Journal of Environmental Management (2002) 65: 383±409
doi:10.1006/jema.2001.0563, available online at http://www.idealibrary.com on
1
A comparative analysis of methods to
represent uncertainty in estimating the
cost of constructing wastewater treatment
plants
Ho-Wen Chen and Ni-Bin Chang*
Department of Environmental Engineering, National Cheng-Kung University, Tainan, Taiwan,
Republic of China
Received 4 May 1999; accepted 29 January 2002
Prediction of construction cost of wastewater treatment facilities could be in¯uential for the economic feasibility of various
levels of water pollution control programs. However, construction cost estimation is dif®cult to precisely evaluate in an
uncertain environment and measured quantities are always burdened with different types of cost structures. Therefore,
an understanding of the previous development of wastewater treatment plants and of the related construction cost
structures of those facilities becomes essential for dealing with an effective regional water pollution control program. But
deviations between the observed values and the estimated values are supposed to be due to measurement errors only
in the conventional regression models. The inherent uncertainties of the underlying cost structure, where the human
estimation is in¯uential, are rarely explored. This paper is designed to recast a well-known problem of construction cost
estimation for both domestic and industrial wastewater treatment plants via a comparative framework. Comparisons were
made for three technologies of regression analyses, including the conventional least squares regression method, the fuzzy
linear regression method, and the newly derived fuzzy goal regression method. The case study, incorporating a complete
database with 48 domestic wastewater treatment plants and 29 industrial wastewater treatment plants being collected
in Taiwan, implements such a cost estimation procedure in an uncertain environment. Given that the fuzzy structure in
regression estimation may account for the inherent human complexity in cost estimation, the fuzzy goal regression
method does exhibit more robust results in terms of some criteria. Moderate economy of scale exists in constructing both
the domestic and industrial wastewater treatment plants. Findings indicate that the optimal size of a domestic wastewater
treatment plant is approximately equivalent to 15 000 m
3
/day (CMD) and higher in Taiwan. Yet the optimal size of an
industrial wastewater treatment plant could fall in between 6000 CMD and 20 000 CMD.

tion of resources conservation motivates the use of
wastewater reclamation units to aid in processing
for wastewater recovery and reuse. Prediction of
construction cost of wastewater treatment facilities
could be in¯uential for the economic feasibility of
various levels of water pollution control programs.
In any circumstances, the costs of wastewater
treatment are going to escalate continuously over
time so that it may require advanced and compre-
hensive evaluation with regard to economic feasi-
bility for those water quality management programs
in the future. However, construction cost estima-
tion is dif®cult to precisely evaluate in an un-
certain environment and measured quantities are
always burdened with different types of cost struc-
tures. Therefore, an understanding of the previous
development of wastewater treatment plants and
of the related construction cost structures of
those facilities becomes essential for dealing with
an effective regional water pollution control pro-
gram. But deviations between the observed values
and the estimated values are supposed to be due
to measurement errors only in the conventional
regression models. The inherent uncertainties of
the underlying cost structure, where the human
estimation is in¯uential, are rarely explored.
Previous cost analyses with respect to wastewater
treatment have focused primarily on capital costs
and associated differences with facility size.
Economy of scale is a major concern. As a result,

and Watada, 1988; Tanaka et al. 1989; Chen, 1988;
Diamond, 1988). Later on, the advances in theory
have been made with respect to the measurement of
vagueness (Bardossy, 1990; Moskowitz and Kim,
1993) and the algorithms of Group Method of Data
Handling (GMDH) (Hayashi and Tanaka, 1990), the
use of a two-stage construction of a linear regression
model (Savic and Pedrycz, 1991), the linkage with
three types of multi-objective programming models
(Sakawa 1992), the development of a fuzzy vector
autoregressive model (Oh et al., 1992), the deriv-
ation of a generalized fuzzy linear regression model
(Wang and Ha, 1992), the application of Monte
Carlo simulation technique for performing a fuzzy
regression analysis (Juang et al., 1992), and the
extension of a fuzzy self-regression model (Lu and
Guang, 1993). On the other hand, many real world
applications, using the fuzzy forecasting technique
as a means, are worthy of further discussions.
Typical examples include the use of fuzzy linear
regression for the forecasting of computer sale in
the market (Heshmaty and Kandel, 1985), for
illustrating cellulose hydrolysis (Gharpuray et al.,
1986), for explaining the potential of fuzzy regres-
sion application in hydrology (Barddossy et al.,
1990), for construction cost analysis of municipal
incinerators using the fuzzy goal regression model
(Chang et al., 1996), and for construction cost
analysis of wastewater treatment plants using
fuzzy linear regression model (Wen and Lee,

