 Open Access
 Total Downloads : 23
 Authors : Santosh Kumar, Praveen Kumar Navin, Yogesh Prakash Mathur
 Paper ID : IJERTCONV6IS11005
 Volume & Issue : RTCEC – 2018 (Volume 6 – Issue 11)
 Published (First Online): 31052018
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Optimization of Sewerage System Using Simulated Annealing
Santosh Kumar
Reaserch Scholer, Dept. of Civil Engg. Malaviya National Institute of Technology, Jaipur, India
Praveen Kumar Navin
Assistant Professor, Dept. of Civil Engg. Vivekananda Institute of Technology, Jaipur, India
Yogesh Prakash Mathur
Professor, Dept. of Civil Engg.
Malaviya National Institute of Technology, Jaipur, India
Abstract Sewer networks are an important part of the infrastructure of any society. Since, the investment needed for construction and maintenance of these large scale networks is so huge and, thus any saving in the cost of these networks may result in considerable reduction of total construction cost. This study focuses on the issues of the design of sewer networks.In this paper, a new and powerful stochastic method, called Simulated Annealing (SA) is adopted for solving the sewer network optimization problem. Simulated Annealing (SA) is a probabilistic method proposed for finding the global minimum of a cost function that may possess several local minima. A sewer network is considered to show the Simulated Annealing algorithm performance, and the results are presented. The results show the capability of the proposed technique for optimally solving the problems of sewer networks.
Keywords – Sewer network, Simulated Annealing, Optimal sewer design

INTRODUCTION
Sewerage or the wastewater system is the system of pipes used to collect and carry rain, wastewater and trade waste away for treatment and disposal. Sewage collection and disposal systems transport sewage through cities and other inhabited areas to sewage treatment plants to protect public health and prevent disease. The design of a sewerage system in general involves selection of a suitable combination of pipe sizes and slopes so as to ensure adequate capacity for peak flows and adequate self cleansing velocities at minimum flow. In a conventional design procedure, efforts are made to analyze several alternative systems (each meeting the physical and hydraulic requirements) and the least cost system is
such as genetic algorithms [8, 9], ant colony optimization algorithms [10, 11], cellular automata [12] and particle swarm optimization algorithms [13], have received significant consideration in sewer network design problems. Recently, Ostadrahimi et al. [14] used multi swarm particle swarm optimization (MSPSO) approach to present a set of operation rules for a multireservoir system. Haghighi and Bakhshipour [15] developed an adaptive genetic algorithm. Therefore, every chromosome, consisting of sewer slopes, diameters, and pump indicators, is a feasible design. The adaptive decoding scheme is set up based on the sewer design criteria and open channel hydraulics. Using the adaptive GA, all the sewer systems constraints are systematically satisfied, and there is no need to discard or repair infeasible chromosomes or even apply penalty factors to the cost function. Moeini and Afshar [16] used tree growing algorithm (TGA) for efficiently solving the sewer network layouts out of the base network while the ACOA is used for optimally determining the cover depths of the constructed layout. Karovic and Mays [17] used simulated annealing within Microsoft Excel to sewer Ssystem design optimization.
In this paper, simulated annealing algorithm is applied to get optimal sewer network component sizes of a predetermined layout.

