Post on 17-Jan-2023
An Interval-Parameter Waste-Load-Allocation Model for RiverWater Quality Management Under Uncertainty
Xiaosheng Qin Æ Guohe Huang Æ Bing Chen ÆBaiyu Zhang
Received: 14 April 2008 / Accepted: 12 January 2009 / Published online: 24 February 2009
� Springer Science+Business Media, LLC 2009
Abstract A simulation-based interval quadratic waste
load allocation (IQWLA) model was developed for sup-
porting river water quality management. A multi-segment
simulation model was developed to generate water-quality
transformation matrices and vectors under steady-state
river flow conditions. The established matrices and vectors
were then used to establish the water-quality constraints
that were included in a water quality management model.
Uncertainties associated with water quality parameters,
cost functions, and environmental guidelines were descri-
bed as intervals. The cost functions of wastewater
treatment units were expressed in quadratic forms. A
water-quality planning problem in the Changsha section of
Xiangjiang River in China was used as a study case to
demonstrate applicability of the proposed method. The
study results demonstrated that IQWLA model could
effectively communicate the interval-format uncertainties
into optimization process, and generate inexact solutions
that contain a spectrum of potential wastewater treatment
options. Decision alternatives can be generated by adjust-
ing different combinations of the decision variables within
their solution intervals. The results are valuable for sup-
porting local decision makers in generating cost-effective
water quality management strategies.
Keywords Interval � Quadratic programming �Water quality management � Uncertainty � Optimization
Introduction
Effective planning for water quality management has been
an important task for facilitating sustainable socio-eco-
nomic development in watershed systems. Over the past
decades, many optimization models have been proposed in
this field (Bouwer 2003; Huang and Qin 2008a). The waste-
load-allocation (WLA) model, also named multi-point-
source waste reduction model, is one of the examples that
have been widely used for limiting the amount of waste
discharged into river water systems (Loucks and others
1981, 1985). It is formulated as a problem of minimizing
treatment or pollution abatement costs while specifying a
minimum allowable water quality in the receiving water
bodies (i.e. water quality standards).Water-quality simula-
tion models are normally embedded within a WLA model
framework in order to describe relationships between river
water-quality levels and wastewater control strategies.
However, water quality planning and management are
complicated with a variety of uncertainties (Loucks 2003).
They may be derived from random nature of hydrodynamic
conditions of river systems and meteorological processes,
as well as shortage of available monitoring data (Burn and
X. Qin (&)
Sino-Canada Center of Energy and Environmental Research,
North China Electric Power University, Beijing 102206, China
e-mail: xiaoshengqin@tom.com
X. Qin
Center for Studies in Energy and Environment, University
of Regina, Regina, Saskatchewan S4S 0A2, Canada
G. Huang
Faculty of Engineering, University of Regina, Regina,
Saskatchewan S4S 0A2, Canada
e-mail: huangg@iseis.org
B. Chen
Faculty of Engineering and Applied Science, Memorial
University of Newfoundland, St. John’s, NL A1B 3X5, Canada
B. Zhang
Department of Civil Engineering, Dalhousie University,
1360 Barrington St, Halifax, NS B3J 1Z1, Canada
123
Environmental Management (2009) 43:999–1012
DOI 10.1007/s00267-009-9278-8
McBean 1987; Revelli and Ridolfi 2004; Karmakar and
Mujumdar 2006; Huang and Qin 2008b). In addition,
representation of system costs for water quality manage-
ment involves a number of nonlinear functions for
projecting environmental-economic interrelationships (Qin
and others 2007a). These complexities lead to difficulties in
formulating and solving the resulting nonlinear optimiza-
tion problems.
Previously, efforts were made in dealing with the
uncertainties and nonlinearities in water quality manage-
ment through stochastic, fuzzy and interval mathematical
programs (Huang 1998; Huang and Loucks 2000; Huang
and others 2002; Karmakar and Mujumdar 2006; Liu and
others 2007, 2008). Stochastic programming is a classical
measure for addressing uncertainties in nonlinear water
quality management systems. For example, Fujiwara
(1988) studied the problem of identifying optimal waste
removals to mitigate the impacts of the waste discharges on
the dissolved oxygen (DO) concentration in a water body
through chance-constrained programming method, where
probability of violating the DO deficit standard was
investigated; Dupacova and others (1991) compared three
stochastic programming methods (i.e., approximation,
stochastic quasi-gradient and joint chance-constrained
methods) in applications of a water quality management
problem in Eastern Czechoslovakia; Mujumdar and Saxena
(2004) developed a stochastic dynamic programming
(SDP) model for dealing with waste load allocation (WLA)
problems, where the random variation of streamflows was
included through seasonal transitional probabilities. In
these studies, several modeling parameters were considered
as random variables and described by probability density
functions (PDFs) (Qin and others 2008a). By means of
stochastic analysis such as Monte Carlo simulation, the
problems could be converted into deterministic sub-models
and then solved by traditional nonlinear programming
methods.
Another approach for water quality management under
uncertainty is based on fuzzy set theory. It is a method that
facilitates the analysis of systems with uncertainties being
derived from vagueness or ‘‘fuzziness’’ rather than ran-
domness alone (Qin and others 2007b, 2008b). It is suitable
for situations when the uncertainties cannot be expressed as
probability density functions (PDFs), such that adoption of
fuzzy membership functions becomes an attractive alterna-
tive (Gen and others 1997; Huang and Chang 2003). For
examples, Lee and Wen (1996) evaluated four types of fuzzy
linear programming (FLP) approaches and applied them in
dealing with the allowable pollution loading problems in a
river basin; Sasikumar and Mujumdar (1998) developed a
fuzzy multi-objective optimization model called fuzzy
waste-load allocation model (FWLAM) for water quality
management of a hypothetical river basin; Mujumdar and
Sasikumar (2002) advanced a fuzzy optimization model for
supporting the seasonal water quality management of river
systems, where fuzzy probability approach was used to
reflect risks of violating water quality standards due to
seasonal variations of river flow; Lee and Chang (2005)
proposed an interactive fuzzy approach for handling multi-
objective water quality optimization problems involving
vague and imprecise information related to data, model
formulation, and the decision maker’s preferences.
