An Optimization Model for Sewer Hydraulic Design
Base on Novel Trenchless Technology
Huahn-Tyng Weng and Shu-Liang Liaw
*Graduate Institute of Environmental Engineering, National Central University,
Chungli, 32054, Taiwan
ABSTRACT
Sewer system is combined with a series of pipes and manholes. It is a multi-stage
multi-option system since various diameters and diggings for every pipe are available.
For obtaining better design alternatives, various diameters with slope the pipes must be
calculated properly for comparison. In the city, the implementation of the sewer system is
always to design the construction mode of a trenchless technology to replace the open-cut
construction mode. In this study, a 0-1 mixed integer optimization model with tunnel
jacking construction mode was the first developed for hydraulic design of gravity sewer
systems. A simple optimization searching method, the Bounded Implicit Enumeration
(BIE) algorithm is used to develop the Sewerage System Optimization Model (SSOM)
for city area. A certain sewer system construction case was selected in Taipei City to
compare the differences between tradition design method and SSOM. It showed that
SSOM with tunnel jacking construction mode was successful in practical design cases,
and showed the effectiveness with saving the cost about 12% in piping, 40% in pumping
water-heads.
KEYWORDS
Bounded Implicit Enumeration, Multi-stage Multi-Option System, Optimal Design, Sewerage
System Optimization Model (SSOM)
INTRODUCTION
In order to reach the goal for higher household connection rate of sewer system in Taiwan area, the
government is going to boost sewerage construction. About 65.5 billion NT dollars will be budgeted
for raising the household connection rate of sewer system from 8.7% to 20.3% in the next six years
(2002~2007). Cost-effectiveness analysis has become an important issue in the planning and
designing of sewer systems. To optimize a gravity sewer system, it is necessary to use the scheme of
a serial multi-stage and multi-option hydraulic design problem, plus screening algorithm to find a
least-cost system. But covering system layout and hydraulic design at the same time is a surely
difficult job for design engineers. Therefore, the optimization of sewer system planning and
designing is divided into two categories. (1) In system layout: In consideration of population,
discharge flow-rate, street layout and topography of the planning area, the attempt of system layout
process is to arrange a network of sewer pipes for collecting and transporting the wastewater
properly. (2) In hydraulic design process: The size and slope of sewer pipes and the amount of
pump stations are determined for a given fixed system layout (Tekeli & Belkaya, 1986). The
optimization models for hydraulic design have been developed for thirty years and
discussed as the followings: Non-linear Programming (Holland, 1966,1967, Swamee,2001);
Linear Programming (Deiniger,1970); Dynamic Programming (Merrit & Bogan,1973);Convex
Separable Mixed Integer Programming (Dajani & Hasit,1974); Linear Programming and Mixed
Integer Linear Programming (Wen & Kuo,1982); Non-uniform state Increment Dynamic
Programming(NIDP) (Orth & Hsu,1984 ); Bounded Implicit Enumeration(BIE) (Liaw &
Lin,1990) ； Heuristic Approach (Charalambous ＆ Elimam,1990); GIS-Based Approach
(Agbenowosi,1995;Greene et al.,1999). The hindrance problems of practical construction were not
considered in urban sewer system, due to the optimization of hydraulic designs still couldn’t be
applied practically unless using more completely developed mathematic models. The construction
hindrance problems in urban area include the smoothness of local traffic, the insufficient space for
construction, the cause of abnormal geology, the hindrance of the other’s existing underground
piping, and etc. The experienced engineers have to consider the appropriate construction methods
first. According to the experiences on the construction of the sewer system in Taipei, the old
open-cut method is replaced with the novel trenchless technology, i. e. Shield Driving Method,
Jacking Method, Micro Tunneling Method, Horizontal Directional Drilling Method and etc.
Particularly, the tunneling jacking mode is one of the most popular methods for the urban narrow
streets of the old communities in Taipei. Therefore, establishing an optimal hydraulic design model
focused on the appropriate construction methods is an essential tool for promoting the municipal
sewer systems in Taiwan. In this study, a Sewerage System Optimization Model (SSOM) based on
a simple and efficient algorithm, Bounded Implicit Enumeration (BIE), is established with the aid
of a personal computer. To approach the most beneficial construction cost, it must meet the
requirements of hydraulic design criteria and response to the real problems existing in the
municipality.
DEVELOPMENT OF THE OPTIMAL DESIGN MODEL
Generally, a sewer system is mainly used to collect and transport sewage from household
connection to sewerage treatment plants through a network of sewer pipes by gravity. A gravity
sewer system usually consists of sewer pipes, manholes, pumping stations and other appurtenances.
