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Optimization of Sewerage System Using Simulated Annealing


Call for Papers Engineering Journal, May 2019

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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

  1. 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 multi-reservoir 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.

  2. SEWER NETWORK DESIGN PROBLEM

    1. Sewer Hydraulics

      In circular sewer steady-state 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 = cross-sectional 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.

    2. 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:

      1. 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.

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

      3. Choosing pipe diameters from the commercial list.

      4. 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.

      5. 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.

    1. 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).

  3. SIMULATED ANNEALING (SA) Simulated Annealing (SA) is a fairly new process for

    numerical optimization of many classes of problems. It is modeled after the centuries-old 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.

      1. Procedure of Simulated annealing algorithm

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

    components.

    1. 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.

    2. 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.

    3. 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 steady-state 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)

    4. 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 above-mentioned steps, a possible structure of the Simulated Annealing algorithm is shown in fig. 1.

        Fig. 1. Flow chart of Simulated Annealing Algorithm

  4. OPTIMIZATION OF SEWER NETWORK The sewer network example (Banjaran sewer network,

    Laxmangarh, Rajasthan, India) is considered to check the

    above-proposed 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 cost of sewer, cost of manholes and cost of 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.

    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

  5. 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 × 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

  6. 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 least-cost design and operation of a sewerage system network. Using the SA approach, the total cost of the sewer system was Rs. 8.505

× 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.

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