are explored in the formulation simultaneously. The
case study, incorporating a complete database
with 48 domestic wastewater treatment plants
and 29 industrial wastewater treatment plants
being collected in Taiwan, is designed to implement
three cost estimation methods in an uncertain
environment.
The principles of fuzzy linear
regression and fuzzy goal
regression
Uncertainty frequently plays an important role in
cost estimation. The random character governing
wastewater generation, treatment, re-use, and dis-
posal are all possible sources of uncertainty. Until
the early 1980's, probability was the only kind of
uncertainty handled by mathematics. The implica-
tion of probability, as symbolized by the concept of
randomness, is based on the `chance' or `opportun-
ity'. In relation to the measurement errors that
exist in a real world event, however, fuzziness takes
on another aspect of uncertainty expression. In
reality, fuzziness is the ambiguity that can be
found in the linguistic description of a concept or
feeling. For example, the uncertainty in expressions
like `the water is dirty' or `the smell in the river is
bad' can be called fuzziness. The degree of fuzziness
to be recognized in such questions is `how dirty is
dirty?' or `how bad is bad?' Therefore, random-
ness and fuzziness considered in decision-making
differ in nature. In mathematics, the probability

cost estimation are constructed based on the well-
known theories of probability and statistics that
have to be subject to a long-standing assumptionÐ
`Independent and Identical Distribution' (IID)Ð for
all observations. With such a limitation in theory,
conventional statistical regression analysis is
sometimes involved in a predicament in the real
world applications. While the database consists of
complex factors, it is sometimes hard to ®nd out
suitable mathematical models to describe the
behaviors of the target system that must be
consistent with the IID assumption. Tanaka et al.
(1982) ®rst mentioned that we must deal with
a fuzzy structure of the systems in the regression
model where human estimation is in¯uential. Fuzzy
regression analysis, representing an alternative to
the statistical regression technique, has been grow-
ing up rapidly in the last few years (Tanaka et al.,
1982). The fuzzy linear regression model may
release the IID assumption in the regression
analysis and allow each predicted value to exhibit
different degree of variation.
The fuzzy linear regression is thus recognized as
a mapping process based on a set of observations.
Fuzzy parameters are used for such a linkage
between the independent variables and the depend-
ent variable. To ®nd out the solution of fuzzy
parameters, an equivalent linear programming
Estimating costs of constructing wastewater treatment plants 385
model has to be solved (see Appendix I). However,

million in population, most of the rivers in this
tiny island have been polluted for a long time. As
the situation of water pollution becomes worse
over time, one of the management strategies is
to install the intercept systems and wastewater
treatment plants to reduce the direct impacts of
ef¯uents on the river systems. Intensive debate in
the society concerns about what is the adequate
level of wastewater treatment process in those
industrial complexes and communities located at
different river reaches or coastal areas and how to
satisfy the overall pollution control requirements
by a cost-effective approach. These questions ini-
tialize a nationwide need to investigate the cost
information of wastewater treatment facilities.
Regression analysis techniques were frequently
applied to estimating engineering cost in many
pollution prevention and environmental quality
control programs (Greenberg, 1995; Chang and
Wang, 1995b; Chang et al., 1996). It is observed
that versatile engineering technologies and com-
petitive bidding processes have made the cost
estimation become relatively vague. An accurate
prediction of the construction cost of industrial and
domestic wastewater treatment plants using
a more reliable approach turns out to be of signi-
®cance for future applications.
Thus, an understanding of the background
information of domestic and industrial wastewater
treatment plants in Taiwan could become a typical

to domestic and industrial wastewater treatment
plants respectively. In general, the primary treat-
ment process only employs the bar screen, grit
removal, and primary sedimentation tank to
ful®ll the basic treatment goals. The secondary
treatment process usually adds more biological
treatment units to enhance the removal of organic
content in the ef¯uents, which can be further
differentiated in terms of biological treatment
technologies, such as the contacts stabilization (CS),
386 H W. Chen and N B. Chang
Figure 1. The geographical distribution of wastewater treatment plants in Taiwan.
Estimating costs of constructing wastewater treatment plants 387
Table 1. Database of all domestic wastewater treatment plants in Taiwan
Plant
no.
Location of
treatment plant
Design
capacity
(10
2
CMD)
Total construction
cost (millions US$)
Normalized total
construction cost
(1995 millions US$)
Level
g

a
SF
a
D1 Hsin-Ying service area
(in highway)
0Á1 0Á48 0Á48 III 
b
 Â
b
      1995 119Á19
D2 Hai-Hu Vacation Center 0Á2 0Á11 0Á14 II
  Â Â Â Â Â Â Â 1990 94Á65
D3 The Home of Charity 0Á2 0Á17 0Á29 II
 Â  Â Â Â Â Â Â 1985 72Á76
D4 Wu-Ling Hotel (Ken-Ting) 0Á27 2Á59 3Á63 III
 Â   Â Â Â Â  1980 96Á15
D5 Kai-Sa Hotel (Ken-Ting) 0Á45 0Á89 1Á47 III
 Â  Â Â Â Â Â  1986 72Á14
D6 Tai-An service area I 0Á5 0Á87 1Á07 III
  Â Â Â Â Â   1990 96Á15
D7 Tai-An service area II 0Á5 0Á54 0Á66 III Â Â
 Â Â Â Â   1990 96Á15
D8 Ken-Ting 2 7Á59 7Á37 III
  Â Â Â Â Â   1993 122Á65
D9 Chung-Cheng University 2Á4 3Á49 4Á15 III Â
 Â Â Â Â Â   1991 100Á34
D10 Kadhsiung County I
Ã
12Á9 14Á81 14Á81 III Â Â Â Â Â  Â   1995 119Á19
D11 Wu-Lai 13 8Á19 8Á19 II Â