Once the layout of a sewer network is determined, the main optimal work for hydraulic design
model is to select the size and slope of piping system under the design criteria and the minimal cost
for construction to achieve the ultimate goal of cost-effectiveness. If only considering a branched
sewer system, the design problem can be viewed as a serial multi-stage multi-option problem. Each
stage of sewer pipes is between manholes and pumping stations. For each of the sewer
pipes, different commercial piping sizes are assigned as different options. Each
commercial piping size is associated with a minimal slope, which is o btained by
comparing the slopes associated with the minimal cover depth, the maximal flow
velocity of partial flow, and the hydraulic of gravity flow. Based on this concept, a Sewer
System Optimization Program (SSOP) developed for the hydraulic design, which is a 0-1 Mixed
Integer Programming (Liaw et.al.1990; 1991). It is the optimal hydraulic design on the
condition of a fixed system layout of gravity sewer system. A case with 73-manhole
was used to illustrate the optimal cost of SSOP and to compare the differences between
DDDP model and AIT model. It showed that SSOP saved the cost 21.8% with DDDP, 3.4% with
AIT. However, SSOP is based on an open-cut construction mode with design variables of diameter
size and excavation depth of pipe and cannot meet the practical situation for the construction of
urban sewer system. According to the Taipei’s sewer system construction experience, the trenchless
technology is used to avoid several hindrances of underground construction and to reach the annual
preset goal of household connection rate. Therefore, a Sewerage System Optimization Model
(SSOM) with a tunnel jacking construction mode is being developed (Liaw & Weng et al. 2001,
2002). One of the trenchless technologies, the tunnel jacking construction mode is not to trench
along piping line but only to dig shaft (pit), a location for thrusting piping work and manhole.
Therefore the main design variables of SSOM are piping size and shaft depth. Fig.1 shows the
SSOM problem scheme of a serial multi-stage multi-option hydraulic design problem. It has
considered in various construction modes and different piping material as well as constraints on
each stage, such as passing through a fixed elevation, designating the site of a pumping station, and
the commercial piping diameter. In Fig1.each combination of options DijSij, representative
certain piping size and slope, is decided as follows: (1) the piping size of i-th stage used be
calculated under a constant flow-rate. Firstly, the scope of feasible diameters from the
minimal feasible diameter Dmin to the maximal feasible diameter Dmax, can be obtained by a given
velocit y from the minimal velocit y V m i n to the maximal velocit y V m a x through
Continunity Equation. Then the commercial piping size, the optimal pipe size Di1,Di2,...Dij ,
can be selected from the previous feasible diameters. (2) If the upstream manhole elevation of i-th
stage was fixed, the four types of design slope could be calculated by Manning Formula, i.e.
the partial flow slope (S p ), the minimum velocity slope (S min ), the minimal slope of
coverage (S c ), the maximal velocity slope (S max ). Through a comparing procedure with the
design criteria and minimized cost, the optimal slope Sij, one of the four types design slope, can
be assigned to optimal pipe size Dij above. Therefore, every commercial diameter is responded to
an optimal slope and other slope raising the cost will be deleted. (Benson,1985; Orth, 1986) In
this study, based on the problem scheme shown in Fig.1, the optimization model developed for the
hydraulic design of the municipal sewer system is a 0-1 mixed integer nonlinear model. For
Stage i=1
option
Case#1
Case#2
Case#3
j(i)=1
D11S11
d11s11
……
j(i)=2
D12S12
d12s12
……
j(i)=3
D13S13
d13s13
……
Stage i=2
Constrain #1
Case#1
Case#2
Case#3
D11S11
d11s11
……
MANHOL
E
D12S12
d12s12
……
LIFT
STATION
D13S13
d13s13
……
APPURTE
N
ANCES
Constrain #2
Constrain #3
Constrain #1
Constrain #2
MANHOL
E
LIFT
STATION
APPURTE
N
ANCES
Constrain #3
Fig.1. SSOM problem scheme of a serial multi-stage multi-option hydraulic
design problem (case#1, case#2.., responding to the variety of construction
mode and the materials of the pipe; constrain#1, constrain#2.., responding
to the limitation for each stage, such as passing through a fixed depth)
practical application, several assumptions are made for the optimization model. (1) a sewer system
means a branched gravity sanitary sewer system, (2) pumping station may be used to lift the
water-head of hydraulic, (3) the piping between any two manholes is a flat surface, (4) the sewer
pipes are circular pipes with the same material, (5) and the flow in pipe is always flow full. The
model is established as follows.
Objective function.
Considering of cost-effectiveness, the single objective of SSOM is to minimize the total cost of a
sewer system, including the cost of pipes Dij, of manholes MHc(Hupi), and of pump stationsPSi (Qi,
Hi). (Dij is as the diameter of the j-th option of the i-th stage, Sij is as the optimal slope to the certain
diameter, Hupi is as the digging of the up stream of the pipe, Xij is as the variety of the option
between 0 and 1)
n
MinZ 
i1
m
 Cost(D , S , Hup ) X
ij
j 1
ij
i
ij
n
n
i 1
i 1
  MHc(Hupi )   PSi (Qi , Hi )
Constraints function.
According to design criteria, assumptions of model, and hindrances of construction, there are
several constraints.
(1) The 0-1 constraints for multi-stage multi-option problems.
Xij = 0, 1 and
n

Xij  1, i  1,2.....n, j  1,2.......m
i 1
(2) The constraints to make assure that the Invert depth of each pipe will not exceed the minimal
and maximal Invert depth.
