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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. # 2002 Elsevier Science Ltd. All rights reserved. Keywords: wastewater treatment, cost analysis, fuzzy linear regression, fuzzy goal regression, forecasting, environmental management. Introduction Only in the last decade have annual per capita wastewater generation rates begun to demonstrate rapid increase due to fast economic development in Taiwan. The concerns with domestic and industrial wastewater disposal have become of signi®cance while the issues of water quality management plans turn out to be critical in many river basins. Centralized treatment facilities have rapidly emerged to aid in processing of domestic and indus- trial ef¯uents in those urban/suburban areas. In many water pollution control programs, the choice of a primary or higher wastewater treatment levels 0301±4797/02/$ ± see front matter # 2002 Elsevier Science Ltd. All rights reserved. * Corresponding author. Email: nibin@giga.net.tw is intimately related to land availability, budget, regulations, and managerial concerns. Two chan- ging situations are worthwhile noticing at this moment. First, with the condition of more stringent ef¯uent standards for river and marine discharge being promulgated in the last decade, more sophisticated treatment processes have to be putting into practice for simply meeting the of®cial requirements. Second, the growing recogni- 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, many regression analyses have been conducted to analyze the capital cost structures for waste treat- ment plants (Graves et al., 1970; Robert, 1975; Chang and Wang, 1995a). But the efforts of those cost estimation techniques for wastewater treat- ment in the earlier stage generally cover various statistical regression models, such as simple linear regression, multiple linear regressions, and other nonlinear regression models. However, measured quantities are always burdened with different types of uncertainties. Due to the in¯uential im- pacts by human estimation, more relevant proper- ties associated with forecasting techniques are worthwhile exploring by a more systematic, com- parative, and comprehensive approach. 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¯u- ential. Fuzzy regression analysis, representing an alternative to the conventional statistical regres- sion techniques, has been becoming a promising approach within the last decade. Its application potentials have been relatively high in many aspects in the way to aid in engineering planning, design, operation, management, and control. Fuzzy linear regression analysis has been the focus of a number of studies utilizing advanced algorithms. Improvement of this method in the earlier stage can be found in the literature (Tanaka and Asia, 1984a, 1984b; Jajuga, 1986; Tanaka, 1987; Tanaka 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, 1999). This practice may grow in importance in the future because of limited resources available as well as rapid economic and population growth in many countries. This paper serves as a companion study of Chang et al. (1996) and Wen and Lee (1999). It is designed to recast a well-known problem of con- struction cost estimation for both domestic and industrial wastewater treatment plants via a com- parative framework. In short, it differs from these earlier studies in several ways. First, this analysis takes both domestic and industrial wastewater 384 H W. Chen and N B. Chang treatment plants into account separately based on a more thorough database. Second, with the aid of fuzzy goal programming technique, advanced information of fuzzy goal regression is coordinated and applied to aiding in a detailed evaluation of forecasting accuracy of constructing wastewater treatment plants in an uncertain environment. Third, the proposed comparative study in relation to three forecasting methods is systematically assessed on the basis of multiple performance indexes. Technologies of