 Â Â Â Â Â Â 1992 115Á72
388 H W. Chen and N B. Chang
D27 Keelung 63 129Á63 126Á56 II Â Â  Â Â Â Â Â Â 1993 122Á10
D28 Yi-Lan 64 50Á85 49Á63 II Â Â
 Â Â Â Â Â Â 1993 122Á10
D29 Tan-Shui 75 104Á00 86Á70 III Â Â
 Â Â Â Â ± ± 1991 100Á00
D30 Guishuic 78 16Á30 15Á91 I Â Â Â Â Â Â Â Â Â 1993 122Á10
D31 Taichung 87Á5 76Á30 74Á48 II Â Â Â Â Â Â
 Â Â 1993 122Á10
D32 Yi-Lan 89Á5 50Á89 50Á22 II Â Â
 Â Â Â Â Â Â 1993 122Á10
D33 Pan-Hsin 95 60Á56 72Á37 II Â Â
 Â Â Â Â Â Â 1991 122Á10
D34 Tei-Ton 100 44Á44 44Á44 II Â Â Â Â Â
 Â Â Â 1995 119Á19
D35 Ping-Ton 130 103Á11 122Á56 III Â Â Â Â Â
 Â ± ± 1991 100Á28
D36 An-Pin 130 39Á63 39Á70 II Â Â
 Â Â Â Â Â Â 1995 118Á82
D37 erh-ien-hsi 132 71Á63 73Á63 II Â Â
 Â Â Â Â Â Â 1993 122Á10
D38 tou-fen 133 29Á00 47Á81 II Â Â
 Â Â Â Â Â Â 1992 72Á30
D39 Hsin-Chu 138
Ã
103Á70 103Á70 III Â  Â Â Â Â Â ± ± 1995 119Á19
D40 Kaohsiung 156
Ã
74Á07 74Á07 III Â  Â Â Â Â Â ± ± 1995 119Á19

Location of
wastewater
treatment
plant
Design
¯owrate
(10
3
CMD)
Total
construction
cost
(millions US$)
Normalized total
construction
cost
(1995 millions US$)
Operation
cost
(10
4
US$/year)
Level
g
of
treatment
Treatment process Year
of
bidding
C.C.I.

b
Â
b
        1987 75Á64
I2 Ta-Wu-Hun 1Á75 0Á56(1980)
2Á89(1991)
4Á40 29Á6 III
   Â  Â Â Â  Â 1980
1991
73Á50
100Á00
I3 Chia-Tai 2Á5 2Á96(1985)
2Á1(1993)
6Á89 54Á8 II
  Â Â Â  Â Â Â Â 1985
1993
72Á19
122Á10
I4 Yung-Kuang 3 2Á34 3Á91 48Á48 III
 Â Â Â Â Â Â  Â  1985 72Á19
I5 Yung-An 3Á2 1Á11(1983)
2Á22(1991)
4Á46 51Á9 III
 Â Â Â Â Â Â  Â  1983
1991
72Á40
100Á00
I6 Tao-Yuan 3Á3 2Á91 3Á01 35Á2 II Â Â Â Â Â
 Â Â Â Â 1992 115Á72
I7 Tou-Liu 3Á5 3Á7(1989) 4Á81 92Á6 II

I18 An-Ping 12(7) 3Á78(1981)
5Á93(1990)
13Á37 70Á37 III   Â Â  Â Â Â  Â 1981
1990
74Á73
96Á15
I19 Min-Hsiung 12 5Á93 18Á33 37Á04 III
 Â Â Â   Â Â Â  1985 72Á19
I20 Tu-Cheng 12 6Á83(1980)
4Á44(1990)
16Á62 83Á04 III
 Â Â Â  Â Â Â  Â 1980
1999
73Á5
96Á15
I21 Ping-Chen 12Á5 14Á07 14Á22 48Á19 III
  Â Â  Â Â Â Â  1994 117Á75
I22 Wu-Ku 12Á5 17Á59 20Á93 55Á56 II
  Â Â Â  Â Â Â Â 1991 100Á00
I23 Kaohsiung 15 13Á33 18Á52 495Á3 III
  Â Â  Â Â Â  Â 1981
1995
74Á73
119Á19
I24 Nan-Kang 16 17Á85 21Á26 109Á89 III
  Â Â  Â Â  Â  1991 100Á00
I25 Hsin-Chu (Hu-Kou) 21 11Á48 12Á41 162Á67 III
  Â Â  Â Â Â  Â 1980 73Á50
I26 Ta-She 22Á6 2Á22(1980)
2Á04(1990)