The stochastic and fuzzy methods have advantages in
their effectiveness in dealing with uncertainties and non-
linearities. However, the stochastic methods are associated
with difficulties in acquiring PDFs for a number of modeling
parameters (Marr 1988); the fuzzy nonlinear programming
methods have been restricted by its complicated mathe-
matical conversions derived from generation of intermediate
models (Wu and others 2006). Interval quadratic program-
ming (IQP) is another alternative for handling both
uncertainty and nonlinearity for optimization problems. The
cost coefficients, constraint coefficients, and right-hand
sides of an IQP model, are represented by interval data.
Since the parameters are interval-valued, the objective value
is interval-valued as well. A pair of two-step mathematical
solution algorithm can be used to calculate the upper bound
and lower bound of the objective values of an interval
quadratic model (Huang and others 1992). The IQP model
will be transformed into a number of deterministic quadratic
programs. Solving these programs produces the intervals of
the objective values and decision variables of the problem.
Applications of IQP have been reported in the field of
solid waste management. For example, Huang and others
(1995) introduced a grey quadratic programming (GQP)
method as a means for municipal solid waste (MSW)
management planning under uncertainty; Wu and others
(2006) proposed an interval quadratic programming model
and applied it to the planning of MSW management system
in the Hamilton-Wentworth Region, Ontario, Canada. The
IQP model is formulated by introducing interval variables
into an ordinary quadratic programming (QP) framework,
and is capable of incorporating uncertainties within the QP
optimization process and the resulting solutions (Huang
and others 1992). Decision alternatives can be obtained by
adjusting decision variables within their solution intervals.
However, applications of IQP model in water quality
management were very limited. Thus, as an extension of
the previous efforts, the objective of this study is to
develop an interval quadratic waste-load-allocation (IQ-
WLA) model for supporting decisions of river water
quality management. A steady-state one-dimensional river
water quality simulation system will be provided to gen-
erate a number of intermediate transformation matrices and
vectors for building the WLP model. Uncertainties related
to water quality parameters will be projected to the
1000 Environmental Management (2009) 43:999–1012
123
optimization model by interval analysis. Nonlinearities and
uncertainties associated with the objective functions will be
handled by formulation of interval quadratic polynomials.
A case study for water quality management planning in the
Xiangjiang river section in Hunan Province, China, will be
used to demonstrate the applicability of the proposed
method.
River Water Quality Simulation
Water quality models are used to address the relationships
between the pollutant loadings and environmental respon-
ses in a stream (or a river), and analyze the potential
impacts of alternative pollution control plans. They have
been used extensively for supporting stream water-quality
management problems over several decades. There are
many different types of water quality models, ranging from
simple models such as Streeter-Phelps, Dobbins and
O’Connor to more sophisticated ones such as QUAL2E and
WASP6 (Rauch and others 1998). In many cases, the
appropriate model and the required data depend on the
purpose of study. Simplifications or assumptions are nor-
mally made in specific planning situations given the limits
of available time and money; in some cases, models should
be relatively simple; in other cases, they may have to be
more complex (Loucks and others 1981). For a typical
WMR problem, the modeling method should be able to
predict the degree of waste removal at various point
sources sites along a water body that will meet both
effluent and water quality standards. In this study, the
O’Connor and Dobbins model is used for supporting
quantification of water quality constraints related to Bio-
logical Oxygen Demand (BOD) and Dissolved Oxygen
(DO) discharges as well as their concentrations in river
waters (O’Connor and Dobbins 1958). A first-order deg-
radation reaction of BOD-DO can be written as:
uxdLc
dt¼ �ðKd þ KsÞLc ð1aÞ
uxdLN
dx¼ �KNLN ð1bÞ
uxdO
dx¼ �kdLc � kNLN þ kaðOs � OÞ; ð1cÞ
where Lc is the ultimate Carbonaceous BOD (CBOD)
concentration, [ML-3]; LN is the ultimate Nitrogenous
BOD (NBOD) concentration, [ML-3]; kd is the CBOD
decay rate in river streams, [T-1]; ks is the CBOD decay
rate due to sedimentation, [T-1]; O is the DO concentra-
tion, [ML-3]; Os is the saturated DO concentration,
[ML-3]; kN is the nitrification rate (NBOD decay rate in
river), [ML-3]; ka is the reaeration rate, [ML-3]; ux is the
average stream flow rate, [L/T]; x is the flow distance along
the x axis, [L].
The solutions of above equations are (Thomann and
Mueller 1987):
Lc ¼ Lc0e�ðkdþksÞx=ux ð2aÞ
LN ¼ LN0e�kN x=ux ð2bÞ
O ¼ Os � ðOs � O0Þe�kax=ux � kdLc0
ka � ðkd þ ksÞ� e�ðkdþksÞx=ux � e�kax=ux
h i
� kNLN0
ka � kNe�kN x=ux � e�kax=ux
h ið2cÞ
where Lc0, LN0, and O0 are initial ultimate CBOD, NBOD
and DO loads in stream, respectively.
Segmentation is necessary since a number of wastewater
discharge outlets scatter along the river, with temporal and
spatial variations of their loadings. Water quality at each
segment is affected by various sources from the upper
stream.