SEWER NETWORK DESIGN PROBLEM

Sewer Hydraulics
In circular sewer steadystate flow is described by the continuity principle (Q= VA) and Mannings equation which is
selected. Obviously, the outcome of such a procedure depends to a large extent on the designer experience and efforts. It is practically almost impossible to incorporate all
v 1 R1/ 3S1/ 2
n
(1)
feasible design alternatives, and an optimal solution is not necessarily reached. Only a resources to computer oriented optimal designing may be a solution.
Many optimization techniques have been applied and developed for the optimal design of sewer networks, such as linear programming [1, 2], nonlinear programming [3, 4] anddynamic programming [57]. Evolutionary strategies,
where Q = sewage flow rate, V = velocity of sewage flow, A = crosssectional flow area, R = hydraulic mean depth, n
= Mannings coefficient and S = slope of the sewer. Common, partially full specifications for circular sewer sections are also determined from the following equations:
D 2 2
d 1 1 cos
r D sin
(2)
(3)
temperature in physical annealing), then find the initial value of the objective function. While these choices of
starting solution and temperature are unique to each application, SA is normally fairly insensitive to the starting
4
a
D2 sin 8
(4)
conditions. In the application to structural optimization, this step establishes the initial physical characteristics of the structural components, ensures that all constraints are met, and determines the initial weight of the structure.
D = sewer diameter, = the central angle in radian and
(d/D) = proportional water depth, a = flow area while running partially full,r = hydraulic mean radius.

Design Constraints
For a given network, the optimal sewer design is defined as a set of pipe diameters, slopes and excavation depths which satisfies all the constraints. Typical constraints of sewer network design are:

Each pipe flow velocity should be greater than the minimum permissible velocityfor self cleaning capability and less than the maximum permissible velocity for preventing from scouring.

Flow depth ratio: wastewater depth ratio of the pipe should be less than 0.8.

Choosing pipe diameters from the commercial list.

Maintaining the minimum cover depth to avoid damage to thesewer line and adequate fall for house connections. The minimum cover depthof 0.9 m and maximum cover depth of 5.0 m has been adopted.

For each manhole, assigning the outlet pipe diameter equal to or greater than the upstream inlet pipes.

The optimal design of a sewer system for a given layout is to determine the sewer diameters, cover depths and sewer slopes of the network in order to minimize the total cost of the sewer system. The objective function can be stated as
n
The term temperature is a holdover from the physical process of annealing, where it refers to the actual heat content of a casting. In simulated annealing, the temperature is a parameter that controls the probability of accepting a new solution that is "worse" than the old one. The higher the temperature, the greater the chance of accepting a "worse" solution. This probability of accepting a worse solution is the feature that allows SA to leave a local minimum and continue to search for the global minimum.

The second step in the algorithm is to randomly perturb the system. In explaining combinatorial optimization, Kirkpatrick, et.al. [18]described a random search method that accepts only lower values of the objective function at each iteration. It usually gets stuck in the local minimum closest to the starting point. This algorithm is often called the Greedy Algorithm because, in its "greed" to find any optimum, it will likely miss the global optimum and accept a local instead (McLaughlin, 1989:25). In 1985, Cerny [19] presented a Monte Carlo algorithm to fid approximate solutions to the traveling salesman problem. "The algorithm generates randomly the permutations of the stations of the traveling salesman trip, with a probability depending on the length of the corresponding route. This offers one method for generating random perturbations to a
Minimize C (TCi PCi )
i1
(5)
system. In structural optimization, this step corresponds to a random change in the physical dimension of one or more
Where i = 1,, n (total number of sewers), TCOSTi (total cost) = (Cost of seweri + Cost of manholei + Cost of earth worki) and PCi = penalty cost (it is assigned if the design constraint is not satisfied).


SIMULATED ANNEALING (SA) Simulated Annealing (SA) is a fairly new process for
numerical optimization of many classes of problems. It is modeled after the centuriesold annealing process for metal and glass castings. Manufacturers anneal castings to make them tougher, by reducing their internal energy (McLaughlin, 1989) between Simulated Annealing and the physical process of annealing. In each case, a system of many variables is minimized. SA uses many steps in a random search to find the optimum of the system. Other random search algorithms are prone to selecting the first local optimum encountered. However, SA has a feature that helps it find the global optimum rather than a local optimum. The many steps required in SA are possible with modern computers, and the more capable computers become, the more useful SA will be.

Procedure of Simulated annealing algorithm

The first step in the algorithm is to choose a starting configuration and control parameter (analogous to

components.

The third step is to evaluate the new solution. The specific mechanics of this evaluation depend on the application. For structural optimization, this step determines the total weight of the structure with the new dimensions.