regression analyses consist of the conventional least square regression (LS) method, the fuzzy linear regression (FLR) method, and the newly derived fuzzy goal regression (FGR) method. And, fourth, the nonlinear cost structures in relation to economy of scale along with the use of dummy variables for the choice of technologies 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 density function and membership function are used to illustrate the situation of `probability' and `fuzziness,' respectively. While the random variable is used for the description of uncertain statistical implication, the fuzzy membership function is therefore de®ned for illustrating the imprecision existing in the real world systems. Various types of linear and nonlinear functions were suggested as probability density functions and fuzzy member- ship functions in the literature. Usually a large number of samples are generally required to iden- tify the probability density function for the occur- rence of a phenomenon, while subjective description of the fuzzy membership function is usually applied in the determination of fuzziness. 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 a membership value is usually called `aspiration level' or `satisfaction level' in decision analysis. The fuzzy sets theory, described by the membership functions, is identi®ed as an alter- native approach to describe the vagueness in the planning goals and the impreciseness of related parameter values, which could be intimately linked to more practical or realistic aspects in decision- making (Chang and Chen, 1997). As a consequence, fuzzy sets theory has received wide attention for illustrating various types of environmental man- agement issues. Most of the traditional regression models for 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, the prediction accuracy of fuzzy linear regression model cannot always be guaranteed better than that of conventional least-squares regression mod- els, although fuzzy regression allows the inclusion of expert knowledge or fuzzy information in the model in advance (Chang et al., 1996). Chang et al. (1996) further proposed the fuzzy goal regression model in order to improve the performance in comparison to fuzzy linear regression outputs. To further upgrade the capability of fuzzy goal regres- sion analysis, this paper provides two revised fuzzy goal regression approaches (see Appendix II), which address the variations or uncertainties in a fuzzy regression analysis by a series of ¯exible formulations in a model. But it still has to identify the most attractive model structure among alter- natives using a fuzzy goal programming procedure (Chang et al., 1996). The following case study is prepared for a comparative analysis of these three methods representing various types of uncertainties involved in the cost estimation of constructing wastewater treatment plants. The methods in- cluded in this case study cover the LS, FLR, and FGR. Both industrial and domestic wastewater treatment plants are included for forecasting practices in this case study. Case study Taiwan is located at the west Paci®c Rim of Asian Continental Shelf. With a small area of about 36 000 square kilometers and over twenty-three 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 procedure for gaining advanced realization of inherent underlying cost structure and for pro- moting forecasting accuracy as well. Figure 1 shows the geographical locations of those planned and existing wastewater treatment plants in Taiwan. Tables 1 and 2 not only present a thor- ough database but also summarize the relevant engineering parameters for those domestic and industrial wastewater treatment plants, respective- ly. The plant numbers marked associated with each treatment plant in Figure 1 are consistent with the numbers listed in Tables 1 and 2 and they will be used thereafter in the context of this paper. To effectively assess the cost structure required for wastewater treatment, the fuzzy linear regression and