Development of an optimization model for water resources systemsplanning

Engineering Applications of Artificial Intelligence 26 (2013) 1061–1071

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Engineering Applications of Artificial Intelligence journal homepage: www.elsevier.com/locate/engappai

Development of an optimization model for water resources systems planning D.Y. Miao a,1, Y.P. Li b,n, G.H. Huang b,2, Z.F. Yang a,3 a b

State Key Laboratory of Water Environment Simulation, School of Environment, Beijing Normal University, Beijing 100875, China MOE Key Laboratory of Regional Energy Systems Optimization, S-C Resources and Environmental Research Academy, North China Electric Power University, Beijing 102206, China

a r t i c l e i n f o

abstract

Article history: Received 16 May 2011 Received in revised form 21 August 2012 Accepted 27 November 2012 Available online 17 January 2013

A number of inexact fuzzy programming methods have been developed for the planning of water-resources-management systems under uncertainty. However, most of them do not allow the parameters in the objective and constraints of a programming problem to be functional intervals (i.e., the lower and upper bounds of the intervals are functions of impact factors). In this study, an interval fuzzy bi-infinite De Novo programming (IFBDP) method is developed in response to the above concern. A case study is also conducted; the solutions are then compared with those obtained from inexact De Novo programming (IDNP) and interval-fuzzy De Novo programming (IFDNP) that takes no account of bi-infinite programming. It is indicated that the IFBDP method can generate more reliable solutions with a lower risk of system failure due to the possible constraints violation and provide a more flexible management planning since the budgets availability can be adjusted with the variations in water price. These solutions are more flexible than those identified through IFDNP since the tolerance intervals are introduced to measure the level of constraints satisfaction. Moreover, it can be used for analyzing various scenarios that are associated with different levels of economic consequences under uncertainty. & 2012 Elsevier Ltd. All rights reserved.

Keywords: De Novo programming Fuzzy sets Functional interval Multiobjective Uncertainty Water resources

1. Introduction

1.1. Literature review

In recent decades, environmental systems analysis and design have become important managerial and operational issues confronting many countries and regions in the world (He et al., 2009). It is an active process that seeks a portfolio of resource levels and optimizes the objective function by allocating a budget according to a resource price, where resource levels are considered as decision variables (Zeleny, 1990). However, such planning efforts are complicated with a variety of uncertain parameters as well as their interactions. In fact, in water resources management and planning problems, many system parameters and their interrelationships are often associated with uncertainties presented in terms of multiple formats (Li et al., 2006). Moreover, these uncertainties may be multiplied by limited budget and resources with a maximized system benefit. Therefore, it is necessary to develop effective optimization methods for supporting water resources management under such complexities and uncertainties.

A number of methods, such as fuzzy, stochastic and interval mathematical programming were developed for dealing with the uncertainties in water resources management problems. For example, Slowinski (1986) proposed an interactive fuzzy multiobjective linear programming method and applied it to water supply planning. Wu et al. (1997) proposed an interactive inexact-fuzzy multiobjective programming model for planning water resources systems, where IPP and fuzzy programming (FP) were incorporated within a multiobjective framework to handle uncertainties presented in terms of discrete intervals and fuzzy sets. Jairaj and Vedula (2000) optimized a multi-reservoir system using fuzzy mathematical programming method, where the uncertainties existing in reservoir inflows were treated as fuzzy sets. Bender and Simonovic (2000) proposed a fuzzy compromise approach to water resources planning under imprecision uncertainty. Lee and Chang (2005) proposed an interactive fuzzy approach for planning a stream water resources management system that involved vague and imprecise information. Li et al. (2009) advanced a multistage fuzzy-stochastic programming model for water-resources allocation and management, where uncertainties expressed as probability distributions and fuzzy sets could be reflected. Lu et al. (2010) developed an interval-valued fuzzy linear programming method based on infinite a-cuts for water resource management. These methods have been applied to various hypothetical and real cases and presented effectiveness in

n

Corresponding author. Tel.:þ 86 10 5197 1255; fax: þ86 10 5197 1284. E-mail addresses: [email protected] (D.Y. Miao), [email protected] (Y.P. Li), [email protected] (G.H. Huang), [email protected] (Z.F. Yang). 1 Tel.: þ86 15120095436; fax: þ 86 10 58802756. 2 Tel.: þ86 10 5197 1215; fax: þ 86 10 5197 1284. 3 Tel.: þ86 10 58807951; fax: þ86 10 58807951. 0952-1976/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engappai.2012.11.010

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accounting for uncertainty in interval parameter program and fuzzy program. However, in real world problems, the constraint resources have imprecise features, which are difficult to or cannot be determined precisely. Compared with the traditional fuzzy programming methods, De Novo programming was effective for dealing with optimal design problems with unknown resource availability and seeking a portfolio of resource availability level to optimize multiple objective functions by allocating a budget according to the resource price (Zeleny, 1981, 1986, 1990). Previously, a number of research works based on the De Novo programming were applied to various system design cases. For example, Zeleny (1990) proposed a basic method to construct the optimal system design for solving a De Novo problem via an ideal system design; one of the important issues in multicriteria De Novo programming was to determine an optimum-path ratio for enforcing a particular budget level of resources so as to establish the optimal system design. Li and Lee (1990) extended Zeleny’s basic method to identify fuzzy system designs for De Novo problems by considering the fuzziness in coefficients, and further treated fuzzy goals and fuzzy coefficients simultaneously, depending on a numerical approach which could be solved as either linear or nonlinear problems (Li and Lee, 1993). Shi (1995) introduced several optimum-path ratios for enforcing different budget levels of resources to identify alternative optimal system designs for solving multicriteria De Novo programming problems. Kotula (1997) used the De Novo programming for control and adjustment of reservoir design and operation characteristics which resulted in optimal or near optimal system performance throughout the life of the reservoir. Zeleny (2005) investigated the evolution of optimality of single and multiobjective programming, where a number of major optimality concepts according to a dual classification were discussed. Chen and Hsieh (2006) presented a fuzzy multistage De Novo programming, where random distribution of budget was analyzed. More recently, Zhang et al. (2009) developed an interval De Novo programming (IDNP) method through introducing intervalparameter programming (IPP) into the De Novo programming framework for the planning of water-resources systems, where uncertainties presented as discrete intervals were addressed. However, the IDNP can only solve the problems containing crisp interval coefficients [a, b], whose lower and upper bounds (i.e., a and b) are both deterministic and definitely known. This is based on the assumption that these interval coefficients are unchanged even if they could be affected by associated impact factors. However, a challenge leads to a need of further improving the aforementioned efforts. It is that traditional crisp intervals in their programs (i.e., the lower and upper bounds are both constants) can hardly address the association of their impact factors and fuzzy programming could be an approach to fill this gap by introducing an intermediate control variable (l) to measure the level of constraints satisfaction. For example, the unit water supply cost and benefit of water users are important parameters determining optimal water allocation and wastewater treatment schemes, and maximum net system benefits in the process of decision making. However, it will vary with the dynamic fluctuation of its impact factors such as water price. In consideration of such an impact, the lower and upper bounds of the unit water supply cost and benefit of water users can hardly be expressed as simple constants any more. Functional intervals represent a type of highly complex uncertainty in comparison to conventional crisp intervals (He and Huang, 2008). It is an extended interval whose lower and upper bounds are both represented as functions of an independent variable (e.g., water price); thus, it could be used to simultaneously account for the parameters’ uncertainty