In the fourth step, accept or reject the new solution. If the new solution gives a lower value for the objective function, accept it. However, if the new solution gives a higher value, consider accepting it. This possibility of accepting the "worse" solution gives the SA algorithm the ability to leave a local optimum, and continue to search for the global optimum. This is the key feature that sets SA apart from other random search algorithms. From statistical mechanics, Kirkpatrick, et.al. [18] described the Metropolis procedure to overcome the Greedy Algorithm's problem of stalling at a local optimum. The Metropolis procedure from statistical mechanics provides a generalization of iterative improvement in which controlled uphill steps can also be incorporated in the search for a better solution [18]. This makes it possible for the algorithm to climb out of a local minimum and find a better local minimum, or the global minimum. Control for the uphill steps is given by the Boltzmann distribution:
Pr (E)
1
Z(T)
E
K T
exp B
(6)
function reaches a stable value for a certain number of iterations [20].

If there is a certain target value of the function (a
Where, () is the probability of accepting the uphill step,
() is a normalizing factor depending on the assigned temperature(), is the average energy level, and is the Boltzmann constant. The value of is a natural constant, determined by experimentation, which adjusts the shape of the Boltzmann distribution to model the physical annealing process. It normally would not represent a valid constant in the SA process, but a different constant may be appropriate. For a given change in temperature, when the temperature is high, the probability of accepting an uphill step is high. As the temperature is reduced, the probability of accepting the uphill step is reduced.


The fifth step in the algorithm is to iterate at a given temperature and, when the system is at a stable average configuration for that temperature, reduces the temperature according to the annealing schedule. This schedule for reducing the temperature is critical to the success of either real or simulated annealing. According to Cerny experiments are done by careful annealing, first melting the substance, then lowering the temperature slowly, and spending a long time at temperatures in the vicinity of the freezing point. If this is not done, and the substance is allowed to get out of equilibrium, the resulting crystal will have many defects [19]. Quenching is the process of deliberately reducing the temperature quickly, without allowing the substance to reach equilibrium. This degenerates the SA algorithm to an ordinary random search like the Greedy Algorithm. In annealing, this process creates a brittle casting, but it is much quicker, and in some cases may be preferred to the slow annealing process. Quenching is not normally used in SA. To get the lowest possible cost with SA, the annealing schedule must allow the system to reach steadystate at each temperature. On the other hand, spending too much time at a given temperature wastes computer resources. So, the annealing schedule must allow the system to stabilize before changing temperature, and then change promptly.
The cooling schedule is often found by trial and error Brooks and Verdini [20]. However, Basu and Fraser [21] suggest that it may be cost effective to spend up to 80 percent of the total CPU time to establish the best cooling schedule. Collins et.al. [22] listed five different schemes for controlling the temperature, T:

A constant value of T; T(t) = C

An arithmetic function of T; T(t) = T(t – 1) C

A geometric function; T(t) = a(t)T(t – 1)

An inverse; T(t) = C/(1 + ta)

A logarithmic function; T(t) = C/In(1 + t)


The last step in the SA algorithm is to iterate until the stopping criteria is met. Several classes of stopping criteria can be used [22].

In the simplest criteria, a fixed amount of CPU time is allocated, and the process stops when the time runs out [20].

Another approach is to compare the value of the objective function at each iteration with the value at previous iterations. Under this criteria, stop when the
known or estimated minimum), stop when the configuration meets the target [20].