According to Qin and others (2007a), a matrix expres-
sion can be established as follows:
AcL~c2 ¼ BL~c þ g~c ð3aÞ
ANL~N2 ¼ BL~N þ g~N ð3bÞ
CO~2 ¼ �DcL~c � DNL~N + BO~þ f~þ h~ ð3cÞ
where Ac, AN, B, C, Dc, and DN are n� n matrixes, and g~c,
g~N , f~, and h~ are n-dimensional vectors. Thus, we have:
L~c2 ¼ UcL~c þ m~c ð4aÞ
L~N2 ¼ UNL~N þ m~N ð4bÞ
O~2 ¼ VcL~c þ VNL~N þ n~ ð4cÞ
where Uc, UN, Vc, and VN are the water-quality
transformation matrices; m~c, m~N , and n~ are the water-
quality transformation vectors; L~c2, L~N2, O~2 are the
predicted CBOD, NBOD and DO levels at different
stream segments, respectively; L~c and L~N are the
observed CBOD and NBOD loads in wastewater sources,
respectively. The transformation matrices and vectors are
defined as follows:
Uc ¼ A�1c B ð5aÞ
m~c ¼ A�1c g~c ð5bÞ
UN ¼ A�1N B ð5cÞ
m~N ¼ A�1N g~N ð5dÞ
Vc ¼ �C�1DcA�1c B ð5eÞ
VN ¼ �C�1DNA�1N B ð5fÞ
Environmental Management (2009) 43:999–1012 1001
123
n~¼ C�1BO~þ C�1ðf~þ h~Þ � C�1DcA�1c g~c
� C�1DNA�1N g~N ð5gÞ
Equations (4) to (5) define the relationships between
input and output of pollution indexes at each river section.
Transformation matrices or vectors are the bases for water
quality predictions, where their elements are functions of
stream hydraulic and water-quality parameters. In water
quality management, L~c2, L~N2, and O~2 are generally rep-
resented as the desired water quality levels (i.e. water
quality standards), and L~c and L~N become the decision
variables.
Interval Quadratic WLA Model Development
Quadratic Cost Function
The main purpose of a waste-load-allocation (WLA)
problem is to determine the degree of treatment for both
CBOD and NBOD levels at each wastewater discharge
source (i) so as to minimize the total treatment cost, while
at the same time, satisfy both wastewater effluent and
surface water-quality standards (Loucks and others 1981;
Karmakar and Mujumdar 2006). The cost functions for
wastewater treatment plants have been extensively inves-
tigated in many countries (Fu and Cheng 1985; Cheng and
Cheng 1990; Qin and others 2007a). Many of them have
been formulated as functions of the removal efficiency of
the total biological oxygen demand (TBOD) and the
wastewater flow rate. Figure 1a shows a typical monoto-
nously increasing nonlinear curve of TBOD removal
efficiency vs. the operational cost for a wastewater treat-
ment plant while the wastewater flow rate keeps constant
(Cheng and Cheng 1990; Zeng and others 2003). The
nonlinearity reflects the economy-of-efficiency where the
treatment cost per unit of pollutant loading will increase
with the treatment efficiency. It is indicated that the non-
linear relationships could be approximated by a quadratic
function with a reasonable degree of error as C = a ? bg2,
where a and b are cost coefficients (Huang and others
1995). Figure 1b presents a typical curve of the operational
cost vs. the treatment scale while the TBOD removal rate
keeps constant; this reflects the economy-of-scale effects.
Therefore, a combined nonlinear cost function can be
expressed as follows (Cheng and Cheng 1990):
C ¼ Qkscale k1 þ k2g2TB
� �ð6Þ
where C is the operational cost for a wastewater treatment
plant, [$]; Q is the design flow rate of wastewater treatment
(treatment scale) [ML-3]; gTB is the removal efficiency of
TBOD; k1 and k2 are the cost-function coefficients (k1,
k2 [ 0); kscale is the economy-of-scale index ranging from
0.7 to 0.9 (Huang and others 1995). The parameters of
kscale, k1, k2 are obtained from statistical analysis and
normally vary with different type of wastewater that is to
be treated. Typical values can be found in Rinaldi and
others (1979), Cheng and Cheng (1990), Lee and Wen
(1997), and Qin and others (2007a).
Since TBOD is the sum of CBOD and NBOD, we can
use the following relationships to correlate gci and gNi with
the cost function (Ci):
Lc0i þ LN0i ¼ LTB0i ð7aÞLci þ LNi ¼ LTBi ð7bÞ
gci ¼Lci
Lc0i; gNi ¼
LNi
LN0i; gTBi ¼
LTBi
LB0ið7cÞ
gTBiLB0i ¼ gciLc0i þ gNiLN0i ð7cÞ
where Lc0i and LN0i are the initial concentrations of CBOD
and NBOD at discharge point i, [ML-3]; Lci and LNi are the
discharge concentrations of CBOD and NBOD at discharge
point i, [ML-3]; LTB0i and LTBi are initial and discharge
concentrations of TBOD at discharge point i, [ML-3]; gTBi
is the TBOD treatment efficiency at discharge point i.
Water Quality Constraints
Constraint equations associated with the TBOD effluent
standards at each discharge point are (Loucks and others
1981):
Lc0ið1� gciÞ þ LN0ið1� gNiÞ�TBEi 8i ð8Þ
where TBEi is the effluent discharge standard for the dis-
charge point i.