(due to imprecise information) and association (with other impact factors). Therefore, one approach to potentially address these uncertainties is to introduce interval bi-infinite programming (IBIP) and fuzzy programming (FP) into the De Novo programming framework; this will lead to an interval fuzzy bi-infinite De Novo programming (IFBDP) method. The developed IFBDP can effectively deal with uncertainties expressed as fuzzy sets and functional interval values in single and multiobjective problems. Besides, techniques of post-optimality analysis (e.g., multicriteria decision analysis, analytical hierarchy process technique, dual programming, and parametric programming) could be used for further supporting fine adjustments of the modeling results and thus for enhancing their applicability to practical situations. Furthermore, intelligent decision support system (IDSS) could be developed based on an integration of optimization modeling, scenario development, user interaction, policy analysis and visual display into a general framework (Li et al., 2010). Uncertainties in water resources planning systems could be effectively reflected and addressed through the interval fuzzy bi-infinite De Novo programming approach, improving the stability of the IDSS for real-world applications. Then the method is applied to a case study of (i) water resources systems planning, which designs an inexact optimal system with budget limit and different weight, and (ii) illustrating its advantages over the previous approach such as inexact De Novo programming (IDNP), and interval-fuzzy De Novo programming (IFDNP) which does not consider bi-infinite programming. A number of scenarios were examined for system uncertainties and decision processes to identify an optimum system design with higher benefits. The results obtained can be used to help decision makers evaluate alternative system designs and to determine which of these designs can most efficiently achieve the desired system objectives.

2. Modeling formulation 2.1. Interval fuzzy bi-infinite De Novo programming Definitions for the concepts of intervals and functional intervals are given before the formulation of the interval fuzzy bi-infinite De Novo programming problem. An interval can be defined as an interval with known upper and lower bounds but unknown distribution information (Chang and Wang, 1997; He et al., 2009): a 7 ¼ ½a ,a þ  ¼ fa9a r a ra þ g 

ð1Þ 7

þ

where a and a are lower and upper bounds of a , respectively. If a  ¼ a þ , then a 7 becomes a deterministic number. As an extension of intervals, the concept of functional intervals is proposed for addressing a kind of more complicated uncertainty. Similar to the definition of intervals, functional interval a 7 (y) can be defined as: a 7 ðyÞ ¼ ½a ðyÞ,a þ ðyÞ ¼ faðyÞ9a ðyÞ r aðyÞ r a þ ðyÞg 

f or

y A ½yl ,yu 

ð2Þ

þ

where a (y) and a (y) are lower- and upper-bounds functions, respectively, and y is independent variable ranging from yl to yu. With definitions (1) and (2), an interval fuzzy bi-infinite De Novo programming problem can be conceptualized as follows: Maxf

7

¼ C 7 ðyÞX 7 ~

for all

y0 A ½yl0 ,yu0 

ð3aÞ

subject to: A 7 ðyÞX 7 b

7

r0

for all

y A ½yl ,yu 

ð3bÞ

D.Y. Miao et al. / Engineering Applications of Artificial Intelligence 26 (2013) 1061–1071

p7 b

7

rB7

7 kj

7

7 ij

and A (y)A{R(y) } are matrices of where C (y)A{R(y) } dimensions k  j and i  j, respectively; R(y) 7 denotes a set of functional intervals associated with an independent variable (y); ckj7 ðyÞ is the element of matrix C 7 (y); aij7 ðyÞ is the element of 7

matrix A (y) at the ith row and the jth column; b

7

7 7 7 design decision variable, b ¼ ðb1 ,b2 ,. . .,bi Þ; X 7 is vector of 7 7 7 management decision variables, X ¼ ðx1 ,x2 ,. . .,xj7 Þ; p 7 is the vector of the unit prices of i resources, p 7 ¼ ðp17 ,p27 ,. . .,pi7 Þ; and 7

is the given total available budget; symbols ¼ and r represent ~

~

fuzzy equality and fuzzy less than, implying that the constraint does not need to be definitely met, but should be satisfied to a certain satisfactory level that is quantified by an introduced control variable (l 7 ) (Zimmermann, 1985). Thus, to better communicate fuzzy objective and constraints, as well as the flow uncertainties, model (3) can be converted into following formulation: Maxl

7

ð4aÞ

subject to: C 7 ðyÞX 7 Z f þ l 

A 7 ðyÞX 7 b 7

7

7 þ

f f

r0

rB þ l



for all

7

B þ B

for all

y0 A ½yl0 ,yu0 

y A ½yl ,yu 



X 7 Z0

t X

ð4bÞ ð4cÞ ð4dÞ ð4eÞ



j ¼ tþ1 þ

bi r 0 xj7 Z0,

for all

yi A ½yil ,yiu 

ð5dÞ

8j

ð5eÞ

þ

0rl r1

ð5fÞ

xjþ ,

j ¼1, 2, y, t, are interval variables with positive

where

coefficients in the objective function; x j , j¼ tþ1, t þ2, y, n, are interval variables with negative coefficients in the objective þ function. Solutions of xjþ , x and l þ can be obtained from j , bi n 7

7

submodel (5). Let f k ¼ maxf k , k¼1, y, q, be the optimal value for Kth objective of submodel with model (5). Let n

7 n 7

n

n 7

f ¼ ð f 1 , f 2 ,. . ., f q Þ be the q-objective value for the ideal

system with respect to B 7 . SignðdÞ is defined as: 8 < 1 ðaij7 ðyÞ Z 0 for all yA ½yl ,yu Þ 7 Sign½aij ðyÞ ¼ : 1 ðaij7 ðyÞ o 0 for all y A ½yl ,yu Þ