When the algorithm is near the optimum the ratio of accepted configurations to total configurations will become very small. The algorithm can stop when this ratio reaches a predetermined value [23].
If none of the other criteria are met, stop when the temperature reaches a value near zero [22]. At this point the algorithm degenerates to a random search, and the cost of further annealing should be compared to the benefit that might be gained. When the correct stopping criteria are met, the algorithm will have a solution closer to the global optimum.
According to the abovementioned steps, a possible structure of the Simulated Annealing algorithm is shown in fig. 1.
Fig. 1. Flow chart of Simulated Annealing Algorithm



OPTIMIZATION OF SEWER NETWORK The sewer network example (Banjaran sewer network,
Laxmangarh, Rajasthan, India) is considered to check the
aboveproposed approach. The Banjaran sewer network as shown in Fig. 2 consists of 105 manholes, 104 pipes and STP is located at Node Number 0.
The following steps were used to optimize the component sizing of sewer system using the Simulated Annealing algorithm:

Start with the first link (I=1) of the first iteration(ITN=1)

Calculate values of Hydraulic Mean Depth, Velocity, Depth of flow, and Discharge in partial flow condition.

Calculate invert levels of upstream and downstream node of a particular link

Calculate no of manholes, depth of excavation and earthwork.

Calculate the total cost of the sewer network (TCOST)

Add the respective penalty cost (PC) in TCOST where constraints are violated.

Calculate feasible solution using SA

Check solutions obtained are feasible or not.

If feasible solution is not obtained repeat the process.

If feasible solution is obtained, then take output.

End.

Calculate cost of sewer, cost of manholes and cost of earthwork.
The cost of pipe (RCC NP4 class), manhole and earth work was taken from theIntegrated schedule of Rates, RUIDP [24].
Fig. 2.Banjaran sewer network


