Water Resour Manage (2015) 29:3015–3036 DOI 10.1007/s11269-014-0902-x
A Generalized Interval Fuzzy Chance-Constrained Programming Method for Domestic Wastewater Management Under Uncertainty – A Case Study of Kunming, China C. Dai & Y. P. Cai & Y. Liu & W. J. Wang & H. C. Guo
Received: 27 May 2014 / Accepted: 9 December 2014 / Published online: 20 May 2015 # Springer Science+Business Media Dordrecht 2015
Abstract In this study, interval mathematical programming (IMP), mλ-measure, and fuzzy chance-constrained programming are incorporated into a general optimization framework, leading to a generalized interval fuzzy chance-constrained programming (GIFCP) method. GIFCP can be used to address not only interval uncertainties in the objective function, variables and left-hand side parameters but also fuzzy uncertainties on the right-hand side. Also, it can reflect the aspiration preference of optimistic and pessimistic decision makers due to the integration of mλ-measure. The developed method is applied to the long-term planning of a domestic wastewater management system in the city of Kunming, China, with consideration of the eco-environmental protection of downstream water body. The solution results of the GIFCP method can generate a series of optimal wastewater allocation patterns and WTPs capacity expansion schemes under different risk levels, provide in-depth insights into the
Electronic supplementary material The online version of this article (doi:10.1007/s11269-014-0902-x) contains supplementary material, which is available to authorized users.
C. Dai : Y. Liu : W. J. Wang : H. C. Guo (*) College of Environmental Science and Engineering, Peking University, Beijing 100871, China e-mail:
[email protected] C. Dai e-mail:
[email protected] Y. Liu e-mail:
[email protected] W. J. Wang e-mail:
[email protected]
Y. P. Cai State Key Laboratory of Water Environment Simulation, School of Environment, Beijing Normal University, Beijing 100875, China e-mail:
[email protected] Y. P. Cai Institute for Energy, Environment and Sustainable Communities, University of Regina, 120, 2 Research Drive, Regina, SK S4S 7H9, Canada
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effects of uncertainties, and consider the proper balance between system cost and risk of constraint violation. Keywords Wastewater treatment plant . Capacity expansion . Possibility measure . Interval optimization . Fuzzy chance constrained programming . Uncertainty
1 Introduction In the past decades, domestic wastewater has rapidly increased due to the rapid population increase and economic development in China. Concerns with domestic wastewater management have become significant and the issues of surface water quality conservation and sustainable economic and social development are critical in many basins (Maeda et al. 2010). However, a variety of complexities exist in such domestic wastewater management systems, including the wastewater generation rate to be treated, the wastewater treatment technologies to be used, the eco-environmental water requirements to be met, and the nutrients emission permits to be offered. Also, from a long-term planning point of view, decision makers are facing problems of insufficiency in terms of available capacities of wastewater treatment plant (WTP) to satisfy the increasing wastewater disposal demands (Chang and Hernandez 2008). Thus, an in-depth cost-benefit analysis for the identification of optimal domestic wastewater management schemes is required. Recently, a number of mathematical models related to domestic wastewater management systems have been developed (Butler and Schütze 2005; Chen and Chen 2014; Chen et al. 2013; Izquierdo et al. 2008; Kang and Lansey 2012; Karuppiah and Grossmann 2006; Park et al. 2013; Rojek 2014; Sousa et al. 2002). For example, Butler and Schütze (2005) described the development of integrated optimization modelling of the operation and control of urban wastewater management systems, and the results indicated that river quality can be significantly improved by implementing an integrated urban wastewater control strategy. Park et al. (2013) developed a bi-objective optimization model to eco-design an existing wastewater treatment system for reducing the environmental impacts and economic costs of industry sectors. Based on previous studies, wastewater management is not a trivial process and there are considerable uncertainties inherent in it, such as multiple forms of uncertainties in public policies, government incentives, and the associated capital and operating costs (Flores-Alsina et al. 2008; Sin et al. 2011; Tjandraatmadja