fuzzy goal regression approaches (FGR II), as elucidated in Appendix I and Appendix II, are thus applied to gaining a better comparative insight and perform more valuable comparisons to the conventional statistical regres- sion method in this practice. Data collection, normalization, and analysis Cost information associated with 48 domestic was- tewater treatment plants and 29 industrial waste- water treatment plants was collected and included in the cost database within a thorough investiga- tion. As listed in Tables 1 and 2, using the primary, secondary, and tertiary treatment levels may fur- ther classify the collected samples with respect 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 of treatment Treatment process Year of bidding C.C.I. f I c II d III e EQ a CS a AS a RBC a VIP a A 2 /O a OD a CO a SF a D1 Hsin-Ying service area (in highway) 0Á1 0Á48 0Á48 III b  b      1995 119Á19 D2 Hai-Hu Vacation Center 0Á2 0Á11 0Á14 II        1990 94Á65 D3 The Home of Charity 0Á2 0Á17 0Á29 II        1985 72Á76 D4 Wu-Ling Hotel (Ken-Ting) 0Á27 2Á59 3Á63 III      1980 96Á15 D5 Kai-Sa Hotel (Ken-Ting) 0Á45 0Á89 1Á47 III       1986 72Á14 D6 Tai-An service area I 0Á5 0Á87 1Á07 III      1990 96Á15 D7 Tai-An service area II 0Á5 0Á54 0Á66 III       1990 96Á15 D8 Ken-Ting 2 7Á59 7Á37 III      1993 122Á65 D9 Chung-Cheng University 2Á4 3Á49 4Á15 III       1991 100Á34 D10 Kadhsiung County I à 12Á9 14Á81 14Á81 III       1995 119Á19 D11 Wu-Lai 13 8Á19 8Á19 II         1995 119Á27 D12 Chi-Mei 13 18Á96 22Á70 III       1991 100Á34 D13 Jui-Fang 16 45Á74 44Á81 II         1993 121Á64 D14 Ton-Kang 17 26Á70 31Á74 III       ± ± 1991 100Á28 D15 Wu-Chia (Kaohsiung) 18 18Á96 31Á30 III       ± ± 1993 72Á19 D16 Tei-Ton 21Á7 20Á37 20Á37 III       ± ± 1995 119Á19 D17 Nei-Pu 22 29Á00 34Á48 III       ± ± 1991 100Á28 D18 Tei-Ton County 22Á8 22Á22 22Á22 III       ± ± 1995 119Á19 D19 Chao-Wan 23 30Á19 35Á89 III       ± ± 1991 100Á28 D20 Nan-Tou 33 18Á52 22Á15 II         1991 99Á60 D21 Pe-Kang 34 54Á56 53Á07 III        1993 122Á55 D22 Chu-Pe 38 30Á56 36Á30 III       ± ± 1991 100Á34 D23 Tei-Ton 38 23Á26 22Á70 II         1993 122Á10 D24 Miao-Li 41 62Á19 60Á67 III        1993 119Á69 D25 Tsao-Tun 44 23Á52 28Á15 II         1991 99Á60 D26 Lin-Hae 54 44Á44 42Á96 II         1992 115Á72 388 H W. Chen and N B. Chang D27 Keelung 63 129Á63 126Á56 II         1993 122Á10 D28 Yi-Lan 64 50Á85 49Á63 II         1993 122Á10 D29 Tan-Shui 75 104Á00 86Á70 III       ± ± 1991 100Á00 D30 Guishuic 78 16Á30 15Á91 I          1993 122Á10 D31 Taichung 87Á5 76Á30 74Á48 II         1993 122Á10 D32 Yi-Lan 89Á5 50Á89 50Á22 II         1993 122Á10 D33 Pan-Hsin 95 60Á56 72Á37 II         1991 122Á10 D34 Tei-Ton 100 44Á44 44Á44 II         1995 119Á19 D35 Ping-Ton 130 103Á11 122Á56 III       ± ± 1991 100Á28 D36 An-Pin 130 39Á63 39Á70 II         1995 118Á82 D37 erh-ien-hsi 132 71Á63 73Á63 II         1993 122Á10 D38 tou-fen 133 29Á00 47Á81 II         1992 72Á30 D39 Hsin-Chu 138 à 103Á70 103Á70 III       ± ± 1995 119Á19 D40 Kaohsiung 156 à 74Á07 74Á07 III       ± ± 1995 119Á19 D41 Hau-Lien 165 68Á52 66Á63 II         1993 102Á42 D42 Nan-Zu 175 26Á30 40Á00 I          1988 78Á36 D43 Yung-Kuang 176 116Á11 113Á33 II         1992 117Á27 D44 Tao-Yuan 263 à 433Á33 440Á41 II         1993 122Á67 D45 Ti-Hau 274 à 3Á48 5Á64 I          1990 73Á5 D46 Chung-Li 331 221Á56 215Á26 II         1993 122Á65 D47 Chung-Chou 560 62Á04 102Á13 I          1983 72Á40 D48 Pa-Li 1320 25Á93 267Á04 I          1992 115Á72 a: EQ: equalization tank; CS: contact-stabilization; AS: activated sludge; OD: oxidation ditch; RBC: rotating biological contactor; CO: coagulation; SF: sand ®lter; A 2 /O: anaerobic-anoxic- aerobic process. b: ` ' represents the inclusion of the designated unit; `Â' represents the exclusion of the designated unit ; `±' represents no information. c: Primary industrial wastewater treatment process. d: Secondary industrial wastewater treatment process. e: Tertiary industrial wastewater treatment process. f: Construction Cost Index in Taiwan. g: Level of treatment: I (primary treatment), II (secondary treatment), and III (tertiary treatment). Estimating costs of constructing wastewater treatment plants 389 Table 2. Database of all industrial wastewater treatment plants in Taiwan Plant no. 