H min  Hupi  H max
H min  Hdmi  H max
; i = 1, 2…….n
(3) The constraints for the requirement that the pipe size of a downstream should be no less than
that of its upstream.
Di 1 , j  Di , j ; i = 1, 2…….n, j = 1, 2………m.
(4) The constraints on design criteria, for example, the minimum pipe size, the minimum and
maximum velocity, the commercial diameter, etc.
THE APPLICATION OF BIE ALGORITHM AND SSOM PROGRAM
The major characteristics of municipal sewer systems include not only the huge system covering
several districts with thousands of manholes but also the limitation of environmental factors. They
are much complicated and are affected by numerous difficult quantifiable factors. It is not easy to
find out the real optimal solution with the computer within a certain interval for the system’s
components. However, the “algorithm” is the key point of the efficient resolution to the
“optimization model”. Enumeration algorithm is the most basically and simply optimal
algorithm but it seems to be the least efficient since it has to assess all alternatives to obtain the best
solution. Implicit Enumeration (IE) algorithm has been developed to cover the deficiency of
Enumeration algorithm. It will store an upper bound and deletes the system, which is during the
proceeding of searching. Besides the upper bound used in IE algorithm, the Bounded Implicit
Enumeration (BIE) algorithm uses an additional set of lower bounds to further eliminate the inferior
combinations of options in search of the optimal solution. It is effective and efficient for solving of
serial multi-stage multi-option optimization problems (Chang and Liaw, 1990). The flow chart of
the BIE algorithm, used in SSOM model, is shown in Fig. 2. This process written with FORTRAN
language includes one main process and six sub-processes. The execution file is about 50 Kbytes by
Microsoft FORTRAN Power-Station Compiler (version 4.1). Therefore, the optimization of sewer
system design can be preceded with personal computer easily.
RESULTS AND DISCUSSION
The results of hydraulic design case study.
For case study, a branch network, which located within the residential and commercial areas in
eastern Taipei administrative, was selected and shown in Fig.3. The layout of a given fixed system
has a total of 98-manhole (97-pipe) and one pumping station at No.1 manhole. Generally, the
branch network sewer constructions in Taipei always construct along urban narrow streets, and
about 2 meters below ground to avoid the hindrances of others piping and structure. Therefore,
designer adopts the tunneling jacking method, to dig 2 meters for starting pit and 1.5 meters for
arriving pit, respectively, and to able to overcome unfitting construction problems in traditional
open-cut mode successfully. The outlet of piping network system should be entered directly into an
existed manhole No.0 in Figure 3 at a fixed elevation of about 2.36 meters with invert of pipes. The
four types of commercial piping sizes are ∮300,∮400,∮500, and∮700mm. The maximun and
minimum limitation velocity are 3 and 0.6 M/sec, respectively. Table 1 shows the hydraulic
calculated sheet of tradition design method in this case study, and a pumping station with up-stream
and down-stream of piping invert elevation of manhole No.1 are “-1” and “+3.36” meters,
respectively. Therefore the pumping station of No.1 must be capable of lifting water-heads about
4.34 meters above. Table 2 presents the hydraulic analysis results of SSOM. The same design
criteria and constrains as tradition design method are utilized in this case study. The up-stream and
down-stream of piping invert elevation of manhole No.1 are “+0.73” and “+3.36” meters, So the
pumping station of No.1 need consider lifting water-heads about 2.63 meters only. It is clearly to
see that SSOM saved the net lifting water-heads about 40% than traditional design in this case
study.
Start
19
Data Input
18
23 24 25 26 27 17 22 21 20
Calculate the flow rate of manhole for a given fixed system layout
16
15
14
Calculate the feasible diameters for each pipe
32
35 36 37 38 39 40 41 13 31 30 29 28
Calculate the system lower bound for each stage with minimal pipe
cost and shaft cost
12
Calculate the initial feasible solution
Algorithm
(Search
procedure
and
alternative
comparison)
Design variables
(piping size & shaft
depth)
Results of
hydraulic and
cost analysis
53
66
33
67
11
68
10
69
9
70
63
8
71
64
72
65
42 43 44 45 46 47 48 49 50 51 52 7 73
Hydraulic Design :
1. Constrain with design
criteria and practical
requirement
2. Hydraulic analysis
34
6 62 61 60 59 58 57 56 55 54
5
93
90 88 87 86 85
4
94
91 89
95 96 97 3 84 83 82 81 80 79 78 77 76 75 74
2
92
1
Data Output
0
End
Fig.2. The Flow Chart of SSOM.
Fig.3.The Layout of A Given Fixed System
The comparison of construction cost.