7

r1

ð4fÞ

where f  and f þ are the lower and upper bounds of the objective’s aspiration level (i.e., f þ is the most desirable system objective value; f  is the least desirable system objective value), respectively; C  (y) and C þ (y) are lower- and upper-bounds functions of C 7 (y), respectively; A  (y) and A þ (y)are lower- and upper-bounds functions of A 7 (y), respectively; l 7 is a control variable, which denotes the degree of satisfaction for the fuzzy objective and/or constraints. A l 7 level close to 1 would correspond to a high possibility of satisfying the constraints/objective under advantageous conditions; conversely, a l 7 value near 0 would be related to a solution that has a low possibility of satisfying the constraints/objective under demanding conditions. The IFBDP model can be solved through a two-step method by transformed into two sets of deterministic submodels, which correspond to the lower and upper bounds of the desired objective function value. This transformation process is based on an interactive algorithm (Huang et al., 2001; He and Huang, 2004), which is different from normal interval analysis and best/ worst case analysis (Huang, 1996). Since the objective is to maximize the value of l 7 , the first step is to formulate the following submodel corresponding to l þ (Huang et al., 2001): maxl

þ

max l 8 9 t   < X      þ=  9pi 9 Sign pi 9aij yi 9 Sign½aij yi xj : ; j¼1 j¼1 8 9 n n X   < X   þ   =  þ 9pi 9 Sign pi 9aij yi 9 Sign½aij yi xj þ : ; j ¼ tþ1 j ¼ tþ1   n for all yi A ½yil ,yiu , 8i r n B l n Bn B

ð7bÞ

n y þ y  y þy  X n  l0 u0 u0 xjþ  x ckjþ l0 j Z fk 2 2 j¼1 j ¼ tþ1 n  n n þ þ l f k  f k , 8k

ð7cÞ

xj7 Z0

ð7dÞ

t X

t X



ckjþ

0 r l r1 n

ð7eÞ

Solving model (7) yields n

b¼

t X

n

n n xj7 , b, l

and r, where: n

    n þ 9aij yi 9 Sign½a ij yi  xj

j¼1

ð5aÞ

n X

þ

  þ   9aij yi 9 Sign½aijþ yi n x j

ð8aÞ

j ¼ tþ1

n y þy  y þ y  X  l0 u0 u0 xjþ  x ckjþ l0 j Zf k 2 2 j¼1 j ¼ tþ1 þ þ  þ l f k f k , 8k

ckjþ

8 9 t t X   < X     þ=   9pi 9 Sign pi 9aij yi 9 Sign½aij yi xj : ;

j¼1

ð7aÞ

subject to:

subject to: t X

ð6Þ

Note that merely functional intervals whose values over the entire range of [yl, yu] are positive or negative are considered in this study. Then, the metaoptimum submodel can be constructed as follows: n

0rl

ð5cÞ

n X      þ   þ   9aij yi 9 Sign½a 9aij yi 9 Sign½aijþ yi x ij yi xj þ j

j¼1

is vector of

7

p7 b

n X

ð3dÞ 7

B

9 8 n   < X   þ   = þ þ 9pi 9 Sign pi 9aij yi 9 Sign½aij yi xj ; : j ¼ tþ1 j ¼ tþ1   þ r B þ l B þ B for all yi A ½yil ,yiu , 8i

ð3cÞ

~

X 7 Z0

1063

j¼1

n

ð5bÞ

P r¼

os Bsþ n

ð8bÞ

B

The value Bsþ is the Sth city budget level of producing n xj7 P with respect to the Jth objective, where os ¼1 and 0r os r1 7 (Shi, 1995). The optimal system design can be established as (xopt , þ

þ

þ

n

þ bopt , f opt ), where lopt ¼ l , xopt ¼ n r  n xjþ , (j ¼1, 2, y, t),

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D.Y. Miao et al. / Engineering Applications of Artificial Intelligence 26 (2013) 1061–1071

Uncertain information Functional intervals

Right hand side variability

Discrete intervals

Imprecise information

De Novo programming

Interval-parameter programming

Fuzzy programming

IBIDNP model

Bi-infinite

IFBDP model

Lower-bound submodel

Upper-bound submodel

Solutions under different scenarios for IFBDP model

Generation of decision alternatives Fig. 1. Framework of the IFBDP approach.

and

where x j , j¼ 1, 2, y, t, are interval variables with positive

f opt ¼ n r  f k . The optimum-path ratio n r provides an effective and fast tool for optimal redesign of large-scale systems. Based on the solutions of the first submodel, the second step is to formulate the following submodel corresponding to l  :

coefficients in the objective function; xjþ , j ¼t þ1, t þ2, y, n, are

n n  x opt ¼ r  xj ,

þ

(j¼t þ1,

tþ2,

y,

n),

þ

n

bopt ¼ n r  b,

þ

n



maxl

ð9aÞ

interval variables with negative coefficients in the objective   þ can be obtained through function. Solutions of x j , xj , bi and l solving submodel (9). Similarly, the metaoptimum submodel can be constructed as follows: n

max l

subject to:

subject to:

n y þy  y þy  X  l0 u0 l0 u0 x xjþ Zf k c j  kj 2 2 j¼1 j ¼ tþ1    þ l f k f k , 8k t X

t X

c kj

 þ 9pi 9 Sign piþ

8 t < X

  þ   9aij yi 9 Sign½aijþ yi x j

ð9bÞ 9 =

j¼1

j ¼ tþ1

r B þ l t X

B þ B



for all

yi A ½yil ,yiu ,

8i

ð9cÞ

  þ   9aij yi 9 Sign½aijþ yi x j

j¼1 n X

þ

     þ  9aij yi 9 Sign½a ij yi xj bi r 0

ð10bÞ

n y þ y  y þy  X n  l0 u0 l0 u0 x xjþ Z f k c j  kj 2 2 j¼1 j ¼ tþ1 n n þ n  þ l f k  f k , 8k

ð10cÞ

xj7 Z0

ð10dÞ

t X

j ¼ tþ1



8 9 t =  þ < X   þ   9pi 9 Sign pi 9aij yi 9 Sign½aijþ yi x j : ; j¼1 j¼1 8 9 = n n X  þ < X      þ þ 9pi 9 Sign pi 9aij yi 9 Sign½a þ ij yi xj : ; j ¼ tþ1 j ¼ tþ1  n n n n r B l B B for all yi A ½yil ,yiu , 8i t X

; 8 9 n n = X  þ < X      þ þ þ 9pi 9 Sign pi 9aij yi 9 Sign½a ij yi xj : ;

j¼1

:

ð10aÞ

for all

þ

c kj

n þ x j r xj ,

j ¼ 1,2,. . .,t

ð10eÞ

ð9dÞ

xjþ Z n x j ,

j ¼ t þ 1,t þ 2,. . .,n

ð10fÞ

ð9eÞ

0 r l r1

yi A ½yil ,yiu 

j ¼ tþ1



n

n þ

fk r fk , xj7 Z 0,

8k

Solving model (10) yields 8j

ð9fÞ n



0rl r1

ð9gÞ

b¼

t X j¼1

ð10gÞ n

n n xj7 , b, l

  þ   9aij yi 9 Sign½aijþ yi n x j

and r, where: n

D.Y. Miao et al. / Engineering Applications of Artificial Intelligence 26 (2013) 1061–1071

þ

n X

3. Case study

    n þ 9aij yi 9 Sign½a ij yi  xj

ð11aÞ

j ¼ tþ1

n

P r¼

os Bs n

A water resources management problem is provided to illustrate applicability of the developed IFBDP approach. Consider a case in which a water manager is responsible for allocating water from a water abundant reservoir for two co-developing cities to three users: a municipality, an industrial unit, and an agricultural sector (Fig. 2). The total water usage budget is [550, 700]  106 dollars and the limited water usage budget includes water supply cost and wastewater treatment cost. The minimum total water needs and minimum proportions of total allocated water to city for three water users are listed in Table 1 (Zhang et al., 2009). In urban water system design, it is necessary to guarantee that the water allocation must meet the least demand of each city and each water user (Jing and Chen, 2011; Kirnbauer and Baetz, 2012). The shrinking water availability has been exacerbating such competitions, particularly under varying natural conditions and deteriorating quantity and quality of water resources. Therefore, all users want to know how much water they can expect. If insufficient water is available, they will try to obtain water from other sources or curtail their development plans. Given a quantity of water that is promised to each user, if this water is delivered, it will result in net benefits to the local economy; however, if the promised water is not delivered, either the water must be obtained from alternative and more expensive sources or the demand must be curtailed, resulting in penalties on the local economy (Li et al., 2008). In order to consider the maximum earnings for three water users, the three criteria will be calculated. The unit system benefits and wastewater treatment costs can be affected by water price of each water user and cities. Thus, unit system benefits and wastewater treatment costs are assumed to

ð11bÞ

B

n 7 The value B with s is the Sth city budget level of producing xj P os ¼1 and 0 r os r1 (Shi, respect to the Jth objective, where  7 1995). The optimal system design can be established as (xopt , bopt ,





n

þ n n  n n þ f opt ), where lopt ¼ l , x opt ¼ r  xj , (j ¼1, 2, y, t), xopt ¼ r  xj , 

n



n 

(j¼ tþ1, t þ2, y, n), bopt ¼ n r  b, and f opt ¼ n r  f k . þ þ þ Therefore, bopt , f opt , xopt ðj ¼ 1,2,. . .,t Þ and x opt ðj ¼ t þ 1,t þ2,. . .,nÞ can be obtained by solving the submodel defined by models (5) to   þ (8), whereas bopt , f opt , x opt ðj ¼ 1,2,. . .,t Þ and xopt ðj ¼ t þ 1,t þ2,. . .,nÞ can be obtained from models (9) to (11). Thus, according to Huang et al. (1993), final solutions for model (4) are: 7 þ xopt ¼ ½x opt ,xopt 

ð12Þ

7  þ lopt ¼ ½lopt , lopt 

ð13Þ

7



7



1065

þ

f opt ¼ ½f opt ,f opt 

ð14Þ

þ

bopt ¼ ½bopt ,bopt 

ð15Þ

Fig. 1 shows the scheme of the modeling methodology. In conclusion, the solution algorithm of the IFBDP model with the objective being maximized is presented as follows: Step 1. Formulate IFBDP model (4). Step 2. Transform the IFBDP model into two submodels, where the objective is to maximize f 7 . Step 3. Formulate f þ submodel (5). Step 4. Solve f þ submodel and metaoptimum submodel n þ (7) and obtain n xj7 , b,lopt and n r. þ þ 7 Step 5. Calculate xopt , bopt and f opt .  Step 6. Formulate f submodel (9). Step 7. Solve f  submodel and metaoptimum submodel (10) n  and obtain,n xj7 , b,lopt and n r.   7 Step 8. Calculate xopt , bopt and f opt 7 þ Step 9. Solutions of the IFBDP model are:xopt ¼ ½x opt ,xopt , 7  þ 7  þ 7  þ f opt ¼ ½f opt ,f opt , lopt ¼ ½lopt , lopt  and bopt ¼ ½bopt ,bopt . Step 10. Stop.

Table 1 Inputs of model (16). Least water demand (106 m3)

Industry

Municipality

Agriculture

City 1 City 2 Least allocation proportion City 1 City 2 Least total water demand (106 m3)

[13, 15] [15, 17]

[13, 15] [15, 17]

[13, 15] [15, 17]

[0.30, 0.32] [0.35, 0.37] City 1 [55, 60]

[0.25, 0.27] [0.35, 0.36] [0.20, 0.23] [0.30, 0.32] City 2 [65, 70]

Agriculture

Industry

Municipality

City 2

WS3 WS4 Reservoir WS1

WS2

City1 Agriculture Industry

Municipality

Fig. 2. Schematic of water allocation to multiple users.

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D.Y. Miao et al. / Engineering Applications of Artificial Intelligence 26 (2013) 1061–1071

Table 2 System benefits as functions of water price j (wpj). Benefit (dollars/m3)

Industry

Municipality

Agriculture

City 1 City 2 Price/unit water of user j ($/m3)

[80 þ 1.5 wpj, 90 þ2.0 wpj] [92þ 1.5 wpj, 100þ2.0 wpj] [1.5, 1.8]

[76þ 1.5 wpj, 80þ 2.0 wpj] [70 þ 1.5 wpj, 75 þ 2.0 wpj] [2.0, 2.1]

[45þ 1.5 wpj, 50þ 2.0 wpj] [40 þ 1.5 wpj, 43þ 2.0 wpj] [1.0, 1.3]

Table 3 Wastewater treatment costs as functions of water price i (wpi). Wastewater treatment efficiency (dollars/m3)

City 1

(6) Minimum proportion of total allocated water to end users in city i:

City 2

xij7 Z eij7

m X

xij7 ,

8i,j

ð16gÞ

i¼1

[0.79þ 0.2 wpi, 0.81þ 0.25 wpi] Municipality [0.65þ 0.2 wpi, 0.70þ 0.25 wpi] Agriculture [0.88þ 0.2 wpi, 0.90þ 0.25 wpi] Price/unit water of city i ($/m3) [2.5, 2.7]

[0.85 þ0.2 wpi, 0.88 þ 0.25 wpi] [0.70 þ 0.2 wpi, 0.75 þ 0.25 wpi] [0.93 þ0.2 wpi, 0.95 þ 0.25 wpi] [2.0, 2.3]