For any water quality segment in the waste-receiving
water body, the stream quality standards for TBOD and DO
concentrations can be expressed as follows (Loucks and
others 1981):
Xn
i¼1
UcjiLc0ið1� gciÞ� �
þ mcj
þXn
i¼1
UNjiLN0ið1� gNiÞ� �
þ mNj�TBWj 8j ð9aÞ
Xn
i¼1
VcjiLc0ið1� gciÞ� �
þXn
i¼1
VNjiLN0ið1� gNiÞ� �
þ nj�DOWj 8j ð9bÞ
where Ucji, UNji, Vcji, and VNji are the elements at the jth
row and the ith column (j = 1, 2, …, n; i = 1, 2, …, n) of
the transformation matrices (Uc, UN, Vc, and VN), respec-
tively; j denotes the index of river segment; mcj, mNj, and nj
are the element at the jth row of transformation vectors (m~c,
m~N , and n~); TBWj and DOWj are the stream water quality
1002 Environmental Management (2009) 43:999–1012
123
standards for TBOD and DO at the discharge point i,
respectively.
Technical constraints are listed as follows:
gminci � gci� gmax
ci ð10aÞ
gminNi � gNi� gmax
Ni ð10bÞ
where gminci and gmax
ci are the minimum and maximum
treatment efficiencies for CBOD, respectively at the dis-
charge point i; gminNi and gmax
Ni are the minimum and
maximum treatment efficiencies for NBOD, respectively at
the discharge point i.
IQWLA Model Formulation
The water quality management system is normally com-
plicated with a variety of uncertainties and nonlinearities,
which may be associated with many components of opti-
mization models. Projecting these uncertainties into water
quality management models is important for analyzing
system behaviors under complexities and providing robust
decision supports for water resources managers (Loucks
and others 1981). In this study, uncertain parameters will
be expressed as intervals where the extreme values at the
upper and lower bounds are required to be specified. The
cost coefficients in Eq. (6) may exhibit uncertain features
due to statistical deviations (Qin and others 2007a), and
could be expressed as the following interval format:
C ¼ Qk�q k�1 þ k�2 g2TB
� �ð11Þ
The transformation matrices or vectors are the bases for
water quality predictions, where their elements are
functions of stream hydraulic and water-quality
parameters. In water quality management, L~c2, L~N2, and
O~2 are generally represented as the desired water quality
levels (i.e. water quality standards), and L~c and L~N are the
decision variables.
Uncertainties associated with the water quality param-
eters (kd, kN, ks and ka) may come from measurement errors
and/or temporal/spatial variations. They can be expressed
as intervals since their distributional information is often
unavailable (Kothandaraman and Ewing 1969). Projection
of these uncertainties into the transformation matrices and
vectors can be performed based on interval analysis (or
Monte Carlo simulation assuming uniform distributions)
(Giri and others 2001). Uncertainties may also be associ-
ated with water quality standards. Different to the water
quality parameters, such type of uncertainties is more
likely derived from biased human judgment or variations of
regulatory requirements. Since the desired strictness of
pollution-control practices are normally different from the
manager’s and the discharger’s perspectives, environmen-
tal standards can be expressed in discrete intervals, with the
upper and lower bounds representing the most conservative
and optimistic regulatory requirements, respectively. The
interval information can be obtained based on discussions
among different stakeholders or from public survey. For
example, the federal government may demand the BOD
level at a surface water body be lower than 6 mg/L, and the
local regulatory agency may require a stricter level, say
4 mg/L; the wastewater dischargers are more likely prefer
a looser standard since the related wastewater treatment
cost would be lower; it is therefore reasonable to use an
interval [4, 6] mg/L as a compromised regulatory standard.
The above treatment leads to the formulation of an
interval quadratic waste-load-allocation model (IQWLA)
as follows:
Min f� ¼Xn
i¼1
Qk�qi k�1 þ k�2
Lc0ig�ci þ LN0ig�Ni
Lc0i þ LN0i
� �2" #
ð12aÞ
Subject to:
Lc0ið1� g�ciÞ þ LN0ið1� g�NiÞ�TBE�i 8iði ¼ 1; 2; . . .; nÞð12bÞ
Fig. 1 A typical nonlinear
curve of cost function
Environmental Management (2009) 43:999–1012 1003
123
Xn
i¼1
U�cijLc0ið1� g�ciÞh i
þ m�cj
þXn
i¼1
U�NjiLN0ið1� g�NiÞh i
þ m�Nj�TBW�j 8jðj
¼ 1; 2; . . .; nÞ ð12cÞXn
i¼1
V�cjiLc0ið1� g�ciÞh i
þXn
i¼1
V�NjiLN0ið1� g�NiÞh i
þ n�j �DOW�j 8jðj
¼ 1; 2; . . .; nÞ ð12dÞ
gminci � g�ci � gmax
ci ð12eÞ
gminNi � g�Ni� gmax
Ni ð12fÞ
Due to the existence of the cross-product terms of gci
and gNi in the cost function (12a), the formulated IQWLA
model is difficult to solve. A dual response regression
technique can be used to transform it to a pure quadratic
form. Detailed procedures can be found in Kennedy and
Gentle (1981) and Kim and Lin (1998). Then, model (12)
turns into a typical inexact quadratic program (IQP) and
can be solved by algorithms proposed by Huang and others
(1995) and Chen and Huang (2001). The constraints (12e)
and (12f) are technical constraints and will be restricted by
the capacities of the currently available treatment
technologies. The constraints may be redundant when the
capacity of wastewater treatment is not a major concern. In
such a case, gminNi and gmax
Ni could be 0 and 1, respectively. In
addition, several conditions need to be satisfied to ensure
existence of optimal solution for a QP problem. When the
objective function to a QP problem is strictly convex for all
feasible points the problem has a unique local minimum
which is also the global minimum (Qin and others 2007c,
2008c). A sufficient condition to guarantee strictly
convexity is for the coefficients of quadratic terms to be
positive definite (Huang and others 1995). As the function
describing operating cost of water treatment is generally
monotone increasing, the objective function (12a) is
concave. This ensures an optimal solution for the model.