et al. 2013; Trinh et al. 2013). In addition, many domestic wastewater system parameters such as wastewater generation rates, capacity of WTPs, lake nutrient emission permits, and eco-environmental water requirements of urban rivers, may appear uncertain in multiple forms that can be expressed as fuzzy sets, probability density functions, and intervals (Cabanillas et al. 2012; Chang and Hernandez 2008). Furthermore, such multiple forms of uncertainties may be further multiplied by the site-specific features of many system components, factors, and parameters (Cai et al. 2011). These multi-form uncertainties have raised concerns in the wastewater management system. Thus, a proper understanding of the sources and effects of uncertainty is needed (Seifollahi-Aghmiuni et al. 2013). Most of the previous methods dealing with the uncertainties and complexities included interval, fuzzy and stochastic mathematical programming (abbreviated as IMP, FMP and SMP) (Shibu and Reddy 2014; Tan et al. 2013; Zhang and Li 2014; Zhang and Huang 2011). Huang (1998) developed the IMP model for dealing with system optimization problems, and the results indicated that IMP is an effective tool in dealing with interval numbers in objective function coefficients and constraint parameters. As a pioneer study upon wastewater planning problem, Chang and Hernandez (2008) used a mixed-integer IMP method to reflect the systematic
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concerns about integrative uncertainties, which could help decision makers to make all-inclusive decisions for WTP expansion in an economically growing region. On the other hand, in fuzzy possibility theory (Zadeh 1978), the membership function of a fuzzy set can be used to encode a possibilistic distribution. Iwamura and Liu (1998) used the possibility to measure the occurrence chance of a fuzzy event in capital budgeting problems. Recently, Huang (2006) advanced a credibility chance-constrained programming method by extending the chance-constrained programming idea to the fuzzy environment based on credibility measure averaging of the possibility and necessity measures. Also, Zhang and Huang (2011) applied the credibility chanceconstrained programming model to water resource management. Particularly, as an improvement upon the conventional methods, mλ-measure was introduced to operational research and was first presented by Yang and Iwamura (2008). It can be used as a linear combination of the possibility measure and necessity measure to balance optimism and pessimism. In this approach, a parameter λ is predetermined by the decision maker according to their personal judgment and experience. Following the idea of stochastic chance-constrained programming, Yang and Iwamura (2008) constructed a generalized fuzzy chance-constrained programming (GFCP) method, after discussing the mathematical properties of mλ-measures, including continuity, monotonicity, subadditivity and boundary. However, no application of GFCP to domestic wastewater management system has been reported. Moreover, seldom researches focus on the multi-formats of uncertainties in parameters and variables, such as right-hand side parameters are fuzzy and variables are interval numbers. In the domestic wastewater management system, multi-types of uncertainties may exist, nevertheless simply presenting information of one type would result in potential errors with the modeling inputs (Korteling et al. 2013). Therefore, to better account for multiple uncertainties that can be expressed as intervals and possibilities associated with domestic wastewater management, one potential approach is to incorporate the concepts of mλ-measure, fuzzy chance-constrained programming and IMP within a general optimization framework. This leads to a generalized interval fuzzy chanceconstrained programming (GIFCP) method. The objective of this research is to develop the GIFCP method and to apply it to domestic wastewater management in the city of Kunming, with consideration of the eco-environmental protection for urban rivers and the eutrophication restoration of Lake Dianchi in the city. GIFCP can reflect not only interval uncertainties in the objective function, variables and left-hand side parameters, but also fuzzy uncertainties on the right-hand side. Aspiration preferences and confidence levels, which represent satisfaction degrees of the constraints, will be effectively analyzed. The developed method can offer a general optimal solution under variable risk parameters, which can be easily used for understanding the trade-offs between system costs and risks.