Industry

(7) Minimum water allocation to end users in city i: xij7 Z qij7 ,

7

where f j be functional intervals associated with the water price. With regard to the estimation of system benefits and wastewater treatment costs, many sources supplied information on the realtime water price. Tables 2 and 3 provide the unit system benefits and wastewater treatment costs used for this study. To guarantee quantity and quality of water, the different water treatment facilities for different water users are investigated. Therefore, the problems under consideration are how to effectively allocate water to the three users to achieve a maximum benefit under uncertainty while incorporating water policies with the least risk of system disruption (Maqsood et al., 2005). Then, the problem can be formulated as follows: maxl

8i,j

ð16hÞ

  is maximize benefit for each water user j; cij7 wpj is

the unit benefit of water user j in city i, which is the function of wpj, i¼1, 2; j ¼1, 2, 3; wpj ¼water price of user j ($/m3), an   independent variable ranging from wpjL to wpjU; aij7 wpi is wastewater treatment efficiency for water usage j in city i, which is the function of wpi; wpi ¼water price of city i ($/m3), an 7

independent variable ranging from wpiL to wpiU; li is the least total water allocation to city i; qij7 and eij7 are the minimum water needs and minimum proportions of total allocated water to city for user j; xij7 is decision variable of water allocation plan to 7

user j in city i; bi is the design variable for total water allocation to city i which is corresponding to unit cost pi within the total 7

7

ð16aÞ

subject to:

water usage budget; bj is the design variable of wastewater treatment capacity for water user j, which correspond to unit cost pj within the total water usage budget.

(1) Maximize benefits for each water user: m X

     þ  7 cij7 wpj xij7 Zf j þ l f j f j

4. Result analysis for all

wpjL

i¼1

rwpj r wpjU ,

8j

ð16bÞ

(2) Treated water demand of city: n X

  7 aij7 wpi xij7 bi r 0

wpiL r wpi rwpiU ,

for all

8i

j¼1

ð16cÞ (3) Total water needs of end user: m X

7

xij7 r bj ,

8j

ð16dÞ

i¼1

(4) Budget limit for water system design: m X

7

pi7 bi þ

i¼1

n X

7

pj7 bj r B þ l

7

B þ B



ð16eÞ

j¼1

(5) Minimum water allocation to city: m X i¼1

7

xij7 Z li ,

8i

ð16fÞ

In this study, a number of scenarios associated with different optimum-path ratios are examined. The optimal water allocation solutions are obtained by solving model (16). Though an increased uncertainty level need to be considered due to the introduction of functional intervals, the optimization solutions do not increase the difficulties in decision making as the solutions are expressed as intervals or constants but not function intervals. According to different o which values from 0.1 to 0.9, we can obtain different optimum-path ratios corresponding to different scenarios. For example, the solutions from model (16) under scenario 5 (i.e., when o1 ¼0.5 and o2 ¼0.5 obtained optimum-path ratios n r ¼0.303 and n r ¼0.302) are shown in Table 4. For instance, for city 1, water flows allocated to industry, municipality and agriculture would be [58.35, 66.23]  106 m3, [37.97, 40.51]  106 m3 and [48.77, 56.59]  106 m3, respectively; in comparison, water supply design budget variable would be [75.28, 91.27]  106 m3. For city 2, water flows allocated to industry, municipality and agriculture would be [52.71, 57.82]  106 m3, [40.37, 43.84]  106 m3 and [56.30, 61.33]  106 m3, respectively; in comparison, water supply design budget would be [74.88, 88.50]  106 m3. Wastewater treatment limit budgets would be [111.06, 115.12]  106 m3 (for industry), [98.34, 116.60]  106 m3 (for municipal), and [158.86, 233.21]  106 m3 (for agriculture), respectively. The results indicate that water supply for city 1 is mainly used to guarantee water demand from industry, then supply to agriculture and municipality; water allocated to city

D.Y. Miao et al. / Engineering Applications of Artificial Intelligence 26 (2013) 1061–1071

1067

Table 4 Solutions from the IFBDP model under scenario 1, 5 and 9.

Water allocation to city 1 (106 m3) Industry Municipality Agriculture Water allocation to city 2 (106 m3) Industry Municipality Agriculture Wastewater treatment limit bj (106 m3) Industry Municipality Agriculture Water supply design bi (106 m3) City 1 City 2 7 lopt 7 7 lmeta (n r  lopt )

scenario 1

scenario 5

scenario 9

[62.99, 71.69] [40.98, 43.86] [52.64, 61.25]

[58.35, 66.23] [37.97, 40.51] [48.77, 56.59]

[53.14, 61.20] [34.57, 37.44] [44.41, 52.29]

[56.89, 62.59] [43.57, 47.46] [60.78, 66.39]

[52.71, 57.82] [40.37, 43.84] [56.30, 61.33]

[47.99, 53.43] [36.76, 40.51] [51.27, 56.67]

[119.88, 124.61] [106.16, 126.22] [171.49, 252.45]

[111.06, 115.12] [98.34, 116.60] [158.86, 233.21]

[101.13, 106.38] [89.55, 107.75] [144.66, 215.50]

[81.26, 98.80] [80.83, 95.81] [0.63, 0.99]

[75.28, 91.27] [74.88, 88.50] [0.63, 0.99]

[68.55, 84.34] [68.19, 81.79] [0.63, 0.99]

[0.21, 0.33]

[0.19, 0.30]

[0.17, 0.28]