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. f I c II d III e EQ a CO a FL a CS a AS a RBC a A 2 /O a OD a CO a SF a I1 Feng-Shan 0Á4 0Á28 0Á45 13Á5 II b  b        1987 75Á64 I2 Ta-Wu-Hun 1Á75 0Á56(1980) 2Á89(1991) 4Á40 29Á6 III      1980 1991 73Á50 100Á00 I3 Chia-Tai 2Á5 2Á96(1985) 2Á1(1993) 6Á89 54Á8 II        1985 1993 72Á19 122Á10 I4 Yung-Kuang 3 2Á34 3Á91 48Á48 III        1985 72Á19 I5 Yung-An 3Á2 1Á11(1983) 2Á22(1991) 4Á46 51Á9 III        1983 1991 72Á40 100Á00 I6 Tao-Yuan 3Á3 2Á91 3Á01 35Á2 II          1992 115Á72 I7 Tou-Liu 3Á5 3Á7(1989) 4Á81 92Á6 II         1989 92Á00 I8 Lung-Te 5 0Á93(1986) 11Á11(1993) 12Á33 42Á7 III       1993 122Á10 I9 Kuang-Hua 5 7Á41 9Á19 66Á7 I         1990 96Á15 I10 Ping-Nan 6 14Á6 14Á22 55Á3 III       1993 122Á10 I11 Ta-Li 6 7Á56 9Á33 ± b III       1990 96Á15 I12 Chuan-Hsing 7 12Á1 11Á81 60Á4 III      1993 122Á10 I13 Kuan-Tien 10 10Á4 15Á15 92Á5 III       1988 81Á57 I14 Kuan-Yin 10Á4 9Á63(1989) 1Á4(1992) 13Á93 103Á7 III       1989 1992 92Á00 115Á72 I15 Hsin-Ying 11 4Á4 7Á00 6Á22 III        1987 75Á64 I16 Taichung (Yu-Shin) 11 6Á67 10Á93 87Á78 III        1984 72Á64 I17 Fang-Yung 12 13Á7 13Á37 62Á22 III       1993 122Á10 390 H W. Chen and N B. Chang I18 An-Ping 12(7) 3Á78(1981) 5Á93(1990) 13Á37 70Á37 III       1981 1990 74Á73 96Á15 I19 Min-Hsiung 12 5Á93 18Á33 37Á04 III       1985 72Á19 I20 Tu-Cheng 12 6Á83(1980) 4Á44(1990) 16Á62 83Á04 III        1980 1999 73Á5 96Á15 I21 Ping-Chen 12Á5 14Á07 14Á22 48Á19 III       1994 117Á75 I22 Wu-Ku 12Á5 17Á59 20Á93 55Á56 II        1991 100Á00 I23 Kaohsiung 15 13Á33 18Á52 495Á3 III       1981 1995 74Á73 119Á19 I24 Nan-Kang 16 17Á85 21Á26 109Á89 III      1991 100Á00 I25 Hsin-Chu (Hu-Kou) 21 11Á48 12Á41 162Á67 III       1980 73Á50 I26 Ta-She 22Á6 2Á22(1980) 2Á04(1990) 9Á26(1995) 15Á48 114Á81 III        1980 1995 73Á50 92Á00 119Á19 I27 Taichung 25 12Á22 19Á48 130Á89 II          1981 74Á73 I28 Ta-Yuan 25 2Á22(1985) 31Á11(1995) 34Á78 333Á33 III       1985 1995 72Á19 119Á19 I29 Chung-Li (Nei-Li) 33Á5 876 32Á44 138Á52 III        1995 119Á19 a: EQ: equalization tank; FL: ¯otation; CS: contact-stabilization; AS: activated sludge; OD: oxidation ditch; RBC: rotating biological contactor; CO coagulation; SF: sand ®lter; A 2 /O: anaerobic-anoxic-aerobic process; FL: ¯oatation tank. b: ` ' represents the inclusion of the designated unit; `Â' represents the exclusion of the designated unit ; `±' represents no information. c: Primary industrial wastewater treatment process. d: Secondary industrial wastewater treatment process. e: Tertiary industrial wastewater treatment process. f: Construction Cost Index in Taiwan. g: Level of treatment: I (primary treatment), II (secondary treatment), and III (tertiary treatment). Estimating costs of constructing wastewater treatment plants 391 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 serve as a useful tool for the subsequent least square and fuzzy regression analyses. Two sets of diagrams (Figs 2±3 and 4±5) were arranged to address the behaviour of the normalized cost datasets. They are addressed in terms of total construction costs and average construction costs versus design ¯ow-rate, respectively. Although there are several in¯uential observations (i.e. outliers) that can be identi®ed in the database, deleting these observations is not encouraged in the applications. In the economic theory, it is anticipated that the return-to-scale curve of construction cost of waste- water treatment facilities actually ¯attens out below some minimum size. As a result, conventional wastewater management facilities for small com- munities often cost signi®cantly more per capita to construct when compared to those for larger cities. Once the small communities become popular along Figure 2. The distribution of total construction cost of all domestic wastewater treatment plants in Taiwan. Figure 3. The distribution of unit construction cost of all domestic wastewater treatment plants in Taiwan. Figure 4. The distribution of total construction cost of all industrial wastewater treatment plants in Taiwan. Figure 5. The distribution of unit construction cost of all industrial wastewater treatment plants in Taiwan. 