In order to suit a bidding process for new sewer engineering, this study adopts the unit-price and
quantity analysis method to get total construction cost of piping. By actual cases analysis, many
combinational unit-prices were obtained for different piping sizes, which is one of important design
variable of SSOM. In the case , three commercial diameters∮300,∮400, and∮500 mm correspond
to the unit-prices of 10060, 11320, and 14720NT$/M , respectively. The cost functions of shaft (pit),
the other design variable of SSOM, have a higher linear relationship between their cost of
construction (Cp NT$) and depth (H meter). The resulted equations are Cp = 32662H + 67045 in
jacking shaft, and Cp = 31097H + 54107 in arrival shaft. It shows that the optimal designs saves the
cost of piping about 12 % by comparing between 78,582,710 NT$ in tradition design method and
69,718,693 NT$ in SSOM. SSOM saves the cost about 12% in piping, and has 40% efficient more
than for pumping water-heads, which mentioned previously. In addition, in this 97-stage pipe case,
it takes the computer only several seconds to figure out the optimal design solution from the total
93312 combinations by BIE algorithms of SSOM. Considering the time saved for optimizing a huge
municipal sewer system hydraulic design, SSOM is a more efficient method for providing a
speeding accurate function.
Table1. The Hydraulic Calculated Sheet of Tradition Design
DGL Velocity Remarks
UGL
DCE
Manhole No. FlowrateLengthSlopeDiameter UCE
(CMS) (M) (%) (mm) up(M)down(M)up(M)down(M) (M/S) drop(M)
to
from
A18 0.0029
A19
1.19
6.50
6.48
3.13
3.36
300
1.0
23
A17 0.0059
A18
1.19
6.55
6.50
2.87
3.13
300
1.0
26
A16 0.0320
A17
1.06
6.70
6.55
2.62
2.87
300
0.8
31
A15 0.0320
A16
1.06
6.85
6.70
2.28
2.62
300
0.8
43
A14 0.0320
A15
1.06
6.86
6.85
2.02
2.28
300
0.8
32
A13 0.0320
A14
0.10
1.06
6.70
6.86
1.76
2.02
300
0.8
32
A12 0.0802
A13
1.11
6.68
6.70
1.39
1.66
400
0.6
45
A11 0.0802
A12
1.11
6.68
6.68
1.13
1.39
400
0.6
43
A10 0.0866
A11
1.11
6.70
6.68
0.90
1.13
400
0.6
39
A9 0.0866
A10
1.11
6.46
6.70
0.72
0.90
400
0.6
30
A8 0.0866
A9
1.11
6.56
6.46
0.50
0.72
400
0.6
36
A7 0.0866
A8
0.10
1.11
6.80
6.56
0.25
0.50
400
0.6
41
A6 0.1062
A7
1.05
7.02
6.80
-0.05
0.15
500
0.4
51
A5 0.1415
A6
1.05
7.20
7.02
-0.05 -0.19
500
0.4
36
A4 0.1475
A5
1.05
7.35
7.20
-0.19 -0.33
500
0.4
36
A3 0.1475
A4
0.20
1.05
7.36
7.35
-0.33 -0.52
500
0.4
48
A2 0.2036
A3
No.A1:
1.32
7.23
7.36
-0.72 -0.87
700
0.3
51
A1 0.2036
A2
1.32 pump st.
7.41
7.23
-0.87 -1.00
700
0.3
45
A0 0.2036
A1
1.32 No.A0:
7.40
7.41
3.26
3.34
700
0.3
25
A17c A17b 0.0024
1.30 existed
6.30
6.26
3.43
3.83
300
1.2
33
A17b A17a 0.0024
1.30 manhole
6.48
6.30
3.17
3.43
300
1.2
22
A17 0.0084
A17a
1.19
6.55
6.48
2.87
3.17
300
1.0
30
A17-5 A17-4 0.0060
1.06
6.80
6.76
3.61
3.83
300
0.8
27
A17-4 A17-3 0.0060
1.06
6.75
6.80
3.41
3.61
300
0.8
25
A17-3 A17-2 0.0092
1.06
6.70
6.75
3.23
3.41
300
0.8
23
A17-2 A17-1 0.0109
1.06
6.64
6.70
3.07
3.23
300
0.8
20
A17-1 A17 0.0177
1.06
6.55
6.64
2.87
3.07
300
0.8
25