2 is mainly for agriculture, then to industry and municipality. Industry users in city 1 bring the highest benefit when water demand is satisfied; whereas municipal users in city 2 correspond to lower benefits. The decision maker would pay attention to the basic demand of the water users in each city when the budget is low; when the budget is abundant, the decision maker can obtain more earnings from each water user. Besides, the water supply to city 1 ([75.28, 91.27]  106 m3) is higher than that to city 2 ([74.88, 88.50]  106 m3). This is because city 1 possesses the higher earnings, higher wastewater treatment efficiency, and lower water supply cost. The amount of wastewater treated for industrial use would be [111.06, 115.12]  106 m3; for municipal use would be [98.34, 116.60]  106 m3; for agriculture use would be [158.86, 233.21]  106 m3. The results obtained can help decision maker design proper wastewater treatment facilities in water purifying industry, municipality and agriculture. Similarly, the solutions of model (16) under scenario 1 (i.e., when o1 ¼0.1 and o2 ¼0.9 obtained optimum-path ratios n r ¼0.328 and n r ¼0.326) and scenario 9 (i.e., when o1 ¼0.9 and o2 ¼0.1 obtained optimum-path ratios n r ¼0.280 and n r ¼0.275) could be interpreted based on the results presented as in Table 4. The results indicate that different scenarios would lead to varied system benefits and budgets and, at the same time, higher benefits would lead to higher budgets for wastewater treatment and supply. The results indicate that different design budgets lead to varied water supply schemes as presented in Table 4. For example, for city 1, water supplies would be [81.26, 98.80]  106 m3 under scenario 1 and [68.55, 84.34]  106 m3 under scenario 9; for city 2, water supplies would be [80.83, 95.81]  106 m3 under scenario 1 and [68.19, 81.79]  106 m3 under scenario 9. For industrial use, the amounts of wastewater treated would be [119.88, 124.61]  106 m3 under scenario 1 and [101.13, 106.38]  106 m3 under scenario 9; for municipal use, the amounts of wastewater would be [106.16, 126.22]  106 m3 under scenario 1 and [89.55, 107.75]  106 m3 under scenario 9, respectively; for agriculture use, the amounts of wastewater would be [171.49, 252.45]  106 m3 under scenario 1 and [144.66, 215.50]  106 m3 under scenario 9, respectively. It is indicated that different optimum-path ratios (i.e., different o) would lead to different budgets; at the same time, different budgets would lead to different water supply quantities and treating wastewater amounts. For example, if scenario 1 is adopted, it means that budget corresponding to water supply and treating wastewater amount would be largest under an advantageous circumstance; but, it also has a highest risk of

system-failure penalty when water shortage occurring. If scenario 9 is adopted, it means that budget corresponding to water supply and treating wastewater amount would be lowest under a disadvantageous circumstance; it may lead to a waste of resources. Generally, a higher budget would lead to higher water supply and treating wastewater amount, at the same time, a higher risk of system design failure when the water flow is low; in comparison, a lower budget would result in lower water supply and treating wastewater amount with a lower risk of penalty. Fig. 3 provide the net system benefits from industry, municipality and agriculture under different scenarios. Different optimum-path ratios correspond to different scenarios, and thus would lead to varied net system benefits. For example, system benefits for the industry would be $[4736.32, 5912.86]  106 and $[3995.36, 5047.56]  106 under scenarios 1 and 9, respectively; system benefits for the municipality would be $[2754.02, 3467.87]  106 and $[2323.18, 2960.38]  106 under scenarios 1 and 9, respectively; system benefits for the agriculture would be $[2354.59, 2770.15]  106 and $[1986.23, 2364.76]  106 under scenarios 1 and 9, respectively. The solutions under the other scenarios could be similarly interpreted based on the results presented in Fig. 3. The results indicate that higher water allocation and wastewater treatment capacity correspond to higher system benefit, and lower water allocation and wastewater treatment capacity correspond to lower system benefit. Moreover, the benefit from the industry is higher than those from municipal and agricultural sectors. For instance, under the best-case scenario (i.e., scenario 1), the wastewater treatment capacity would be 119.88  106 to 124.61  106 m3 for industry, 106.16  106 to 126.22  106 m3 for municipality and 171.49  106 to 252.45  106 m3 for agriculture (as shown in Table 4); it means that higher water allocation and higher wastewater treatment efficiency and lower water supply cost could achieve higher benefits, but at the same time, a higher risk of penalty would generate when the promised water is not delivered under demanding conditions. Under the worst-case scenario (i.e., scenario 9), the wastewater treatment capacity would be 101.13  106 m3 to 106.38  106 m3 for industry, 89.55  106 m3 to 107.75  106 m3 for municipality and 144.66  106 m3 to 215.50  106 m3 for agriculture (as shown in Table 4). The results indicate that lower water allocation and lower wastewater treatment efficiency and high water supply cost could bring about lower system benefit, less shortage, and lower penalty but, at the same time, more extra cost for wastewater treatment would be needed when future water flow level is high.

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D.Y. Miao et al. / Engineering Applications of Artificial Intelligence 26 (2013) 1061–1071

7000 industry

municipality

agriculture

Maximum benifits ($106)

6000 5000 4000 3000 2000 1000 lower upper lower upper lower upper lower upper lower upper lower upper lower upper lower upper lower upper

S1

S2

S3

S4

S5

S6

S7

S8

S9

Fig. 3. Maximum benefits under different scenarios (Symbol ‘‘S’’ denotes ‘‘Scenario’’).

7500

upper bound lower bound

industry

0.30 Net benefits (106)

Meta-satisfaction degree

0.35

0.25 0.20

industry 5500 municipal agriculture

municipal

3500

agriculture

0.15 lower level

upper level

0.10 S1

S2

S3

S4

S5

S6

S7

S8

S9

Fig. 4. Meta-satisfaction degree under different scenarios (Symbol ‘‘S’’ denotes ‘‘Scenario’’).

Through solving model (16), different lmeta levels can be obtained. Here, lmeta can be defined as meta-satisfaction degree 7 7 corresponding to l, and lmeta ¼ n r  lopt . The solutions of metasatisfaction degree are presented in Fig. 4. For example, the 7 solution of lmeta would be [0.21, 0.33] under scenario 1, which is the highest meta-satisfaction degree. This is because the raised strictness and the admissible violation of the uncertain flows under advantageous conditions. It is indicated that a plan with higher allocated flows is generated, resulting a higher system benefit and but, at the same time, a potentially higher penalty when the promised water is not delivered under demanding conditions (e.g., when the water flow level is low or low7 medium). In comparison, under scenario 9, the result of lmeta would be [0.17, 0.28]; the meta-satisfaction degree would be lower than those under the other scenarios. This is because the admissible violations of the uncertain flows under demanding conditions. It is indicated that there would be lower benefit and thus lower system penalty but, at the same time, potentially more waste of resources when the water flow level is medium-high or 7 high. Under scenario 5, the result of lmeta would be [0.19, 0.30]; the solution would correspond to a situation when water availability stands between conservative and optimistic scenarios. The relaxations of system constraints would imply raised constraintviolation risks. Tightened resources availability limitations would then be associated with lower constraint-violation risks. Decisions with lower constraint-violation risks would be associated with a lower system benefit but increasing system reliability; a

1500 IFBDP

IDNP

Fig. 5. Net benefits from IFBDP and IDNP models.

desire for higher benefit could result in raised risks of violating the system constraints. Therefore, different scenarios corresponding to different water supplies are associated with different options in handling the tradeoffs among system benefit, metasatisfaction degree, and constraint-violation risk.