the activated sludge (AS), the oxidation ditch (OD),
the rotating biological contractor (RBC), the unit
developed in Virginia Initiative Plant in Norfolk,
Virginia (VIP), and the relevant anaerobic-
anoxic-aerobic processes (A
2
/O). Besides, the
tertiary treatment process involves the removal of
nitrogen and phosphorus content such that add-
itional expensive units, such as coagulation (CO)
and sand ®lter (SF), are required. These methods
are frequently applied to preventing the eutrophi-
cation issues and maintaining the essential water
quality in the reservoirs/lakes close to the water
intakes.
Since the cost structures of constructing waste-
water treatment plants may cover various economic
implications at different time frames, all the col-
lected cost data must be subject to a normalization
process before the regression practices are per-
formed. The Construction Cost Index (CCI) was
thus selected as an economic factor to normalize
these cost data into a common temporal basis.
Normalization for the location difference, however,
was neglected because Taiwan is a spatially smaller
area. The currency ratio used in this analysis is
approximately 27NT$/lUS$ in 1995. Such a normal-
ized cost database, representing a large-scale cali-
brated effort to integrate the nationwide baseline
information of wastewater treatment plants, may

remains unclear. Thus, this regression analysis can
be extended to identify the degree of economy of
scale and cost elasticity based on different types
of treatment facilities. Further observation in
Figures 2 and 4 con®rms that the variations of
total construction cost become larger as facility size
increases. This phenomenon does imply the exist-
ence of `heteroscedasticity' in statistics. Yet the
degree of heteroscedasticity of industrial waste-
water treatment plants is relatively smaller than
that of domestic wastewater treatment plants. To
identify the economy of scale easily and remove the
effect of heteroscedasticity in the least squares
regression analysis, the scatter plots of average
construction cost were therefore prepared for per-
forming the least squares regression analysis. Such
an alternation is equivalent to use the weighted
regression analyses in statistics. As shown in
Figures 3 and 5, the heteroscedasticity impact can
be reduced to a minimum extent as long as the
regression analysis is based on the plots of average
cost versus facility size. However, the heteroscedas-
ticity impact is not an issue in both fuzzy linear
regression and fuzzy goal regression practices since
they are designed to explore or explain such uncer-
tain cost structure in theory. As a consequence,
no pretreatment for the normalized datasets is
required before performing those fuzzy regression
analyses.
Model formulation and regression

practice using all the collected data of domestic
wastewater treatment plants as a whole to provide
an overall insight of the cost structure would
also become achievable. Yet the total numbers of
samples of the industrial wastewater treatment
plants do not support basic differentiation in
terms of primary, secondary, and tertiary treatment
levels such that it is impossible to conduct a sep-
arate regression analysis with respect to different
treatment levels. But dummy variables still can be
used for differentiating the cost impact due to using
different biological treatment technologies.
Results and discussions
LINDO
1
and SAS
1
software packages were used as
computer solvers in dealing with the required linear
or nonlinear regression analyses. Tables 4 and 5
address all the regression outputs in this study. The
sign conventions in these models are generally
acceptable in most of the cases. The regression
results appear not well ®tted to the least squares
regression model in the integrated case of industrial
wastewater treatment plants. But the situation
turns out to be better in the case of domestic
wastewater treatment plants. The numbers in par-
entheses attached below those estimates in the least
squares practices are t-ratios in statistics. Some of

1
 Q
The case for secondary
treatment process
CC  a
0
 Q
a1
EXP(a
2
 CS  a
3
 AS  a
4
 RBC  a
5
OD  a
6
A
2
/O)
The case for tertiary
treatment process
CC  a
0
 Q
a1
EXP(a
2
 CS  a

2
/O  a
7
 ETU)
Industrial CC  b
0
 Q
b1
EXP(b
2
 EQ  b
3
 CO  b
4
 FL  b
5
 AS  b
6
RBC  b
7
 OD  b
8
CO  b
9
 SF)
CC: total construction cost of wastewater treatment plants (1995Â10
6
US$); Q: design ¯ow-rate (tons/day; TPD). EQ: dummy variable designating the option of equalization tank; FL:
dummy variable designating the option of ¯otation; CS: dummy variable designating the option of contact-stabilization; AS: dummy variable designating the option of activated sludge
process; OD: dummy variable designating the option of oxidation ditch; RBC: dummy variable designating the option of rotating biological contactor; CO: dummy variable designating the

8
(CO) b
9
(SF)
LS 22Á26
(5Á76)
Ã
0Á85
(6Á49)
0Á77
(1Á64)
0Á44
(1Á83)
0Á76
(0Á78)
À0Á16
(À0Á44)
0Á66
(1Á88)
0Á09
(0Á06)
À0Á29
(À1Á08)
0Á01
(À0Á39) 0Á69
7Á84
FLR 0Á82 0Á99 0Á71 0Á01 1Á22 À0Á33 0Á33 À0Á36 À0Á25 0Á36
FGR 17Á39 0 97 0Á71 0Á39 0Á76 À0Á18 0Á38 0Á09 À 0Á29 0Á01
Ã
t-ratios corresponding to the estimates of repressors in the regression models.