A13a4 A13a3 0.0016
1.30
6.11
6.09
2.52
2.78
300
1.2
22
A13a3 A13a2 0.0046
1.30
6.49
6.11
2.20
2.52
300
1.2
27
A13a2 A13a1 0.0135
1.19
6.55
6.49
1.99
2.20
300
1.0
21
0.1
A13a1 A13 0.0202
1.19
6.70
6.55
1.76
1.99
300
1.0
23
0.6
A13a2-1 A13a2 0.0030
1.19
6.49
6.50
2.80
3.10
300
1.0
30
A13a2-bA13a2-a0.0042
1.19
6.40
6.38
2.83
3.17
300
1.0
34
0.3
A13a2-a A13a2 0.0059
1.19
6.49
6.40
2.50
2.83
300
1.0
33
A13-7 A13-6 0.0040
1.06
6.64
6.65
3.65
3.97
300
0.8
40
A13-6 A13-5 0.0071
1.06
6.60
6.64
3.39
3.65
300
0.8
33
A13-5 A13-4 0.0099
1.06
6.65
6.60
3.13
3.39
300
0.8
32
0.5
A13-4 A13-3 0.0141
1.06
6.70
6.65
2.93
3.13
300
0.8
25
A13-3 A13-2 0.0217
1.06
6.75
6.70
2.26
2.43
300
0.8
21
A13-2 A13-1 0.0281
1.06
6.80
6.75
2.10
2.26
300
0.8
20
0.1
A13-1 A13 0.0281
1.06
6.70
6.80
1.76
2.10
300
0.8
42
A7-11 A7-10 0.0032
1.19
6.72
6.80
2.66
3.03
300
1.0
37
A7-10 A7-9 0.0032
1.19
6.70
6.72
2.26
2.66
300
1.0
40
A7-8 0.0050
A7-9
1.06
6.78
6.70
2.01
2.26
300
0.8
31
A7-7 0.0050
A7-8
1.06
6.84
6.78
1.74
2.01
300
0.8
34
A7-6 0.0131
A7-7
1.06
6.80
6.84
1.57
1.74
300
0.8
21
A7-5 0.0161
A7-6
1.06
6.70
6.80
1.41
1.57
300
0.8
20
A7-4 0.0161
A7-5
1.06
6.50
6.70
1.13
1.41
300
0.8
35
A7-3 0.0161
A7-4
1.06
6.60
6.50
0.89
1.13
300
0.8
30
DGL VelocityRemarks
UGL
No. FlowrateLength SlopeDiameterUCE DCE
Manhole
(%) (mm) up(M)down(M)up(M)down(M) (M/S) drop(M)
(CMS) (M)
to
from
A7-2
A7-3
1.06
6.58
6.60
0.68
0.89
300
0.8
26
0.0199
A7-1
A7-2
1.06
6.70
6.58
0.52
0.68
300
0.8
20
0.0199
0.2
A7
A7-1
1.06
6.80
6.70
0.35
0.52
300
0.8
21
0.0242
0.9
A7-7
A7-7a
1.06
6.84
6.90
2.64
2.81
300
0.8
21
0.0081
A6-8
A6-9
1.06
6.08
6.00
1.99
2.34
300
0.8
44
0.0030
A6-7
A6-8
1.06
6.15
6.08
1.67
1.99
300
0.8
40
0.0030
A6-6
A6-7
1.06
6.20
6.15
1.32
1.67
300
0.8
44
0.0157
A6-5
A6-6
1.06
6.25
6.20
1.15
1.32
300
0.8
21
0.0157
A6-4
A6-5
1.06
6.35
6.25
0.99
1.15
300
0.8
20
0.0229
A6-3
A6-4
1.11
6.38
6.35
0.82
0.99
400
0.6
28
0.0312
A6-2
A6-3
1.11
6.75
6.38
0.54
0.82
400
0.6
46
0.0383
A6-1
A6-2
1.11
6.80
6.75
0.40
0.54
400
0.6
23
0.0446
0.2
A6
A6-1
1.11
7.02
6.80
0.15
0.40
400
0.6
42
0.0464
A6-7a3 A6-7a2 0.0079
1.06
6.00
6.04
2.17
2.48
300
0.8
39
A6-7a2 A6-7a1 0.0079
1.06
6.20
6.00
1.81
2.17
300
0.8
45
A6-7
A6-7a1
1.06
6.15
6.20
1.67
1.81
300
0.8
17
0.0127
A6-4a7 A6-4a6 0.0035
1.19
6.14
6.08
2.67
2.89
300
1.0
22
A6-4a6 A6-4a5 0.0084
1.06
6.20
6.14
2.41
2.67
300
0.8
32
A6-4a5 A6-4a4 0.0084
1.06
6.25
6.20
2.22
2.41
300
0.8
24
A6-4a4 A6-4a3 0.0084
1.06
6.15
6.25
1.95
2.22
300
0.8
34
A6-4a3 A6-4a2 0.0084
1.06
6.08
6.15
1.57
1.95
300
0.8
47
A6-4a2 A6-4a1 0.0084
1.06
6.20
6.08
1.31
1.57
300
0.8
33
0.1
A6-4
A6-4a1
1.06
6.35
6.20
1.09
1.31
300
0.8
28
0.0084
0.9
A6-1
A6-1a
1.30
6.80
6.75
1.30
1.70
300
1.2
33
0.0017
A3-10 0.0099
A3-11
1.06
6.45
6.40
2.95
3.21
300
0.8
32
0.1
A3-9
A3-10
1.06
6.40
6.45
2.68
2.95
300
0.8
34
0.0186
A3-8
A3-9
1.11
6.64
6.40
2.33
2.58
400
0.6
41
0.0282
A3-7
A3-8
1.11
6.70
6.64
2.20
2.33
400
0.6
22
0.0282
A3-6
A3-7
1.11
6.75
6.70
2.10
2.20
400
0.6
17
0.0310
A3-5
A3-6
1.11
6.80
6.75
1.96
2.10
400
0.6
23
0.0387
A3-4
A3-5