RESULTS
The performance of the proposed Simulated Annealing procedure for optimization of the sewer system is now tested against Banjaran sewer network. The result exhibit a final total cost ofRs.8.505 Ã— 106. 100000 evaluations were done for a system having 100 iterations for each evaluation. Then after accepting the higher as well as lower
values of the function the global best solutions were achieved. The pipe diameter and slopes have been shown for the best solution. Accordingly the total cost of the sewerage system has been shown in the results. Table 1 shows the solution obtained by Simulated Annealing approach.
Table 1 Results of the Banjaran sewer network obtained by Simulated Annealing
Pipe no.
Node no.
Length (m)
Design
flow (m/s)
Diameter (mm)
Slope (1 in)
vp (m/s)
d/D
Cover depths (m)
Up
Down
Up
Down
24
23
22
30
0.0001
200
250
0.17
0.05
1.12
1.422
39
37
36
28
0.0002
200
250
0.19
0.06
1.426
1.12
41
38
39
20
0.0001
200
80
0.2
0.03
1.14
1.12
42
39
40
24
0.0001
200
250
0.18
0.05
1.434
1.12
44
40
42
28
0.0003
200
250
0.24
0.08
1.12
6.487
45
41
28
29
0.0001
200
250
0.16
0.04
1.12
1.338
46
42
35
28
0.0004
200
250
0.26
0.09
6.487
2.182
47
43
44
30
0.0001
200
60
0.26
0.03
1.12
1.184
48
44
27
38
0.0002
200
250
0.21
0.07
1.184
1.538
52
49
48
35
0.0001
200
250
0.17
0.05
1.12
1.489
54
50
51
35
0.0001
200
250
0.17
0.05
1.125
1.12
55
51
52
34
0.0002
200
250
0.21
0.07
1.12
1.343
56
52
53
30
0.0621
300
200
1.09
0.73
1.343
1.781
57
53
54
35
0.0622
300
200
1.09
0.73
1.781
1.969
69
64
63
30
0.0001
200
250
0.17
0.05
1.12
1.541
83
69
68
30
0.0001
200
200
0.18
0.05
1.12
1.126
80
70
67
30
0.0001
200
250
0.17
0.05
1.12
1.259
77
71
66
30
0.0001
200
250
0.17
0.05
1.12
1.373
74
72
65
30
0.0001
200
250
0.17
0.05
1.12
1.164
107
87
88
30
0.0001
200
250
0.16
0.05
1.12
1.415
102
88
83
33
0.0002
200
250
0.2
0.07
1.415
2.981
117
97
96
16
0.0002
200
250
0.21
0.07
1.12
1.593
120
98
99
30
0.0001
200
250
0.16
0.05
1.12
1.333
127
99
104
34
0.0002
200
250
0.21
0.07
1.333
1.297
122
100
101
30
0.0001
200
250
0.16
0.05
1.12
1.3
123
101
102
26
0.0002
200
250
0.2
0.06
1.3
1.362
126
104
103
30
0.0003
200
250
0.24
0.08
1.297
1.312
23
22
21
30
0.0002
200
250
0.21
0.07
1.422
1.701
36
36
35
27
0.0002
200
250
0.21
0.07
1.12
1.388
51
48
47
35
0.0002
200
250
0.21
0.07
1.489
1.832
71
63
79
30
0.0004
200
250
0.26
0.09
1.541
1.371
75
65
84
30
0.0004
200
250
0.26
0.09
1.164
1.498
78
66
89
30
0.0004
200
250
0.26
0.1
1.373
1.57
97
79
80
30
0.0005
200
250
0.27
0.1
1.519
1.12
98
80
81
17
0.0005
200
250
0.28
0.11
1.12
1.266
99
81
82
35
0.0007
200
250
0.31
0.13
1.266
1.493
101
82
83
30
0.0008
200
250
0.32
0.14
1.493
2.667
95
83
77
35
0.0011
200
250
0.36
0.16
2.981
2.849
103
84
85
30
0.0005
200
250
0.27
0.11
1.846
1.12
104
85
86
17
0.0005
200
250
0.28
0.11
1.12
1.238
106
86
91
35
0.0007
200
80
0.46
0.1
1.266
1.12
109
89
90
30
0.0005
200
80
0.4
0.08
1.57
1.3
110
90
91
18
0.0005
200
250
0.28
0.11
1.303
1.12
111
91
92