)
F values
The case for
primary
treatment
level
LS À406Á5
(À0Á91)
Ã
5Á68
(8Á33)
± ± ± ± ± ± ± 0Á94 69Á35
FLR À43Á4 0Á4 ± ± ± ± ± ± ± ± ±
FGR 1Á9 0Á18 ± ± ± ± ± ± ± ± ±
The case for
secondary
treatment
level
LS 187025
(1Á22 Â 10
6
)
0Á84
(11Á43)
À9Á16
(À9Á2 Â 10
À7
)
À8Á53
(À8Á6 Â 10

)
1Á66
(5Á81 Â 10
À7
)
1Á06
(6Á08 Â 10
À8
)
1Á11
(9Á05 Â 10
À8
)
± 0Á82 13Á18
FLR 0Á10 0Á91 1Á56 1Á24 1Á38 0Á79 0Á93 1Á32 ± ± ±
FGR 33Á95 0Á92 0Á38 0Á00 0Á54 0Á48 0Á06 1Á03 ± ± ±
The integrated
case for all
treatment
levels
LS 0Á6
(7Á35)
Ã
0Á84
(17Á89)
0Á83
(2Á08)
0Á94
(3Á56)
1Á55

illustrates the principles for the estimation of sam-
ples variance when applying the least squares and
fuzzy regression analyses. In particular, the estima-
tion of samples variance when applying the FLR and
Figure 6. The prediction of construction cost of primary
domestic wastewater treatment process by different
regression models.
Figure 7. The prediction of construction cost of second-
ary domestic wastewater treatment process by different
regression models.
Figure 8. The prediction of construction cost of tertiary
domestic wastewater treatment process by different
regression models.
Figure 9. The prediction of construction cost of mixed
types of domestic wastewater treatment process by
different regression models.
Figure 10. The prediction of construction cost of mixed
types of industrial wastewater treatment process by
different regression models.
396 H W. Chen and N B. Chang
FGR methods must rely on the use of linear and goal
programming models, respectively. While Figure 12
shows the bandwidth of con®dence interval over
samples based on the LS method, Figures 13 and 14
describe the fuzzy trajectories over observations
based on the FLR and FGR methods. Note that
discrepancies always exist in the presentation of
variations via the use of FLR and FGR analyses. The
numbers shown on each small axis in Figure 14
correspond to the degree of ®tting for each predicted

treatment levels of industrial wastewater treatment
plants. Table 10 con®rms this conclusion. When
taking Tables 9 and 10 into account simultaneously,
it can be recognized that using the FGR method may
always generate the lowest value of average of the
absolute error percentage but exhibit relatively
higher values of sum of absolute residual and sum
of squared residual. In any circumstances, the FGR
method can always achieve a better performance
than the FLR method.
It can be concluded that the fuzzy linear regres-
sion model might not be suitable to deal with the
nonlinear regression issues although the proposed
model in this study can be linearized in advance
before the regression analysis is performed; on the
other hand, a higher forecasting capability of cost
estimation can be con®rmed via the use of the FGR
method in most of the cases. This is much due to the
fact that the FGR method may handle the complex-
ity of observations with smaller facility sizes, poten-
tial outliers, and/or non-homogeneous features in
sample variance over facility sizes. Conventional
Figure 11. The expression of uncertainties in relation to different regression models.
Estimating costs of constructing wastewater treatment plants 397
Figure 12. The prediction results and associated con®dence interval for industrial wastewater treatment plants based on
the LS method.
Figure 13. Fuzzy trajectory of fuzzy linear regression model for industrial wastewater treatment plants.
398 H W. Chen and N B. Chang
least squares method cannot control fragmental and
uncertain datasets easily. This ®nding, coupled with

48 1320 267Á04 262Á43 À4Á61 480Á09 213Á05 236Á69 À30Á35
Sum of absolute residual

80 391 82
Sum of squared residual

2130 55808 3001
Average of absolute error percentage

166% 285% 166%
aX e
i
 0; bX

residual
j j
; gX

residual
2
; dX

Y À residual
j j
residual
 
6100%X
Estimating costs of constructing wastewater treatment plants 399
Table 7. The comparative study of secondary wastewater treatment plants
Plant