1.11
6.80
6.80
1.82
1.96
400
0.6
23
0.0410
A3-3
A3-4
1.11
6.90
6.80
1.65
1.82
400
0.6
28
0.0432
0.5
A3-2
A3-3
1.11
7.04
6.90
1.36
1.65
400
0.6
48
0.0541
0.5
A3-1
A3-2
1.11
7.20
7.04
0.73
0.86
400
0.6
21
0.0606
0.7
A3
A3-1
1.11
7.36
7.20
-0.02
0.23
400
0.6
42
0.0636
A3-10a5 A3-10a4 0.0043
1.19
6.12
6.10
3.73
3.85
300
1.0
12
A3-10a4 A3-10a3 0.0043
1.19
6.14
6.12
3.57
3.73
300
1.0
16
A3-10a3 A3-10a2 0.0043
1.19
6.50
6.14
3.41
3.57
300
1.0
16
A3-10a2 A3-10a1 0.0086
1.06
6.45
6.50
3.17
3.41
300
0.8
30
A3-10a1 A3-10 0.0086
1.06
6.45
6.45
2.95
3.17
300
0.8
27
A3-9a2 A3-9a1 0.0022
1.30
6.38
6.45
3.52
3.89
300
1.2
31
0.6
A3-9
A3-9a1
1.30
6.40
6.38
3.18
3.52
300
1.2
28
0.0043
0.9
A3-9
A3-9b
1.06
6.40
6.60
3.48
3.75
300
0.8
34
0.0053
0.5
A3-4a2 A3-4a1 0.0022
1.30
6.90
7.00
3.60
3.91
300
1.2
26
0.9
A3-4
A3-4a1
1.30
6.80
6.90
2.72
3.10
300
1.2
32
0.0022
0.9
A3a2
A3a3
1.06
7.30
7.34
2.46
2.68
300
0.8
28
0.0084
0.9
A3a1
A3a2
1.06
7.32
7.30
1.34
1.56
300
0.8
27
0.0115
0.9
A3
A3a1
1.06
7.36
7.32
0.18
0.44
300
0.8
33
0.0177
TOTAL COST = Σ UNIT PRICE * QUANTITY = 78582710 NT$
Table2. The Hydraulic Analysis Results of SSOM
Manhole No. FlowrateLengthSlopeDiameterUCE DCE UGL DGL Velocity Cost
Manhole No. FlowrateLengthSlopeDiameter UCE DCE UGL DGL Velocity Cost
from
to (CMS) (M) (%)
(M) up(M)down(M)up(M)down(M) (M/S) (NT$)
from
to (CMS) (M) (%)
(M) up(M)down(M)up(M)down(M) (M/S) (NT$)
No.1 No.0 0.2036
25
0.39
0.5
7.41
7.4
1.04 4570253 No.50 No.51 0.0199
26
0.29
0.3
2.55
2.48
6.6
6.58
0.6
650448
3.36
3.26
No.2 No.1 0.2036
45
0.39
0.5
0.91
7.23
7.41
1.04 1125607
51
52 0.0199
20
0.29
0.3
2.48
2.42
6.58
6.7
0.6
570589
0.73
3
2
0.2036
51
0.39
0.5
1.1
0.91
7.36
7.23
1.04 1343644
52
7
0.0242
21
0.25
0.3
2.42
2.37
6.7
6.8
0.6
607798
4
3
0.1475
48
0.67
0.4
1.42
1.1
7.35
7.36
1.17 993605
53
46 0.0081
21
0.56
0.3
4.4
4.28
6.9
6.84
0.6
452507
5
4
0.1475
36
0.67
0.4
1.66
1.42
7.2
7.35
1.17 821465
54
55 0.0030
44
1.15
0.3
3.5
2.99
6
6.08
0.6
683887
6
5
0.1415
36
0.61
0.4
1.89
1.66
7.02
7.2
1.13 910442
55
56 0.0030
40
1.15
0.3
2.99
2.53
6.08
6.15
0.6
759953
7
6
0.1062
51
0.35
0.4
2.06
1.89
6.8
7.02
0.84 1044781
56
57 0.0157
44
0.35
0.3
2.53
2.38
6.15
6.2
0.6
875283
8
7
0.0866
41
0.23
0.4
2.16
2.06
6.56
6.8
0.69 864684
57
58 0.0157
21
0.35
0.3
2.38
2.31
6.2
6.25
0.6
592785
9
8
0.0866
36
0.23
0.4
2.24
2.16
6.46
6.56
0.69 780606
58
59 0.0229
20
0.26
0.3
2.31
2.25
6.25
6.35
0.6
565660
10
9
0.0866
30
0.23
0.4
2.31
2.24
6.7
6.46
0.69 739783
59
60 0.0312
28
0.21
0.3
2.25
2.19
6.35
6.38
0.6
750701
11
10 0.0866
39
0.23
0.4
2.4
2.31
6.68
6.7
0.69 816476
60
61 0.0383
46
0.18
0.3
2.19
2.11
6.38
6.75
0.6
834726
12
11 0.0802
43
0.2
0.4
2.48
2.4
6.68
6.68
0.64 880595
61
62 0.0446
23
0.28
0.3
2.11
2.05
6.75
6.8
0.63 639600
13
12 0.0802
45
0.2
0.4
2.57
2.48
6.7
6.68
0.64 1036226
62
6
0.0464
42
0.31
0.3
2.05
1.92
6.8
7.02
0.66 890456
14
13 0.0320
32
0.21
0.3
2.64
2.57
6.86
6.7
0.6
716583
63
64 0.0079
39
0.57
0.3
3.54
3.32
6.04
6
0.6
652202
15
14 0.0320
32
0.21
0.3
2.7
2.64
6.85
6.86
0.6
692706
64
65 0.0079
45
0.57