Case Study
Background
The Xiangjiang River section (from Houzishi to Chen-
guanzhen areas) flowing through the City of Changsha,
Hunan Province, China, will be provided as a case to dem-
onstrate the applicability of the proposed methodology. This
river section, with a length of about 25 km, receives the
majority of wastewater discharged from the City of Chang-
sha. Six main discharging sources exist along the section,
including the Changsha Leatheroid Plant, the Xiangya Hos-
pital, the Hunan Paper Mill, the No. 1 Wastewater Treatment
Plant, the Changsha Textile Plant and the Changsha Chemical
Plant (Qin and others 2007a). In order to meet the environ-
mental requirement, the raw wastewater from each industry
must be treated with specific facilities before discharge. The
operating costs of these facilities were directly related to the
inflow of wastewater and the level of its treatment. The water
quality in the river was related to contaminant concentrations
and flow rate of the discharged wastewater. It is desired that a
waste-load-allocation model be developed in order to mini-
mize the operating cost and satisfy the national
environmental standard, through identification of desired
wastewater treatment efficiencies (or allowable discharge
amounts) for the six discharging sources.
Details of the related water-quality parameters were
provided in Zeng and others (2004). Figure 2 is a sche-
matic diagram of the study system. Table 1 shows water
quality data in different river segments. Uncertainties
associated with BOD deoxygenation and decay rates (kdi,
kNi and ksi) and reaeration rate (kai) due to dynamic and
fluctuating features were handled as intervals. Table 2
presents the wastewater discharge data from various sour-
ces. The cost function of each discharge source is listed in
Table 3. These functions were derived from the data as
reported in Qin and Zeng (2002) and Zeng and others
(2004). The coefficients of these functions are expressed as
intervals due to the uncertainties in the obtained informa-
tion, including parameters kq, k1 and k2.
The right-hand sides of optimization-model constraints
(e.g. water quality standards) are also expressed as intervals.
Table 4 shows the national surface-water standards for
TBOD and DO and the national wastewater-discharge
standards of TBOD for industrial sectors (SEPA 1996,
2002). According to Table 4, the intervals of allowable
TBOD and DO levels in the river system are identified as [4,
6] and [3, 5] mg/L, respectively. The allowable interval for
BOD concentrations in the leatheroid plant, paper mill and
textile plant are determined as [150, 600] mg/L. Those for
chemical and hospital wastewater, the intervals are [60, 300]
mg/L. Since there is no Class III standard available for
wastewater treatment plant, we assume it is 60 mg/L; then,
the corresponding interval becomes [30, 60] mg/L. Due to
capacity restrictions of available wastewater treatment
technologies, we assume the maximum possible treatment
efficiencies for both CBOD and NBOD are 95%. The river
water level has large variations during different seasons, with
a maximum level at 39 m (1998) and minimum one at 25 m
(1999). The study period is assumed at the drought season
when the point-source pollution problem is most serious. The
width of the river system is ranging from 400 to 1000 m and
the ratio of the river width to length is negligible, such that
the river system will be considered as one-dimensional.
1004 Environmental Management (2009) 43:999–1012
123
Cost Function Conversions
From Table 3, the operating costs of wastewater treat-
ment are related to the TBOD levels. Since TBOD
are functions of CBOD and NBOD, the resulted cost
functions for the optimization model are not in pure
quadratic forms and must be converted. The conversion
is realized by using the quadratic regression function
provided by MATLAB7.0 Statistic Toolbox (Kim and
Lin 1998). The transformed cost functions are listed as
follows:
gTB1 ¼ � 0:1307þ 0:2780gc1 þ 0:2670gN1
þ 0:3166g2c1 þ 0:1316g2
N1
ð13aÞ
Fig. 2 The study river system
Table 1 Inexact water quality parameters in different river segments
Parameters River segments
I II III IV V VI
River deoxygenation rate for CBOD (kdi, d-1) [0.52, 0.56] [0.33, 0.38] [0.81, 0.87] [0.81, 0.87] [0.68, 0.74] [0.68, 0.74]
Sediment deoxygenation rate for CBOD (ksi, d-1) [0.28, 0.32] [0.15, 0.20] [0.31, 0.37] [0.31, 0.37] [0.28, 0.34] [0.28, 0.34]
NBOD decay rate (kNi, d-1) [0.40, 0.46] [0.26, 0.35] [0.60, 0.64] [0.60, 0.64] [0.54, 0.62] [0.54, 0.62]
Reaeration rate kai (d-1) [0.48, 0.55] [0.42, 0.48] [0.78, 0.92] [0.78, 0.92] [0.51, 0.60] [0.51, 0.60]
Environmental Management (2009) 43:999–1012 1005
123
gTB2 ¼� 0:1077þ 0:2195gc2 þ 0:1991gN2
þ 0:3370g2c2 þ 0:2414g2
N2
ð13bÞ
gTB3 ¼� 0:1234þ 0:2328gc3 þ 0:2618gN3
þ 0:1888g2c3 þ 0:3178g2
N3
ð13cÞ
gTB4 ¼� 0:1075þ 0:2026gc4 þ 0:2262gN4 þ 0:1566g2c4
þ 0:4122g2N4 ð13dÞ
gTB5 ¼ �0:1168þ 0:2248gc5 þ 0:2219gN5 þ 0:3979g2c5
þ 0:1647g2N5
gTB6 ¼� 0:1213þ 0:2145gc6 þ 0:2719gN6 þ 0:2055g2c6
þ 0:3063g2N6 ð13eÞ
A standard R-square test is performed using 200
randomly generated samples of gc1 and gN1. Figure 3
shows the detailed regression performance, where the
observed results denote the calculated gTBi using
equations provided in Table 3, and the regression results
denote those from using the converted pure quadratic
equations. It is found that most of the R-square levels are
higher than 0.96. The R-square level is calculated by the
following equation:
R2 ¼½Pns
i¼1 ðypi � �ypÞðyoi � �yoÞ�2Pns
i¼1 ðypi � �ypÞ2Pns
i¼1 ðyoi � �yoÞ2ð14Þ
where yoi denotes observed data; ypi means regressed data;
�yo is mean value of yo; �yp is mean value of yp; ns is number
of samples. The results imply that these functions can
reasonably approximate values of gTBi (i = 1, 2, …, 6).