36 130 39Á70 82Á00 42Á30 48Á37 8Á67 86Á77 47Á07
37 132 73Á63 83Á00 9Á37 58Á94 À14Á69 87Á94 14Á31
38 133 47Á81 83Á60 35Á79 59Á51 11Á70 88Á53 40Á72
41 165 66Á63 100Á00 33Á37 59Á79 À6Á84 107Á09 40Á46
43 176 113Á33 106Á00 À7Á33 68Á49 À44Á84 113Á37 0Á04
46 331 215Á26 180Á00 À35Á26 71Á34 À143Á92 198Á00 À17Á26
Sum of absolute residual

327 496 314
Sum of squared residual

13618 37076 13812
Average of absolute error percentage

37% 75% 29%
aX e
i
 0; bX

residual
j j
; gX

residual
2
; dX

Y À residual
j j
residual

14 17 31Á74 24Á45 À 7Á29 25Á94 À 5Á80 17Á26 À 14Á48
15 18 31Á30 40Á29 8Á99 31Á47 0Á17 26Á70 À 4Á60
16 21Á7 20Á37 20Á87 0Á50 20Á86 0Á49 20Á37 0Á00
17 22 34Á48 29Á71 À 4Á77 32Á83 À 1Á65 21Á90 À 12Á58
18 22Á8 22Á22 21Á67 À 0Á55 21Á82 À 0Á40 21Á32 À 0Á90
19 23 35Á89 30Á73 À 5Á16 34Á19 À 1Á70 22Á82 À 13Á07
21 34 53Á07 53Á07 0Á00 53Á24 0Á17 53Á07 0Á00
22 38 36Á30 44Á93 8Á63 54Á07 17Á77 36Á29 À 0Á01
24 41 60Á67 47Á59 À 13Á08 57Á95 À 2Á72 38Á93 À 21Á74
29 75 86Á70 75Á17 À 11Á53 100Á59 13Á89 68Á03 À 18Á67
35 130 122Á60 113Á98 À 8Á62 122Á15 À 0Á45 113Á10 À 9Á50
Sum of absolute residual

82 54 113
Sum of squared residual

675 573 1579
Average of absolute error percentage

30% 19% 27%
aX e
i
 0; bX

residualj j; gX

residual
2
; dX


value
Residual Predicted
value
Residual
1 0Á1 0Á48 0Á31 À0Á17 0Á29 À0Á19 0Á27 À0Á21
2 0Á2 0Á14 0Á29 0Á15 0Á25 0Á11 0Á23 0Á08
3 0Á2 0Á29 0Á33 0Á04 0Á34 0Á06 0Á29 0Á00
4 0Á27 3Á63 1Á46 À2Á17 2Á83 À0Á80 0Á95 À2Á68
5 0Á45 1Á47 2Á25 0Á78 1Á50 0Á03 1Á47 0Á00
6 0Á5 1Á07 1Á20 0Á12 1Á21 0Á13 1Á08 0Á01
7 0Á5 0Á66 1Á34 0Á68 1Á64 0Á98 1Á35 0Á69
8 2 7Á37 3Á85 À3Á52 4Á14 À3Á23 3Á55 À3Á82
9 2Á4 4Á15 4Á49 0Á35 4Á87 0Á72 4Á15 0Á00
10 12Á9 14Á81 14Á81 0Á00 16Á45 1Á63 14Á82 0Á00
11 13 8Á19 9Á98 1Á80 10Á35 2Á16 8Á19 0Á00
12 13 22Á70 20Á96 À1Á74 29Á87 7Á17 22Á14 À0Á56
13 16 44Á81 13Á33 À31Á49 16Á98 À27Á84 12Á25 À32Á56
14 17 31Á74 26Á29 À5Á45 37Á93 6Á19 27Á87 À3Á87
15 18 31Á30 50Á70 19Á41 30Á16 À1Á13 34Á85 3Á55
16 21Á7 20Á37 20Á84 0Á47 22Á94 2Á57 21Á30 0Á93
17 22 34Á48 32Á68 À1Á80 47Á71 13Á23 34Á77 0Á29
18 22Á8 22Á22 21Á72 À0Á50 23Á97 1Á75 22Á23 0Á01
19 23 35Á89 33Á93 À1Á96 49Á64 13Á75 36Á13 0Á24
20 33 22Á15 24Á55 2Á41 32Á33 10Á18 22Á80 0Á65
21 34 53Á07 57Á75 4Á68 60Á50 7Á42 71Á50 18Á42
22 38 36Á30 51Á84 15Á54 77Á60 41Á31 55Á58 19Á29
23 38 22Á70 27Á66 4Á96 36Á66 13Á95 25Á73 3Á03
24 41 60Á67 55Á28 À5Á39 83Á03 22Á37 59Á33 À1Á34
25 44 28Á15 31Á30 3Á16 41Á77 13Á62 29Á18 1Á03
26 54 42Á96 37Á21 À5Á75 50Á12 7Á15 34Á79 À8Á17

residual
j j
Y g X

residual
2
; dX

Y À residual
j j
residual
 
6100%X
Estimating costs of constructing wastewater treatment plants 401
disposal programs. All of them require accurate
prediction of cost information to support a success-
ful planning and assessment. But cost analysis
involves many uncertain factors that could become
one of the impediments in decision-making. Apart
from providing direct investigation of the issues of
cost structures for building wastewater treatment
plants in an uncertain environment using different
model formulations, this study also tries to answer
what is the optimal size of small community that
would suffer the least degree of higher cost in
wastewater treatment and what are the possible
uncertainties in estimating the cost of constructing
wastewater treatment plants when using different
estimation methods. The case study starts from the
illustration of the latest development information of

