
12
12
20
2
21
5
16
6
7
Pipe
Length
(m)
385
275
165
138
248
303
193
330
330
165
770
248
220
275
220
83
165
715
193
Table 12. Node details for Bengali camp zone WDN
Node No.
Figure 3. Layout of Bengali Camp Zone WDN
Table 10. Pipe diameters, and nodal pressure as per existing
design for Bengali camp zone WDN
Pipe
No.
Existing
Pipe
Diameter
(mm)
Pipe
No.
Existing
Pipe
Diameter
(mm)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
500
500
500
300
100
100
150
150
450
450
450
450
450
450
400
400
400
350
300
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
250
200
150
200
200
100
100
100
150
150
150
100
150
150
100
100
150
300
300
Cost of the WDN as
per Existing Design
(` )
25735031
Node
No.
Available
Nodal
Pressure
(m)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
10.98
11.66
17.33
18
18.26
14.63
16.43
20.12
16.13
15.03
14.8
16.64
14.17
16.25
11.51
18.27
17.97
Node
No.
Available
Nodal
Pressure
(m)
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
107
18.75
18.89
19.29
19.64
20.69
19.41
22.48
24.22
27.18
25.11
24.88
24.46
21.94
21.48
25.06
15.95
18.75
-
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
107(Resvr)
Nodal
Elevation (m)
195
194
188.5
187.8
187.5
191
189.2
185.5
189.5
190.5
190.8
188.8
191.4
189.2
194
187.5
187.8
187
186.8
186.4
186
184.8
186
182.8
181
178
180
180.2
180.5
183
183.8
180
189.5
195
(LPS)
0
1.27
0
0.607
2.469
3.517
2.063
1.031
0.55
1.27
0.989
2.61
2.259
1.34
1.059
0.211
0.382
1.186
2.116
1.27
0.564
4.515
8.903
10.055
1.91
2.364
2.939
10.309
11.145
3.901
0.91
1.673
1.27
-
Base Demand
(m3/h)
0
4.572
0
2.1852
8.8884
12.6612
7.4268
3.7116
1.98
4.572
3.5604
9.396
8.1324
4.824
3.8124
0.7596
1.3752
4.2696
7.6176
4.572
2.0304
16.254
32.0508
36.198
6.876
8.5104
10.5804
37.1124
40.122
14.0436
3.276
6.0228
4.572
-
Shibu and Reddy
315
Table 13. Commercially available pipe diameters and unit
cost of pipe for Bengali camp zone WDN
inch
mm
1
4
100
Unit Cost of
Pipe
(`/m length)
860
2
3
4
5
6
7
8
9
10
11
12
6
8
10
12
14
16
18
20
24
28
32
150
200
250
300
350
400
450
500
600
700
800
1077
1374
1840
2333
2885
3442
4142
4826
6375
8141
10161
Available Pipe Diameter
Sl. No.
1) Model Run and Output for Case Study III
At the start of the run, it is assumed that all the candidate
diameters have equal probability of selection (i.e., P0,r=1/12).
The performance function used for solving the model is given
by (13). As the iteration progresses, some of the candidate
diameters become superior to the others based on the
performance values and their probability increases, while for
others the probability gets reduced. This step-by-step iterative
procedure for updating the probability of selecting a candidate
diameter for each pipe will continue until they satisfy the
stopping criteria. At the end, the probability of selecting a
option for a pipe will be approximately equal to ones and
zeros. This means that only 38 decisions (i.e., total number of
pipes) will be having probability equal to one which forms the
optimal solution set, and the remaining will be having a
probability equal to zero. The stopping criteria is arrived in
38,400 objective function evaluations with smoothing
parameter α = 0.35 and PN =108. The output of the model run
for Bengali Camp Zone WDN is given in Table 14.
Table 14. Cross Entropy Model Output For Bengali Camp
Zone WDN
Pipe
No.
Optimum
Pipe
Diameter
(mm)
Pipe
No.
1
600
20
2
600
21
3
600
22
4
300
23
5
100
24
6
150
25
7
100
26
8
100
27
9
500
28
10
500
29
11
500
30
12
450
31
13
450
32
14
400
33
15
400
34
16
350
35
17
400
36
18
350
37
19
300
38
Optimum Cost (`)
Optimum
Pipe
Diameter
(mm)
200
100
100
150
200
100
100
100
150
150
100
100
150
150
100
100
100
300
150
25235630
Node
No.
Available
Nodal
pressure
(m)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
10.99
11.8
17.43
18.19
18.38
14.74
16.52
20.12
16.21
15.11
14.82
16.65
14.19
16.27
11.58
18.38
18.10
Node
No.
Availabl
e Nodal
pressure
(m)
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
107
18.89
19.05
19.43
19.78
20.86
19.57
22.55
24.29
27.20
25.13
24.90
24.48
21.94
21.49
25.00
15.97
18.89
-
On comparing the results of the CE method for Bengali
Camp zone WDN with the existing design, it is noticed that the
optimal solutions of CE is better than existing design, resulting
in 1.94% lesser cost than the existing design. The solution is
obtained in 38,400 function evaluations. Also the minimum
nodal pressure requirements are well satisfied. The results of
present study amply demonstrate that the CE method is an
effective optimization method for WDN and has capability to
handle larger number of discrete decision variables and
various constraints. Thus, CE method is well suited for
optimal design of larger water supply networks.
V. Conclusions
This study presented Cross Entropy (CE) method for
solving water distribution network optimization problems.