Formulation of IQWLA Model
Based on water quality simulation models with interval
analysis, the interval-parameter transformation matrices
U�c ;U�N ;V
�c and V�N
� �and vectors m~�c ;m~
�N and n~�
� �are
obtained as follows:
Table 2 Wastewater data from various discharge sources
Parameters Sources
1 2 3 4 5 6
Leatheroid
plant
XiangYa
hospital
Changsha
paper mill
Wastewater
treatment plant
Textile
plant
Changsha
chemical plant
Raw CBOD Lci (mg/L) 960 450 735 100 860 420
Raw NBOD LNi (mg/L) 640 350 1015 180 540 580
Discharge rate Qi (m3/s) 0.40 0.45 0.15 2.0 0.60 0.14
Table 3 Cost functions for different industries
Source Industry Quadratic cost function (¥ 104)
1 Leatheroid
plant
Q1[0.7, 0.8]([346, 496] ? [3025, 3263]gTB1
2)
2 Hospital Q2[0.7, 0.8]([308, 458] ? [3214, 3452]gTB2
2)
3 Paper mill Q3[0.7, 0.8]([280, 420] ? [2987, 3209]gTB3
2)
4 WT plant Q4[0.7, 0.8]([422, 578] ? [3432, 3680]gTB4
2)
5 Textile plant Q5[0.7, 0.8]([324, 474] ? [2860, 3098]gTB5
2)
6 Chemical plant Q6[0.7, 0.8]([262, 412] ? [2846, 3084]gTB6
2)
Table 4 China surface-water
and wastewater-discharge
standards
Standards Class Suitable areas BOD
(mg/L)
DO
(mg/L)
Surface water quality standard
(SEPA 2002)
Class III Residential and fisheries areas 4 5
Class IV Industrial and recreational purposes 6 3
Wastewater discharge standard
(SEPA 1996)
Class II Industrial wastewater from leatheroid
plants, paper mill, textile plants
150
Wastewater treatment plant 30
Other industrial wastewater 60
Class III Industrial wastewater from leatheroid
plant, paper mill, textile plant
600
Wastewater treatment plant N/A
Other industrial wastewater 300
1006 Environmental Management (2009) 43:999–1012
123
The objective of the IQWLA model can be formulated as:
The related constraints can be established based on
Eq. (12) and the obtained transformation matrices and vectors.
Result Analysis
Table 5 presents the solutions obtained through the
IQWLA model. The solutions corresponding to the lower
bounds of the total costs (e.g. 47.87 million RMB) are
obtained when the intensities of wastewater treatment at
the discharge sources g�i ; i ¼ 1; 2; . . .; 6� �
are at their
lower levels; the solutions corresponding to the upper
bounds of the total costs (e.g., 124.54 million RMB)
corresponds to the conditions when the treatment inten-
sities reach their upper-bound levels. Thus, when different
U�c ¼ 10�3 �
½1:831; 2:016� 0 0 0 0 0
½1:582; 1:809� ½2:056; 2:263� 0 0 0 0
½1:337; 1:564� ½1:725; 1:969� ½0:685; 0:754� 0 0 0
½1:308; 1:528� ½1:687; 1:924� ½0:670; 0:738� ½9:050; 9:950� 0 0
½0:744; 0:961� ½0:954; 1:210� ½0:374; 0:468� ½5:052; 6:318� ½2:708; 2:976� 0
½0:646; 0:840� ½0:833; 1:063� ½0:325; 0:410� ½4:395; 5:537� ½2:341; 2:656� ½0:631; 0:694�
26666664
37777775
U�N ¼ 10�3 �
½1:831; 2:016� 0 0 0 0 0
½1:671; 1:895� ½2:056; 2:263� 0 0 0 0
½1:528; 1:753� ½1:877; 2:100� ½0:685; 0:754� 0 0 0
½1:503; 1:725� ½1:847; 2:065� ½0:674; 0:742� ½9:050; 9:950� 0 0
½1:089; 1:313� ½1:323; 1:585� ½0:478; 0:571� ½6:425; 7:667� ½2:708; 2:976� 0
½1:007; 1:218� ½1:221; 1:477� ½0:441; 0:534� ½5:921; 7:163� ½2:493; 2:787� ½0:631; 0:694�
26666664
37777775
m�c ¼ f½0:897; 1:081�; ½0:778; 0:962�; ½0:662; 0:825�; ½0:647; 0:808�; ½0:367; 0:501�; ½0:323; 0:443�gT
m�N ¼ f½0:434; 0:555�; ½0:397; 0:517�; ½0:365; 0:478�; ½0:359; 0:471�; ½0:261; 0:360�; ½0:240; 0:336�gT
V�c ¼ �10�3 �
0 0 0 0 0 0
½0:119; 0:174� 0 0 0 0 0
½0:254; 0:340� ½0:179; 0:243� 0 0 0 0
½0:262; 0:347� ½0:192; 0:256� ½0:005; 0:007� 0 0 0
½0:511; 0:630� ½0:540; 0:673� ½0:153; 0:198� ½2:013; 2:617� 0 0
½0:544; 0:657� ½0:585; 0:718� ½0:175; 0:220� ½2:310; 2:908� ½0:197; 0:272� 0
26666664
37777775
V�N ¼ �10�3 �
0 0 0 0 0 0
½0:098; 0:162� 0 0 0 0 0
½0:209; 0:294� ½0:137; 0:186� 0 0 0 0
½0:216; 0:298� ½0:147; 0:196� ½0:004; 0:005� 0 0 0
½0:482; 0:618� ½0:487; 0:637� ½0:136; 0:184� ½1:778; 2:423� 0 0
½0:529; 0:659� ½0:549; 0:695� ½0:160; 0:209� ½2:106; 2:761� ½0:161; 0:233� 0
26666664
37777775
n� ¼ f½5:953; 6:860�; ½5:977; 6:805�; ½6:003; 6:767�; ½5:962; 6:713�; ½6:029; 6:646�; ½6:064; 6:649�g
Min f� ¼ �121:46; 609:30½ � þ 404:03; 477:64½ �g�c1 þ 388:05; 458:74½ �g�N1 þ 460:13; 543:96½ �ðg�c1Þ2
þ 191:26; 226:11½ �ðg�N1Þ2 þ 372:43; 433:27½ �g�c2 þ 337:82; 393:00½ �g�N2 þ 571:80; 665:20½ �ðg�c2Þ