5. Discussion 5.1. Comparison with IDNP method The problem can also be solved through an inexact De Novo programming (IDNP) method by simplifying the fuzzy membership functions into discrete intervals. This leads to a lack of systemreliability information as defined the control variable (l 7 ). Fig. 5 presents a comparison of the net benefit values obtained through the IFBDP and IDNP methods under scenario 5. The net benefits from INDP for industry, municipality, and agriculture would be $[4121.4, 7682.1]  106, $[2558.2, 4441.7]  106, and $[1927.2, 3449.8]  106, respectively. In comparison, the net benefits from the IFBDP for industry, municipality and agriculture would be $[4387.63, 5462.18]  106, $[2551.27, 3203.55]  106, $[2181.25, 2559.01]  106, respectively. Compared with IDNP, the IFBDP approach provides more information regarding trade offs among system benefits, certainty and reliability. Prior to further discussion, the uncertainty degree (UD) of an interval number (x 7 ) can be

D.Y. Miao et al. / Engineering Applications of Artificial Intelligence 26 (2013) 1061–1071

defined as follows (Huang and Moore, 1993): 

UD x

 7

x þ x  100% ¼ þ ðx þ x Þ=2

ð17Þ

where x þ is the upper bound of x 7 ;x  is the lower bound of x 7 . This definition of an interval number can be understood as the ratio of its width to its difference. It demonstrates that when system is highly uncertain, UD becomes close to 100%; when system turns out to be more deterministic, UD goes toward 0%. Therefore, the concept of UD is useful for quantitatively evaluating the levels of uncertainties or the input/output data qualities for mathematical models (Qin et al., 2007). The corresponding UD of IDNP for industry, municipality, and agriculture would be 60.34%, 53.82% and 56.64%, respectively. In comparison, the uncertainty degrees from IFBDP (i.e., 21.82%, 22.67% and 15.94%) are much lower than those from IDNP. Because the system benefits under other scenarios have similar tendency to Fig. 5, the solutions under the other scenarios could be similarly interpreted. Under scenarios 1and 9, the uncertainty degrees from IFBDP are 22.10% and 23.27% (for industry), 22.95% and 24.12% (for municipality), and 16.22% and 17.40% (for agriculture), respectively. The resulting intervals from IDNP model are much larger than those from IFBDP, and have larger width than those of IFBDP. It implies that solutions from IDNP are more uncertain than those from IFBDP. The above results demonstrate that incorporating fuzzy information within IDNP can effectively reduce the uncertain degree of solutions. 5.2. Comparison with IFDNP method If the effect of water price on system benefit was neglected, the study problem can be formulated as an interval-fuzzy De Novo programming (IFDNP) model, where bi-infinite programming technique was not employed for dealing with interval functions. Tables 5 and 6 present the solutions from the IFDNP under scenario 5. For example, the net benefits from IFDNP would be $[3403.07, 4048.16]  106 for industry, $[1991.62, 2361.41]  106 for municipality, and $[1538.73, 1834.10]  106 for agricul7 ture, lower than those from IFBDP. Meanwhile, the optimal lmeta value obtained from IFBDP (i.e., [0.19, 0.30]) is larger than that obtained through IFDNP (i.e., [0.15, 0.21]). It is shown that the interval range between the bounds can be significantly narrowed Table 5 Comparisons of the optimized objective value through the two methods.

Net benefits ($106) Industry Municipality Agriculture 7

meta-satisfactory level (lmeta )

IFBDP

IFDNP

[4387.63, 5462.18] [2551.27, 3203.55] [2181.25, 2559.01] [0.19, 0.30]

[3403.07, 4048.16] [1991.62, 2361.41] [1538.73, 1834.10] [0.15, 0.21]

down by the IFBDP and IFDNP model, and IFDNP has a more narrowed interval range. Since the variation of marketing factors is not addressed in IFDNP, the constraints could be violated under certain water price level. For example, if not considering the fluctuation of water price, the problem only contains a finite number of constraints. Such treatment could be over-optimistic, possibly leading to the increase in the risk of constraints violation. Different from IFDNP, IFBDP method can consider all possible values of the water price within their variation ranges, where any of the values can generate a constraint; thus the number of constraints is infinite. This provides a more flexible means of making decision than the IFDNP method. Therefore, IFBDP could be considered as a more favorite approach than IFDNP in reflecting system dynamic feature.

5.3. Comparisons with IDNP and IFDNP methods Fig. 6 presents the objective function values obtained through the IDNP, IFDNP and IFBDP approaches, which are significantly different from each other. It is shown that IFBDP and IFDNP have a narrower interval than IDNP. The shrunk interval demonstrates that IFBDP and IFDNP can address uncertainties without unrealistic simplifications or information losses, such that the obtained solutions would possess increased certainty. It is also shown that the IFBDP model leads to a higher value than the IFDNP. The raised benefit corresponds to a reduced possibility in satisfying the constraints and aspiration; and the increased system certainty is based on a reduced certainty on the possibility of satisfying the constraints and aspiration. Although further sensitivity analyses can be undertaken to address the imprecise information existing in the water resources systems, each analysis can only represent one of many potential responses to the uncertain inputs. In particular, sensitivity analyses can hardly reflect interactions among various uncertain parameters. Therefore, compared with the IDNP and IFDNP, the IFBDP can incorporate more uncertain information within its modelling framework and resulting solution. It can deal with uncertainties presented in terms of fuzzy sets and interval numbers, and thus avoids over-simplification of fuzzy membership functions into discrete intervals. The obtained interval solutions can be used to generate multiple alternatives and help decision makers identify desired policies under various environmental and system-reliability constraints. Generally, the previous inexact De Novo program approach can only deal with decision-making problems with infinite constraints but a single or multi-objective. The main limitation of the IDNP is its over-simplification of fuzzy membership information into intervals. It is difficult to select the final or ‘‘best’’ decision alternatives from a number of solution options, especially for the problems of designing a new optimal system where the budget is unknown. This leads to the lack of system reliability information 7 as defined by lopt in the obtained solutions. IDNP can only solve

Table 6 Solutions from the IFDNP model. Water user

Industry

Municipality

Agriculture

Water allocation to city 1 (106 m3) Water allocation to city 2 (106 m3) Water treatment limit bj (106 m3) Net benefits (106 dollars)

[43.22, 51.05] [38.90, 44.39] [69.24, 78.07] [3403.07, 4084.16]

[28.11, 31.07] [30.00, 33.73] [41.72, 44.28] [1991.62, 2361.41]

[34.94, 40.88] [40.62, 45.85] [70.03, 78.62] [1538.73, 1834.10]

Water supply design bi (106 m3)

City 1 [86.13, 96.51] [0.73, 0.99]

City 2 [95.31, 103.99]

7 lopt 7 7 lmeta (n r  lopt )