2 Methodology In fuzzy set theory, a possibility measure and necessity measure are employed to describe the chance of a fuzzy event. Let et be a fuzzy variable with membership function μ, and let u and r be real numbers. The possibility- and necessity-measure of a fuzzy even (i.e., et ≤r ) can be expressed as follows, respectively : ð1aÞ Pos et≤ r ¼ sup μðuÞ ¼ supfμ ~tðuÞju∈R; u≤rg; u≤r
Nec et ≤r ¼ 1−Pos et > r ¼ 1− sup μðuÞ: u> r
ð1bÞ
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Consider a triangular fuzzy variable (i.e., et ) that is fully determined by the triplet ðt; t; tÞ of crisp numbers with t < t < t , whose membership function is given as follows: 8 > r−t > > ; if t ≤r ≤t; > > 0 ; if r ≤t or r ≥t; > > > > : r − t ; if t ≤r ≤t: t−t
ð2Þ
Based on the membership function, the possibility and necessity measure of et ≤r are as follows, respectively: 8 > 0 ; if r ≤t; > > < r−t Pos et ≤r ¼ ; if t ≤r ≤t; > > >t−t : 1 ; if r ≥t:
ð3aÞ
8 ; if r ≤t; >0 < r−t ; if t ≤r ≤t; Nec et ≤r ¼ > :t−t 1 ; if r ≥t:
ð3bÞ
In a decision-making system, the possibility measure is more suitable for an optimistic decision maker who mainly cares about the inherent opportunity cost; conversely, a pessimistic decision-maker considering potential risks may utilize the necessity measure as a chance measure. However, neither absolutely optimistic nor absolutely pessimistic decision makers exists in the real word. For an eclectic decision maker, the following mλ-measure can reflect the tradeoff between the optimism and pessimism (Azadegan et al. 2011): ð4Þ mλ et ≤r ¼ λPos et ≤r þ ð1−λÞNec et ≤r ; where the parameter λ∈[0, 1] is a weight to trade-off chance cost with risk, which is specified by the decision maker according to the aspiration preferences. mλ-measure is a general method that implies the possibility (i.e., λ=1), necessity (i.e., λ=0) and credibility (i.e., λ=0.5) introduced by Liu (2006). In general, the larger the parameter λ is the more optimistic the decision-maker is. On the basis of the functions of Eq. (3a) and (3b), the mλ-measure of et ≤ r can be expressed as follows: 8 > 0 > > > > > < λr − λt > mλ et ≤ r ¼ t−t > > ð1−λÞr þ λt − t > > > > > t−t : 1
; if r ≤t; ; if t≤ r ≤ t;
ð5Þ
; if t ≤r ≤t; ; if r ≥t:
Figure 1 presents the differences among possibility, necessity and mλ-measure of a fuzzy event (i.e., et≤ r ), where a triangular fuzzy set et is designed as (16, 18, 22). On the basis of the
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Fig. 1 Measures of a fuzzy event: a fuzzy set, b possibility, c necessity, and d mλ-measure
concept of using mλ-measure to represent the chance of a fuzzy event, a generalized fuzzy chance-constrained programming (GFCP) method can be generated as follows (Yang and Iwamura 2008): min f ¼
J X
c jx j
ð6aÞ
j¼1
subject to: ( mλ
J X
) ai j x j ≥ e bi ≥γi ; i ¼ 1; 2; …; I
ð6bÞ
j¼1
x j ≥0; j ¼ 1; 2; …; J
ð6cÞ
where x=(x1,x2,…, xJ) is a vector of non-fuzzy decision variables; cj are cost coefficients; aij are technical coefficients; e bi are right-hand side coefficients determined by the triplet bi ; bi ; bi of crisp numbers with bi < bi < bi ; γi denote the predetermined confidence levels. Constraints (6b) are fuzzy chance constrains constraining the failure risk based on the
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mλ-measure. Let ∑ ai j x j be replaces by si. Then, the constraints (6b) can be represented as j¼1
follows: n o bi ≤si ≥γi ; i ¼ 1; 2; …; I mλ e
ð7Þ Let μe ¼ mλ e bi . If si > bi , then the bi ≤si denote the mλ-measure of variable si for e bi element mass is completely valid due to μe ¼ 1 . If si < bi , then the element mass is bi
completely invalid due to μe ¼ 0 . When bi ≤si ≤bi , μe varies between 0 and 1 based on the bi bi increasing function. Thus, only a single si exists in bi ; bi corresponding to a given confidence level (i.e., μe ¼ γi ) (Zhang and Huang 2010). Normally, a confidence level bi
should be greater than 0.5 (Herrera-Viedma and López-Herrera 2010). Thus, constraint (7) is equivalent to 1≥μe ≥γi ≥0:5 , for i=1,2,…,I. Figure 2 presents the calculation rules of μe ≥ bi
bi
γi according to the preference of decision makers [i.e.. (a) 0≤λ≤0.5 and 0.5≤γi ≤1, (b) 0.5