35
0.0015
200
70
0.6
0.13
1.12
1.624
113
92
93
30
0.0016
200
70
0.62
0.14
1.624
2.214
114
93
94
30
0.0017
200
70
0.63
0.14
2.214
2.919
115
94
21
29
0.0018
200
80
0.61
0.15
2.919
3.257
116
96
95
30
0.0003
200
250
0.22
0.08
1.593
1.637
124
103
102
33
0.0004
200
250
0.26
0.09
1.312
1.374
22
21
20
12
0.002
200
80
0.64
0.16
3.257
3.451
37
35
34
30
0.0007
200
250
0.31
0.13
2.182
2.433
50
47
46
27
0.0003
200
250
0.24
0.09
1.832
2.042
81
95
67
30
0.0003
200
250
0.25
0.09
1.637
1.855
125
102
57
29
0.0007
200
250
0.31
0.13
1.374
1.407
4
20
4
30
0.0021
200
80
0.64
0.16
3.451
3.853
38
34
30
18
0.0008
200
250
0.32
0.13
2.433
2.288
49
46
45
10
0.0011
200
250
0.36
0.16
2.042
2.131
79
67
68
34
0.0007
200
250
0.31
0.13
1.855
1.561
82
68
54
24
0.0011
200
250
0.35
0.16
1.561
1.819
68
45
62
36
0.0063
200
200
0.64
0.37
2.131
2.888
58
54
55
30
0.0634
300
200
1.1
0.75
1.969
1.858
59
55
56
30
0.0635
300
200
1.1
0.75
1.858
1.971
60
56
57
15
0.0635
300
200
1.1
0.75
1.971
1.913
61
57
58
30
0.0643
300
200
1.1
0.76
1.913
1.792
62
58
59
30
0.0644
300
200
1.1
0.76
1.792
2.257
63
59
60
30
0.0645
300
200
1.1
0.76
2.257
2.555
64
60
24
34
0.0646
300
200
1.1
0.76
2.555
2.77
65
62
61
30
0.0064
200
200
0.64
0.37
2.888
2.9
26
24
25
30
0.0647
300
200
1.1
0.76
2.77
2.93
27
25
26
30
0.0648
300
200
1.1
0.76
2.93
2.866
28
26
27
32
0.0649
300
200
1.1
0.76
2.866
2.366
29
27
28
32
0.0652
300
200
1.1
0.76
2.366
1.939
30
28
29
30
0.0654
300
200
1.1
0.77
1.939
2.19
31
29
30
25
0.0655
300
200
1.1
0.77
2.19
3.038
32
30
31
30
0.0663
300
200
1.1
0.78
3.038
3.197
33
31
32
30
0.0664
350
250
1.04
0.63
3.197
2.744
34
32
33
30
0.0665
350
250
1.04
0.63
2.744
2.02
35
33
17
20
0.0666
350
250
1.04
0.63
2.02
1.888
67
61
73
30
0.0067
200
200
0.65
0.38
2.9
2.97
89
73
74
30
0.0068
200
200
0.65
0.38
2.97
3.05
90
74
75
17
0.0068
200
250
0.6
0.41
3.05
2.379
91
75
76
35
0.0077
200
250
0.62
0.43
2.379
2.819
93
76
77
30
0.0078
200
250
0.62
0.43
2.819
3.817
94
77
78
30
0.0091
200
250
0.65
0.47
3.817
4.304
96
78
2
28
0.0092
200
250
0.65
0.47
4.304
3.154
1
2
3
30
0.0153
200
250
0.72
0.63
3.154
3.336
2
3
4
30
0.0154
200
250
0.72
0.64
3.336
2.991
3
4
5
30
0.0176
200
250
0.74
0.7
3.853
3.58
5
5
6
30
0.0177
200
250
0.74
0.7
3.58
3.593
6
6
7
30
0.0178
200
250
0.74
0.7
3.593
3.044
7
7
105
30
0.0179
200
250
0.74
0.71
3.044
3.132
128
105
8
7
0.0179
200
250
0.74
0.71
3.132
3.159
8
8
9
28
0.0192
200
250
0.75
0.74
3.159
3.595
10
9
10
28
0.0193
200
250
0.75
0.75
3.595
2.943
11
10
11
30
0.0195
200
250
0.75
0.75
2.943
3.105
13
11
12
22
0.0195
200
250
0.75
0.76
3.105
2.384
14
12
13
30
0.0266
250
250
0.83
0.62
2.384
2.078
15
13
14
21
0.0266
250
250
0.83
0.62
2.078
1.917
16
14
15
30
0.0267
250
250
0.83
0.62
1.917
1.737
17
15
16
30
0.0268
250
250
0.83
0.62
1.737
2.039
18
16
17
28
0.0269
250
250
0.83
0.62
2.039
2.601
19
17
18
30
0.0936
400
350
0.99
0.69
2.601
3.22
20
18
19
30
0.0936
400
350
0.99
0.69
3.22
3.531
21
19
1
26
0.0937
400
350
0.99
0.69
3.531
3.734