2
þ 409:59; 476:49½ �ðg�N2Þ2 þ 152:44; 197:98½ �g�c3 þ 171:43; 222:64½ �g�N3 þ 123:63; 160:56½ �ðg�c3Þ
2
þ 208:10; 270:26½ �ðg�N3Þ2 þ 1129:56; 1298:11½ �g�c4 þ 1261:13; 1449:32½ �g�N4 þ 873:09; 1003:38½ �ðg�c4Þ
2
þ 2298:14; 2641:07½ �ðg�N4Þ2 þ 427:25; 487:06½ �g�c5 þ 421:74; 480:78½ �g�N5 þ 756:24; 862:11½ �ðg�c5Þ
2
þ 313:03; 356:86½ �ðg�N5Þ2 þ 126:64; 167:04½ �g�c6 þ 160:53; 211:75½ �g�N6 þ 121:32; 160:04½ �ðg�c6Þ
2
þ 180:84; 238:53½ �ðg�N6Þ2
ð15Þ
Environmental Management (2009) 43:999–1012 1007
123
waste treatment settings are selected from their interval
solutions, the relevant system costs will vary within
intervals correspondingly.
The interval solutions of CBOD and NBOD removal
rates reflect the decision alternatives under both high-risk-
tolerant and conservative conditions. The solutions at the
lower bounds represent a situation when the decision
makers are optimistic about the system under study. Thus,
solutions with lower treatment requirements are generated,
which may imply a higher risk of violating the relevant
water-quality criteria under disadvantageous conditions.
Conversely, the solutions of the upper bounds correspond
to a situation when the decision makers prefer a conser-
vative policy that could guarantee that the water-quality
Fig. 3 Regression performance
Table 5 IQWLA model
solutionsSources CBOD removal rate (%) NBOD removal rate (%)
Lower bound Upper bound Lower bound Upper bound
Leatheroid plant 60.34 87.71 65.74 95.00
Hospital 62.72 91.63 62.22 93.62
Paper mill 67.27 95.00 64.58 88.84
WT plant 85.06 95.00 74.96 86.11
Textile plant 55.53 85.70 59.72 95.00
Chemical plant 70.72 95.00 69.48 93.28
Total cost (¥104) 4,787 12,454
1008 Environmental Management (2009) 43:999–1012
123
criteria be met. In such a case, higher treatment require-
ments are desired. The obtained interval solutions can help
decision makers analyze the tradeoffs between system cost
and potential risk. Generally, a post-modeling analysis
based on public survey, round-table discussion, and expert
consultation would be necessary to help reach a final
decision.
For example, when the decision makers aim toward a
lower system cost, only primary treatment (i.e., 55.53%
and 59.72% for CBOD and NBOD) would be needed for
the textile plant; when the decision makers aim towards a
lower risk of water quality standard violation (a higher
system cost), tertiary treatment facilities would have to be
used in order to reach high pollutant-removal efficiencies
(i.e., 85.70% and 95.00% for CBOD and NBOD, respec-
tively). Except for the wastewater treatment plant, a
secondary treatment would be sufficient for most of the
discharge sources under an optimistic consideration.
However, when environmental protection is of severe
concern by local government, all discharge sources would
require tertiary treatment.
The result in Table 5 indicates that different treatment
intensities are allocated to the discharge sources in order to
obtain a minimum system cost. It appears that the required
CBOD and NBOD treatment (i.e. [85.06, 95.00] and
[74.96, 86.11]%, respectively) for the wastewater treatment
plant are the most stringent, followed by the chemical
plant, paper mill, and leatheroid plant. This is due to the
fact that the TBOD discharge standard for wastewater
treatment plant (i.e. [30, 60] mg/L) is strictest among all
discharge sources. The high treatment requirement for
chemical plant may due to its relatively stricter discharge
criteria and low unit treatment cost (see Table 3). The
relatively higher treatment requirements for the leatheroid
plant and paper mill may be the result of their higher
CBOD and NBOD levels in raw wastewater.
The above results demonstrate that the optimization
model can effectively communicate the interval-format
uncertainties into the optimization process, and generate
inexact solutions that contain a spectrum of potential
wastewater treatment options. The decision alternatives can
be obtained by adjusting different combinations of the
decision variables within their solution intervals. The results
are valuable for supporting local decision makers in gener-
ating cost-effective water quality management schemes.