Residual Predicted
value
Residual
1 0Á4 0Á45 0Á55 0Á54 0Á48 0Á03 0Á45 0Á00
2 1Á75 4Á40 4Á40 5Á30 5Á55 1Á14 4Á40 0Á00
3 2Á5 6Á89 5Á91 0Á86 5Á79 À1Á10 6Á89 0Á00
4 3 3Á91 4Á27 À0Á52 4Á92 1Á01 4Á19 0Á27
5 3Á2 4Á46 4Á53 À0Á92 5Á25 0Á79 4Á46 0Á00
6 3Á3 3Á01 3Á69 0Á38 3Á70 0Á69 3Á01 0Á00
7 3Á5 4Á81 4Á20 À0Á10 4Á00 À0Á82 4Á81 0Á00
8 5 12Á33 9Á70 À1Á70 8Á96 À3Á38 10Á09 À2Á25
9 5 9Á19 8Á30 0Á72 8Á27 À0Á92 9Á18 0Á00
10 6 14Á22 9Á77 À3Á10 10Á20 À4Á02 9Á23 À4Á99
11 6 9Á33 9Á77 1Á79 10Á20 0Á87 9Á23 À0Á10
12 7 11Á81 10Á36 6Á84 8Á29 À3Á52 11Á72 À0Á09
13 10 15Á15 15Á49 0Á19 16Á92 1Á77 15Á15 0Á00
14 10Á4 13Á93 12Á05 3Á62 9Á56 À4Á37 11Á64 À2Á28
15 11 7Á00 10Á37 1Á59 9Á68 2Á68 10Á93 3Á93
16 11 10Á93 10Á37 À2Á34 9Á68 À1Á25 10Á93 0Á00
17 12 13Á37 13Á71 5Á84 11Á01 À2Á36 13Á38 0Á01
18 12 13Á37 13Á71 5Á84 11Á01 À2Á36 13Á38 0Á01
19 12 18Á33 18Á39 À10Á20 27Á83 9Á50 17Á99 À0Á34
20 12 16Á62 10Á38 À9Á12 10Á87 À5Á75 9Á06 À7Á56
21 12Á5 14Á22 18Á95 3Á44 21Á10 6Á88 18Á80 4Á58
22 12Á5 20Á93 25Á26 0Á42 28Á49 7Á56 32Á80 11Á87
23 15 18Á52 16Á77 3Á59 13Á74 À4Á78 16Á61 À1Á91
24 16 21Á26 21Á85 10Á14 18Á80 À2Á46 26Á12 4Á87
25 21 12Á41 22Á73 14Á92 19Á16 6Á76 23Á01 10Á61
26 22Á6 15Á48 18Á38 À4Á30 20Á34 4Á86 16Á74 1Á26
27 25 19Á48 15Á91 À4Á18 13Á77 À5Á71 15Á97 À3Á51

of the centralized treatment facilities require more
comprehensive evaluation. This study con®rms that
moderate economy of scale does exist in construct-
ing both the domestic and industrial wastewater
treatment plants. The optimal size of a domestic
wastewater treatment plant is approximately
equivalent to 15 000 CMD and higher in Taiwan.
The greater the design capacity, the less the unit
construction cost appears. On the other hand, the
optimal size of an industrial wastewater treatment
plant turns out to be more ¯exible. It falls in
between 6000 CMD and 20 000 CMD. Small com-
munities do not bene®t the economy of scale that are
possible with the construction of wastewater man-
agement facilities for larger communities. Thus,
integrated planning with neighboring sewage sys-
tems can be encouraged when shipping cost is not
an in¯uential factor.
Acknowledgements
The authors acknowledge the ®nancial support from the
National Science Council (NSC89-2211-E-006-005) in
Taiwan and helpful comments raised by all anonymous
referees in this study.
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Appendix I: the theory of fuzzy
linear regression models
In fuzzy sets theory (Zedah, 1965), a fuzzy set A can
be characterized by a membership function, usually
denoted m, which assigns to each object of a domain
its grade of membership in A. The more an element
or object can be said to belong to a fuzzy set A,
the closer to 1 is its grade of membership. Such
an uncertainty description has been successfully
applied to many real-world problems for engineer-
ing planning, design, operation, management, and
control. The following sections provide a brief
review of previous fuzzy linear regression model
(Tanaka et al., 1982) and the derivation of the
proposed fuzzy goal regression model.
Tanaka et al. (1982) introduced the idea of fuzzy
linear regression. According to their discussions,
fuzziness must be considered in systems where
human estimation is in¯uential. The fuzzy regres-
sion structure actually focuses on ®nding a set of
fuzzy parameters to achieve a speci®c mapping
between the explanatory variables (independent
variables) and the explained variable (dependent