392 H W. Chen and N B. Chang [...]... technologies, therefore, can be formulated as an aggregate exponential term in the model Except the separate analysis designed for each treatment level, an integrated 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.. .Estimating costs of constructing wastewater treatment plants the river basin, this would contribute to the increased per capita cost in the society But the sensitivity of the rate of increased per capita cost 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... turns out to be better in the case of domestic wastewater treatment plants The numbers in parentheses attached below those estimates in the least squares practices are t-ratios in statistics Some of them pass the inference test with respect to 5% level of signi®cance But many of the dummy variables chosen in those separate cases of least squares forecasting analysis are not statistically signi®cant This... variable designating the option of oxidation ditch; RBC: dummy variable designating the option of rotating biological contactor; CO: dummy variable designating the option of coagulation; SF: dummy variable designating the option of sand ®lter; A2 /O: dummy variable designating the option of anaerobic-anoxic-aerobic process; FL: dummy variable designating the option of ¯oatation tank, ETU: dummy variable... ®ndings clearly indicate that the FLR method cannot always be favoured over the others except for the case of analyzing tertiary wastewater treatment plants Although the LS method always has better performance in terms of the index of `sum of squared residual', the FGR method may present relatively better outputs in terms of the other two indexes considered Table 9, representing an integrated analysis. .. 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... 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 variable) If Y and x are the set of dependent vari- Figure A1 Fuzzy set of parameter A: A approximates a ables and independent variables respectively, the mapping can be performed by using fuzzy function Y (x, A) , where A. .. signi®cant in search of the better management strategy in the real world applications Estimating costs of constructing wastewater treatment plants The inherent uncertainties when making choice of the centralized treatment facilities require more comprehensive evaluation This study con®rms that moderate economy of scale does exist in constructing both the domestic and industrial wastewater treatment plants... is probably due to the insuf®cient cost structures used to re¯ect the inherent uncertainties in the cost estimation On the other hand, the impact of three outliers included in the database of industrial wastewater treatment plants could be in uential in the cost estimation when using the LS model The modi®ed coef®cients of determination (R2), representing the measurement of the `goodness -of- ®t' in regression,... recorded in the literature In this section, we will present a typical approach that eventually leads to a full illustration of the theory of fuzzy goal regression model for use in various forecasting practices In a goal-programming problem, planning goals are stated precisely and algebraic equations are formulated to correspond to the stated goal The realization of goals can be described by a set of goal . literature (Tanaka and Asia, 198 4a, 1984b; Jajuga, 1986; Tanaka, 1987; Tanaka and Watada, 1988; Tanaka et al. 1989; Chen, 1988; Diamond, 1988). Later on, the. is prepared for a comparative analysis of these three methods representing various types of uncertainties involved in the cost estimation of constructing wastewater