0.3
3.32
3.06
6
6.2
0.6
777930
16
15 0.0320
43
0.21
0.3
2.79
2.7
6.7
6.85
0.6
816964
65
56 0.0127
17
0.41
0.3
3.06
2.99
6.2
6.15
0.6
530279
17
16 0.0320
31
0.21
0.3
2.85
2.79
6.55
6.7
0.6
825171
66
67 0.0035
22
1.05
0.3
3.58
3.35
6.08
6.14
0.6
462567
18
17 0.0059
26
0.71
0.3
3.71
3.52
6.5
6.55
0.6
609555
67
68 0.0084
32
0.55
0.3
3.35
3.17
6.14
6.2
0.6
669761
19
18 0.0029
23
1.20
0.3
3.98
3.71
6.48
6.5
0.6
472627
68
69 0.0084
24
0.55
0.3
3.17
3.04
6.2
6.25
0.6
577345
20
21 0.0024
33
1.35
0.3
3.76
3.32
6.26
6.3
0.6
591842
69
70 0.0084
34
0.55
0.3
3.04
2.86
6.25
6.15
0.6
703524
21
22 0.0024
22
1.35
0.3
3.32
3.02
6.3
6.48
0.6
555922
70
71 0.0084
47
0.55
0.3
2.86
2.6
6.15
6.08
0.6
817080
22
17 0.0084
30
0.55
0.3
3.02
2.85
6.48
6.55
0.6
671526
71
72 0.0084
33
0.55
0.3
2.6
2.42
6.08
6.2
0.6
702448
23
24 0.0060
27
0.7
0.3
4.26
4.07
6.76
6.8
0.6
531482
72
59 0.0084
28
0.55
0.3
2.42
2.26
6.2
6.35
0.6
641165
24
25 0.0060
25
0.7
0.3
4.07
3.9
6.8
6.75
0.6
578130
73
62 0.0017
33
1.74
0.3
4.25
3.68
6.75
6.8
0.6
591842
25
26 0.0092
23
0.51
0.3
3.9
3.78
6.75
6.7
0.6
581227
74
75 0.0099
32
0.48
0.3
3.9
3.75
6.4
6.45
0.6
581782
26
27 0.0109
20
0.45
0.3
3.78
3.69
6.7
6.64
0.6
533784
75
76 0.0186
34
0.31
0.3
2.9
2.8
6.45
6.4
0.6
772437
27
17 0.0177
25
0.32
0.3
3.69
3.61
6.64
6.55
0.6
604543
76
77 0.0282
41
0.23
0.3
2.8
2.71
6.4
6.64
0.6
943782
28
29 0.0016
22
1.85
0.3
3.59
3.18
6.09
6.11
0.6
462567
77
78 0.0282
22
0.23
0.3
2.71
2.66
6.64
6.7
0.6
585447
29
30 0.0046
27
0.85
0.3
3.18
2.95
6.11
6.49
0.6
623924
78
79 0.0310
17
0.21
0.3
2.66
2.62
6.7
6.75
0.6
559789
30
31 0.0135
21
0.39
0.3
2.95
2.87
6.49
6.55
0.6
719619
79
80 0.0387
23
0.18
0.3
2.62
2.58
6.75
6.8
0.6
601589
31
13 0.0202
23
0.29
0.3
2.87
2.81
6.55
6.7
0.6
608185
80
81 0.0410
23
0.17
0.3
2.58
2.54
6.8
6.8
0.6
625934
32
30 0.0030
30
1.15
0.3
4
3.65
6.5
6.49
0.6
561662
81
82 0.0432
28
0.27
0.3
2.54
2.47
6.8
6.9
0.61 734256
33
34 0.0042
34
0.91
0.3
3.88
3.57
6.38
6.4
0.6
583287
82
83 0.0541
48
0.42
0.3
2.47
2.27
6.9
7.04
0.77 884416
34
30 0.0059
33
0.71
0.3
3.57
3.34
6.4
6.49
0.6
681116
83
84 0.0606
21
0.52
0.3
2.27
2.16
7.04
7.2
0.86 601523
35
36 0.0040
40
0.94
0.3
4.15
3.78
6.65
6.64
0.6
662262
84
3
0.0636
42
0.58
0.3
2.16
1.91
7.2
7.36
0.9
843965
36
37 0.0071
33
0.62
0.3
3.78
3.57
6.64
6.6
0.6
662846
85
86 0.0043
12
0.89
0.3
3.6
3.49
6.1
6.12
0.6
380582
37
38 0.0099
32
0.48
0.3
3.57
3.42
6.6
6.65
0.6
677563
86
87 0.0043
16
0.89
0.3
3.49
3.35
6.12
6.14
0.6
484446
38
39 0.0141
25
0.37
0.3
3.42
3.32
6.65
6.7
0.6
593871
87
88 0.0043
16
0.89
0.3
3.35
3.21
6.14
6.5
0.6
508760
39
40 0.0217
21
0.27
0.3
3.32
3.26
6.7
6.75
0.6
578288
88
89 0.0086
30
0.54
0.3
3.21
3.05
6.5
6.45
0.6
645951
40
41 0.0281
20
0.23
0.3
3.26
3.22
6.75
6.8
0.6
551378
89
75 0.0086
27
0.54
0.3
3.05
2.9
6.45
6.45
0.6
639444
41
13 0.0281
42
0.23
0.3
3.22
3.12
6.8
6.7
0.6
796170
90
91 0.0022
31
1.47
0.3
3.95
3.49
6.45
6.38
0.6
571722
42
43 0.0032
37
1.12
0.3
4.3
3.89
6.8
6.72
0.6
632082
91
76 0.0043
28
0.89
0.3
3.49
3.24
6.38
6.4
0.6
613266
43
44 0.0032
40
1.12