CONCLUSION
The optimization technique adopted in this work proved to be successful in optimal designing of the sewerage network. In this study, the Simulated Annealing (SA) method of optimization a stochastic approach was applied to the problem of finding optimal pipe diameters and slopes for the conjunctive leastcost design and operation of a sewerage system network. Using the SA approach, the total cost of the sewer system was Rs. 8.505
Ã— 106. The results indicated that the proposed approach is very promising and reliable, that must be taken as the key alternative to solve the problem of optimal design of the sewer network.
REFERENCES

J. S. Dajani, R. S. Gemmell, and E. K. Morlok, Optimal Design of Urban Wastewater Collection Networks, J. Sanit. Eng. Div., vol. 98, no. 6, pp. 853867, 1972.

A. A. Elimam, C. Charalambous, and F. H. Ghobrial, Optimum Design of Large Sewer Networks, J. Environ. Eng., vol. 115, no. 6, pp. 11711190, Dec. 1989.

R. K. Price, Design of storm water sewers for minimum construction cost, in 1st International Conference on Urban Strom Drainage, 1978, pp. 636647.

P. K. Swamee, Design of Sewer Line, J. Environ. Eng., vol. 127, no. 9, pp. 776781, Sep. 2001.

S. Walsh and L. C. Brown, Least Cost Method for Sewer Design, J. Environ. Eng. Div., vol. 99, no. 3, pp. 333345, 1973.

G. A. Walters and A. B. Templeman, Nonoptimal dynamic programming algorithms in the design of minimum cost drainage systems, Eng. Optim., vol. 4, no. 3, pp. 139148, Oct. 1979.

G. Li and R. G. S. Matthew, New Approach for Optimization of Urban Drainage Systems, J. Environ. Eng., vol. 116, no. 5, pp. 927944, Sep. 1990.

G. A. Walters and T. Lohbeck, Optimal Layout of Tree Networks Using Genetic Algorithms, Eng. Optim., vol. 22, no. 1, pp. 2748, 1993.

M. H. Afshar, Application of a Genetic Algorithm to Storm Sewer Network Optimization, Sci. Iran., vol. 13, no. 3, pp. 234244, 2006.

M. H. Afshar, Partially constrained ant colony optimization algorithm for the solution of constrained optimization problems: Application to storm water network design, Adv. Water Resour., vol. 30, no. 4, pp. 954965, 2007.

M. H. Afshar, A parameter free Continuous Ant Colony Optimization Algorithm for the optimal design of storm sewer networks: Constrained and unconstrained approach, Adv. Eng. Softw., vol. 41, no. 2, pp. 188195, 2010.

Y. Guo, G. A. Walters, S. T. Khu, and E. Keedwell, A novel cellular automata based approach to storm sewer design, Eng. Optim., vol. 39, no. 3, pp. 345364, 2007.

J. Izquierdo, I. Montalvo, R. PÃ©rez, and V. S. Fuertes, Design optimization of wastewater collection networks by PSO, Comput. Math. with Appl., vol. 56, no. 3, pp. 777784, 2008.

L. Ostadrahimi, M. a. MariÃ±o, and A. Afshar, Multireservoir Operation Rules: Multiswarm PSObased Optimization Approach, Water Resour. Manag., vol. 26, no. 2, pp. 407427, 2012.

A. Haghighi and A. E. Bakhshipour, Optimization of Sewer Networks Using an Adaptive Genetic Algorithm, Water Resour. Manag., vol. 26, no. 12, pp. 34413456, 2012.

R. Moeini and M. H. Afshar, Sewer Network Design Optimization Problem Using Ant Colony Optimization Algorithm and Tree Growing Algorithm, in EVOLVEA Bridge between Probability, Set Oriented Numerics, and Evolutionary Computation IV, pp. 91 105, SpringerInternational Publishing (2013)

O. Karovic and L. W. Mays, Sewer System Design Using Simulated Annealing in Excel, Water Resour. Manag., vol. 28, no. 13, pp. 45514565, 2014.

S. Kirkpatrick,C.D. Gelatt and M.P. Vecchi, Optimization by simulated annealing Science, 220, 671680, 1983.

V. Cerny, A theromo dynamical approach to the traveling salesman problem: An efficient simulation algorithm, Journal of Optimization: Theory and Applications, 45, 4151, 1985.

D.G. Brooks and W.A. Verdini , Computational experience with generalized simulated annealing Over continuous variables, American Journal Mathematics and Management Sciences 8: 425 449, 1988.

A. Basu and L.N. Frazer, Rapid Determination of the Critical Temperature in Simulated Annealing Inversion, Science,249 4975, pp. 14091412 (1990).

N.E. Collins, R.W. Eglese andB.L. Golden,Simulated Annealing An Annotated Bibliography, American Journal of Mathematical and Management Sciences Volume. 8, 34 (1988)

L. Ingber, Simulated annealing: Practice versus theory, Mathematical and computer modeling. Vol. 18, pp. 2957, (1993).

Integrated Schedule of Rates: Rajasthan urban infrastructure development project (RUIDP),Government of Rajasthan (2013)