j j
c
j
if a
j
À c
j
a
j
a
j
 c
j
0 otherwise
@
A1
Given fuzzy parameter A  {a,c}, described by the
center a and the width c, the fuzzy linear function is
Y  A
1
x
1
 A
2
x
2
 Á Á Á  A
n
x
n

y
i
 
1 À
y
i
Àx
t
a
j j
c
t
x
j j
if x=0
1 if x  0, y  0
0 if x  0, y=0
V
b
`
b
X
A3
where and jxj  jx
1
jY jx
2
j, F F F jx
n
j





i
c
t
jx
i
j


i

j
c
t
j
x
ij




X
A4
which can be minimized subject to h
i
! H, where
H is the prescribed degree of ®tting and h





À e
i
A5
where e
i
is the subjective measure of vagueness for
the observed Y value. The following linear program-
ming model, in (A6)±(A9), is therefore used to ®nd
fuzzy parameter A
*
i
 (a
i
,c
i
):
minimize J 

i
c
t
x
i
j j A6
subject to
a

x
ij




! Ày
i
 1 À He
i
Vi
A8
c
j
! 0 Vj A9
in which H is the minimum required degree of
®tting.
Appendix II: the theory of fuzzy
goal regression models
Fuzzy goal regression model is derived from the
concept of fuzzy goal programming. Since the
Figure A2. The fuzzy type of input-output data in fuzzy
linear.
Non fuzzy input-fuzzy output data
Sample number Fuzzy output, Y
i
Inputs, X
ij
1 Y
1

ming is generally designed to satisfy the fuzzy
objectives and constraints simultaneously, a deci-
sion space in a fuzzy environment is thus de®ned as
the intersection of those membership functions
corresponding to those fuzzy objectives and con-
straints (Zimmermann, 1985). Therefore, if {m
G1
,
m
G2
, F F F , m
G2m
} and {m
C1
, m
C2
, F F F , m
Cp
} are denoted
as the fuzzy membership functions for those fuzzy
objectives {G
l
, G
2
, F F F , G
m
} and fuzzy constraints
{C
1
, C

in decision-making. The aspiration level of each
fuzzy membership function is actually a common
membership value achieved in the decision set,
m
D
(s), ®nally. Thus, the max-min convolution
requires maximizing the minimum membership
values of those elements, as below (Zimmermann,
1978):
max
s
m
D
max
s
minfm
G
1
, m
G
2
, F F F , m
G
2m
;
m
C
1
, m
C

ever, in many circumstances, the linguistically
vague statement such as `around b' is more
acceptable and comfortable in the decision-making
process. For instance, `the annual cost should be
about $20 000'. Such a goal programming with
linguistic goals can be formulated as:
find x
such that (Ax)
i

~
b
i
, i  1,2, F F F ,m
x ! 0 A11
where
~
b
i
express linguistic goals such as `the cost
should be around
~
b
i
.' We may have the following
similar problem:
find x
such that (Ax)
i
%b

i
x

Ax
i
À b
i
À d
i

Â
ad
i
b
i
 d
i
 À Ax
i
ÃÂ
ad
i
0
V
b
`
b
X
if b
i

Where d
i
are the maximum acceptable deviations
(subjectively determined) from the center b
i
. On
the other hand, triangular membership functions
can also be expressed (b
i
, b
i
À d
i
, b
i
 d
i
) where b
i
is the most preferred value and b
i
À d
i
(the most
pessimistic value) and b
i
 d
i
(the most optimistic
value) are the least preferred values, Vi.

b
i
 d
i
 À Ax
i
ÂÈ
ad
i
g
, for other i
such that b
i
Ax
i
b
i
À d
i
and x ! 0
A15
By linking Equations (A14) and (A15) together, we
obtain:
max a
subject to
Ax
i
À b
i
À d

frequently used for those fuzzy constraints with
`approximately smaller than or equal to' and
`approximately larger than or equal to' relation-
ships, respectively, as shown in most previous liter-
ature (Bellman and Zadeh, 1970; Zimmermann,
1978; Zimmermann, 1985).
Hence, the fuzzy structure illustrated in (A16)
can be applied for solving the fuzzy goal regression
model. In a fuzzy goal regression analysis, the
proposed linear regression model (

a
i
x
i
) can be
viewed as a linear function that is designed to
approach the target value or the observed value
of dependent variable Y. The regression effort
for each set of observation, corresponding to one
dependent variable and several independent vari-
ables, is to achieve the goal where the target value
is approximated by a set of fuzzy parameters, a
i
,
multiplied by the observed input data, x
i
. For n sets
of observations, we have n set fuzzy goals to be
handled in the fuzzy goal-programming model. The