The interval solutions of the IQWLA model contain a
full spectrum of possible decision variables under uncer-
tainty, leading to a wide decision spaces for decision
makers. The right-hand sides of the constraints are envi-
ronmental guidelines, where the associated uncertainties
could be derived from human judgment or variations of
regulatory requirements. Through round-table discussion
among various stakeholders, the uncertainty can be
effectively reduced, and the wideness of the regulatory
intervals can be properly controlled. For example, if a
stricter pollution control strategy is preferred (i.e. strict
scenario), the allowable TBOD and DO intervals of surface
water quality can be decided as [4, 5] and [4, 5] mg/L,
respectively; they belong to the stricter half parts of the
intervals [4, 6] and [3, 5] mg/L. Similarly, the less strict half
parts of intervals can be determined as [5, 6] and [3, 4] mg/
L, respectively. This corresponds to a loose scenario. A
similar treatment can be performed to the wastewater dis-
charge standards. Figure 4 presents the calculated
optimization solutions under these two scenarios. It is
indicated that the original solution intervals to the IQWLA
model are divided into two different parts. The upper parts
of intervals are obtained under strict treatment require-
ments, and the lower parts are from loose ones. Obviously,
the system uncertainties have been considerably decreased.
Other techniques such as chance-constrained programming
or fuzzy programming techniques are also potential
approaches for reducing the right-hand side uncertainties
(Li and others 2006; Qin and others 2007a).
For a typical wastewater treatment unit, the removal
efficiency is dependent on specific treatment processes.
The primary treatment is usually comprised of preliminary
treatment followed by primary clarifiers which remove
approximately 30 to 40% of TBOD. Enhanced primary
treatment can be performed by the addition of a coagulant,
improving the degree of TBOD removal up to 70%. Sec-
ondary treatment involves a primary process and a
biological treatment stage such as activated sludge, trick-
ling filters or aerated lagoons, and can generally reach up to
90% removal rates of TBOD. Tertiary treatment is com-
prised of a three stage process which further improves
effluent quality past the primary and secondary treatment
phases (see above examples) and can reach usually 95–
Fig. 4 Solutions under the strict and loose scenario
Environmental Management (2009) 43:999–1012 1009
123
98% removal rates of TBOD. The solution from the opti-
mization model represents a group of theoretically ideal
removal rates for TBOD (including both CBOD and
NBOD) at various discharge points. In practical applica-
tions, it is normally difficult to accurately control the
treatment efficiency at a specific level. In many cases,
combinations of different treatment processes could be
used to meet the required standards, since each type of
treatment has a certain treatment efficiency range for dif-
ferent quality parameters. For example, the treatment
processes for the six discharging industries under the most
conservative conditions (corresponding to the upper
bounds of the solutions as shown in Table 5) could be
secondary, tertiary, secondary, secondary, secondary and
tertiary, respectively; under the most optimistic conditions,
the treatment processes could be enhanced primary,
enhanced primary, enhanced primary, secondary, enhanced
primary, and secondary, respectively.
Generally, the solutions from the IQWLA model
demonstrate tradeoffs between the overall wastewater
treatment cost and the system-failure risk due to inherent
uncertainties that exist in various system components.
Planning with a higher system cost may guarantee that the
water-quality management requirements and environmen-
tal regulations be met (i.e., higher system reliability); if
the plan aims towards a lower system cost, the possibility
of meeting the requirements by the suggested treatment
requirements decreases (i.e., higher system risk). In
practical applications, the allowable levels of environ-
mental violations could be decided based on discussions
among stakeholders, and the decision variables could be
adjusted continuously within their solution intervals
according to specific system conditions. This would allow
decision makers to incorporate more implicit knowledge
(such as socio-economic conditions), if required for the
decision.
Conclusions
A simulation-based interval quadratic waste load allocation
(IQWLA) model was developed in this study for river water
quality management. Uncertainties associated with water
quality parameters, cost functions, and environmental
guidelines were reflected as interval parameters. The cost
functions of wastewater treatment plant were expressed as
quadratic forms. A water quality planning problem in the
Changsha section of the Xiangjiang River in China was
used as a study case to demonstrate the applicability of the
proposed method. The study results demonstrated that the
proposed simulation-based IQWLA model could effec-
tively communicate the interval-format uncertainties into
the optimization process, and generate inexact solutions that
contain a spectrum of potential wastewater treatment
options. Decision alternatives can be obtained by adjusting
different combinations of the decision variables within their
solution intervals. The results are valuable for supporting
local decision makers in generating cost-effective water
quality management schemes.
The research work was advantageous over the previous
studies in the following aspects: (i) a multi-segment water
quality model was developed to generate water-quality
transformation matrices and vectors, which could be
embedded directly into a water quality management model;
(ii) uncertainties associated with water quality parameters
could be projected to optimization models through interval
analysis; (iii) the nonlinearity problem associated with the
objective function of wastewater treatment could be miti-
gated by quadratic regression technique.
The water simulation system proposed in this study is
suitable for one-dimensional steady-state river systems, with
CBOD, NBOD and DO levels being the main water-quality
indicators. In further studies, additional water-quality indi-
cators such as chemical oxygen demand (COD) and
phosphates can also be handled through developing similar
transformation matrices/vectors. In practical applications,
the solutions from the proposed water quality management
model are suitable for a preliminary evaluation of various
alternatives and for identifying the important data require-
ment before initiation of more extensive or expensive data
collection and simulation studies. The model solution would
be more applicable, if post-modeling analyses such as multi-
criteria decision analysis, group decision making, and public
survey can be performed.
Acknowledgments This research was supported by the Major State
Basic Research Development Program of MOST (2005CB724200 and
2006CB403307), the Canadian Water Network under the Networks of
Centers of Excellence (NCE), and the Natural Science and Engi-
neering Research Council of Canada. The authors deeply appreciate
the editor and the anonymous reviewers for their insightful comments
and suggestions.
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