For hydraulic simulation of WDNs, EPANET tool kit is
adopted and carried out simulation-optimization modeling for
design of WDNs. Initially, the CE method is applied for two
benchmark WDN design problems, namely Hanoi WDN and
Newyork city tunnel WDN. To evaluate the performance of
CE optimization method, the results are compared with the
past studies and it is found that the CE method is giving good
quality optimal solutions in a few number of objective function
evaluations. The results also demonstrated that the CE method
can be used effectively for optimal design of new WDN as
well as for rehabilitation of existing WDN (i.e., for capacity
expansion of WDNs, in terms of adding parallel pipes without
disturbing the existing pipes). It is also found that the CE
method is capable of handling larger number of discrete
decision variables and different types of constraints. After
successful validation to standard WDNs, the CE method is
applied to a real WDN in India and the results are compared
with the existing solutions. It is found that CE method is giving
minimum cost solutions (i.e., good quality optimal solutions)
in quicker time (i.e., rapid convergence to optimum). Thus, the
study concludes that the cross entropy optimization method is
an effective optimization method for solving WDN problems,
and which can be applied for optimal design of any practical
WDN problems.
References
[1] Gupta, I., Gupta, A., and Khanna, P., 1999, “Genetic
algorithm for optimization of water distribution systems,”
Environmental Modelling & software, 24(4), pp.437-446.
[2] Goldberg, D.E., and Kuo, C.H., 1987, “Genetic
algorithms in pipeline optimization,” Journal of Computing in
Civil Engineering, 1(2), pp. 129-141.
[3] Simpson, A. R., Dandy, G. C., and Murphy, L. J.,1994,
“Genetic algorithms compared to other techniques for pipe
optimization,” J. Water Resour. Plang. and Mgmt., ASCE,
120(4), pp. 423-443.
[4] Savic, D.A. and Walters, G.A., 1997, “Genetic
algorithms for least-cost design of water distribution
networks,” J. Water Resour. Plang and Mgmt., ASCE, 123(2),
pp. 67-77.
Cross Entropy Optimization for Optimal Design of Water Distribution Networks
[5] Dandy, G.C., Simpson A.R., and Murphy L.J., 1996,
“An improved genetic algorithm for pipe network
optimization,” Water Resour. Res., 32(2), pp. 449 - 458.
[6] Wu, Z. Y., and Simpson, A. R., 2001, “Competent
genetic-evolutionary optimization of water distribution
systems,” Journal of Computing in Civil Engineering, 15(2),
pp. 89-101.
[7] Cunha, M.D.C., and Sousa, J., 1999, “Water distribution
network design optimization: simulated annealing approach,”
J. Water Resour. Plang. and Mgmt., 125(4), pp. 215-221.
[8] Eusuff, M. M., and Lansey, K.E., 2003, “Optimization of
water distribution network design using the shuffled frog
leaping algorithm,” J. Water Resour. Plang. and Mgmt.,
ASCE, 129(3), pp. 210-225.
[9] Maier, H. R., Simpson, A.R., Zecchin, A. C., Foong,
W.K.,Phang, K.Y., Seah, H.Y., and Tan, C.L.,2003, “Ant
Colony Optimization for Design of Water Distribution
Systems,” J. Water Resour. Plang. and Mgmt., ASCE, 129(3),
pp. 200-209.
[10] Shibu, A. Reddy, M.J., 2011, “Least cost design of water
distribution network by Cross entropy optimization,” World
Congress on Information and Communication Technologies
(WICT), vol.,no., pp..302-306,11-14Dec.2011
doi: 10.1109/WICT.2011.6141262
[11] Rubinstein, R.Y., 1997, “Optimization of computer
simulation models with rare events,” European Journal of
Operations Research, 99, pp. 89-112.
[12] Shannon, C.E., 1948, “A Mathematical theory of
communication,” Bell System Tech. Journal, 27, pp. 379 423.
[13] Kullback, S, and Leibler, R.A., 1951, “On information
and sufficiency,” Ann. Math. Statics, 22, pp. 79-86.
[14] Fujiwara, O., and Khang, D.B., 1990, “A two-phase
decomposition method for optimal design of looped water
distribution networks,” Water Resour. Res., 26(4), pp.
539-549.
[15] Dijk, M. V., Vuuren, S. V., and Van, Z., 2006,
“Optimizing water distribution systems using a weighted
penalty in a genetic algorithm,” ISSN, Water SA, 34(5), pp.
378 - 478.
[16] Vairavamoorthy, K., and Ali, M., 2000, “Optimal design
of water distribution systems using genetic algorithms,”
Computer-Aided Civil and Infrastructure Engineering,
Blackwell, 15(4), pp. 374–382.
[17] Schaake, J. and Lai, D., 1969, “Linear programming
and dynamic programming applications to water distribution
network design,”, Rep. No. 116, Dept. of Civil Engineering,
Massachusetts Institute of Technology, Cambridge, Mass.
Author Biographies
1
Shibu A. is Research Scholar in the Department of Civil
Engineering, Indian Institute of Technology Bombay, India. He
completed his M.Tech.(Hydraulics Engineering) in 2000 from
University of Kerala, India. His field of research is water distribution
system modeling under uncertainty by using evolutionary
techniques.
2
M. Janga Reddy is Assistant Professor in the Department of Civil
Engineering, Indian Institute of Technology, Bombay, India. He
316
completed his Ph.D. in 2006 from Indian Institute of Science,
Bangalore, India. His research interests include water resource
systems: development of simulation and optimization models;
optimal operations of single and multi reservoir systems; irrigation;
hydropower; flood control;
water distribution systems;
multi-criterion decision making.