0.3
3.89
3.44
6.72
6.7
0.6
732288
92
76 0.0053
34
0.76
0.3
4.1
3.84
6.6
6.4
0.6
583287
44
45 0.0050
31
0.79
0.3
3.44
3.2
6.7
6.78
0.6
674998
93
94 0.0022
26
1.47
0.3
4.5
4.12
7
6.9
0.6
502807
45
46 0.0050
34
0.79
0.3
3.2
2.93
6.78
6.84
0.6
695322
94
81 0.0022
32
1.47
0.3
4.12
3.64
6.9
6.8
0.6
669539
46
47 0.0131
21
0.4
0.3
2.93
2.84
6.84
6.8
0.6
674338
95
96 0.0084
28
0.55
0.3
4.84
4.69
7.34
7.3
0.6
541542
47
48 0.0161
20
0.34
0.3
2.84
2.77
6.8
6.7
0.6
566087
96
97 0.0115
27
0.43
0.3
4.69
4.57
7.3
7.32
0.6
594707
48
47 0.0161
35
0.34
0.3
2.77
2.66
6.7
6.5
0.6
737038
97
3
0.0177
33
0.32
0.3
4.57
4.46
7.32
7.36
0.6
678546
49
48
0.0161
30
0.34
0.3
2.66
2.55
6.5
6.6
0.6
663171
TOTAL NUMBER OF COMBINATIONS= 93312 . MAL OFV = 69718693 NT$
CONCLUSIONS AND PERSPECTIVES
In this study, a 0-1 mixed integer optimization model with tunnel jacking construction mode was
first developed for hydraulic design of gravity sewer systems. The Bounded Implicit Enumeration
(BIE) algorithm is used to develop the Sewerage System Optimization Model (SSOM). The SSOM
can provide a set of various options and constraints, which can be selected, based on the practical
requirement and construction environment, to apply for optimal designs of trunk sewer and
branch networks system. The results generated are easy to understand and practical to use. It is
more suitable for personal computer since the memory-space required is small for process, and
more convenient for design engineering to evaluate an optimal alternative of decision-making. In
the case study in Taipei City, the SSOM was successful used in practical design cases. It showed
the effectiveness with saving the cost about 12% in piping, 40% in pumping water-heads. The
traditional design is expected to be replaced with SSOM as it is easy to input and more users
friendly. Therefore, the SSOM is the effective tools for improving the overall efficacy of
the investment and promoting the efficiency of the household connection rate of sewer
system in Taiwan.
ACKNOWLEDGMENTS
Many thanks to William Chung, Project Manger of CTCI Corporation of Taiwan, who has
contributed to the cost function of shaft and to Dr. Bi-Liang Lin, Assistant Prof. of Department of
Civil Engineering in Ming-Hsin University of Science & Technology of Taiwan, for his efforts on
programming throughout the development of model.
REFERENCES
Benson, R.E., "Self-Cleaning Slope for Partially Full Sewers", Journal of the Environmental
Engineering Division, ASCE, Vol. 111, No. 6, pp. 925-928, 1985.
Chang, S.Y. and Liaw, S.L., " An Efficient Implicit Enumeration Algorithm for Multistage Systems
", for presentation at the TIMS/ORSA Joint National Meeting, New Orleans, LA, May 4-6, 1987.
Liaw S.L., B.L. LIN, " Optimization of Sewer System, (II) Hydraulic Design ", International
Conference on Computer Application in Water Resources, Tamkang University, Vol. 2, July,
1991,pp. 77-84.
Orth, H.M., Model-Based Design of Water Distribution and Sewage Systems, 1nd Ed., John Wiley
& Sons, 1986.