[0.15, 0.21]

1069

1070

D.Y. Miao et al. / Engineering Applications of Artificial Intelligence 26 (2013) 1061–1071

lower bound

8000

upper bound

Net benefit ($106)

7000 6000 5000 4000 3000 2000

IFBDP

ul tu re ric ag

y

IDNP

m un ic ip al

in du str

ul tu re ric ag

y m un ic ip al

in du str

in du str

y m un ic ip al ag ric ul tu re

1000

IFDNP

Fig. 6. Comparison of net benefits obtained through IFBDP, IDNP and IFDNP models.

the problems containing crisp interval coefficients [a, b], whose lower-and upper-bounds (i.e., a and b) are both deterministic and definitely known. This is based on the assumption that these interval coefficients are unchanged even if they could be affected by associated impact factors. In actual systems, this definition is not suitable for all cases where the two bounds may be associated with the external impact factors. As an extension of these efforts, an IFBDP problem is proposed in this study by allowing the parameters in the objective and constraints to be functional intervals. To solve the IFBDP problem, it is step by step converted to two deterministic single-objective programming problems by assigning weights to each objective. Thus the solutions of IFBDP can be easily obtained through conventional mathematical programming approaches. As the actual value of each variable or parameter varies within its two bounds, the system benefit may  þ change correspondingly between f opt and f opt with a variety of reliability levels. In the IFBDP method, the available budgets and costs for wastewater treatment and water supply will be adjusted with the variations in water price. Therefore, the IFBDP approach has advantages over IFDNP approach which takes no account of biinfinite programming as follows: (1) better reflecting the association of the total system benefits with water price; (2) generating more reliable solutions with a lower risk of system failure due to the possible constraints violation; and (3) providing a more flexible management planning since the budgets availability can be adjusted with the variations in water price. Outputs of the IFBDP model can reflect fluctuations in system benefit (or budget) due to implementing different water-management policies; moreover, the IFBDP solutions contain information of system failure risk under varying water-management conditions. While the programming approach is merely applied to a hypothetical water resources management system, useful methodological and technical decision supports could be provided to decision makers. The modeling results indicate that different scenarios can be generated with time due to the variation of human behaviors and economic conditions. The proposed approach could be valid to real-world applications. As the IFBDP problem can be finally divided into two sets of deterministic submodels based on an interactive algorithm, conventional mathematical solvers can be employed to find the optimal solutions.

Therefore, the computational burdens would not be significantly increased when the approach is applied to practical large-scale problems where numerous fuzzy constraints and decision variables need to be handled.

6. Conclusions As there is a variety of impact factors (e.g., water price) associated with many parameters (e.g., wastewater treatment efficiency) in water resources management systems, incorporating conventional interval parameters into the programming frameworks can hardly reflect real-world problems. Instead of the previous concept of crisp intervals, functional intervals are proposed by allowing the lower and upper bounds to be the functions of their impact factors. This leads to a new interval fuzzy bi-infinite De Novo programming (IFBDP) problem, where the numbers of both objective and constraint are infinite. To solve the IFBDP problem, a two-step interactive algorithm is developed based on the previous approaches. For each submodel, the design variables in the IFBDP model are constrained by the total budget. By function transformation, the maximum value of each objective function can be obtained, corresponding to the performance of the ideal design under a given B. A metaoptimum model can be constructed and solved to get optimal design solutions for all scenarios. The IFBDP approach is then applied to a water resources management system to illustrate its performance in dealing with various uncertainties in modeling parameters. It is indicated that the IFBDP solutions can provide reasonable schemes for effectively allocating water from cities to water users, and designing proper wastewater treatment facilities in water purifying industry, municipality and agriculture. The obtained scenarios are more flexible than those through IDNP as tolerance intervals are allowed for all objectives and constraints. Decision makers can develop a scheme in terms of their preferences to the tradeoff between subjective satisfactory level and system reliability level. The IFBDP solutions are also compared to IFDNP solutions which take no account of bi-infinite programming. The satisfactory level and net system benefits through IFBDP are larger than those through IFDNP, but IFDNP has a more narrowed interval range

D.Y. Miao et al. / Engineering Applications of Artificial Intelligence 26 (2013) 1061–1071

than that of IFBDP. As not considering the fluctuation of water price in IFDNP, the problem only contains a finite number of constraints. Such treatment could be over-optimistic, possibly leading to the increase in the risk of constraints violation. While IFBDP problem considers all possible values of the water price within their variation ranges, where any of the values will generate a constraint; thus the number of constraints is infinite. This provides a more flexible means of making decision than the IFDNP method. Although this study is the first attempt for planning water resources system through the developed IFBDP approach, the obtained results suggest that the IFBDP method is applicable and can be extended to other practical problems involving optimal environmental systems design under uncertainty.

Acknowledgement This research was supported by the National Natural Science Foundation for Distinguished Young Scholar (51225904), the Natural Sciences Foundation of China (51190095), and the Program for Innovative Research Team in University (IRT1127). The authors are grateful to the editors and the anonymous reviewers for their insightful comments and suggestions. References Bender, M.J., Simonovic, S.P., 2000. A fuzzy compromise approach to water resources planning under uncertainty. Fuzzy Sets Syst. 115 (1), 35–44. Chang, N.B., Wang, S.F., 1997. A fuzzy goal programming approach for the optimal planning of metropolitan solid waste management systems. Eur. J. Oper. Res. 99 (2), 303–321. Chen, Y.W., Hsieh, H.E., 2006. Fuzzy multi-stage De Novo programming problem. Appl. Math. Comput. 181 (2), 1139–1147. He, L.and Huang, G.H., 2004. An Interval-parameter Semi-infinite Programming Method for Municipal Solid Waste Management. Technical Report, Environmental Informatics Laboratory, University of Regina, Saskachewan, Canada. He, L., Huang, G.H., 2008. Optimization of regional waste management systems based on inexact semi-infinite programming. Can. J. Civ. Eng. 35 (9), 987–998. He, L., Huang, G.H., Lu, H.W., 2009. Flexible interval mixed-integer bi-infinite programming for environmental systems management under uncertainty. J. Environ. Manage. 90, 1802–1813. Huang, G.H., Baetz, B.W., Patry, G.G., 1993. A grey fuzzy linear programming approach for municipal solid waste management planning under uncertainty. Civ. Eng. Environ. Syst. 10, 123–146. Huang, G.H., Moore, R.D., 1993. Grey linear programming, its solving approach, and its application. Int. J. Syst. Sci. 24, 159–172. Huang, G.H., 1996. IPWM: an interval parameter water quality management model. Eng. Optim. 26 (2), 79–103.

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