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Journal of Environmental Management 92 (2011) 1986e1995

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Journal of Environmental Management

journal homepage: www.elsevier .com/locate/ jenvman

Interactive two-stage stochastic fuzzy programming for water resourcesmanagement

S. Wang, G.H. Huang*

Faculty of Engineering and Applied Science, University of Regina, Regina, Saskatchewan S4S 0A2, Canada

a r t i c l e i n f o

Article history:Received 11 August 2010Received in revised form10 February 2011Accepted 14 March 2011Available online 12 April 2011

Keywords:IntervalInteractive fuzzy resolutionPolicy analysisTwo-stageUncertaintyWater resources management

* Corresponding author. Tel.: þ1 306 585 4095; faxE-mail addresses: wangshuo725319@gmail.com

uregina.ca (G.H. Huang).

0301-4797/$ e see front matter � 2011 Elsevier Ltd.doi:10.1016/j.jenvman.2011.03.024

a b s t r a c t

In this study, an interactive two-stage stochastic fuzzy programming (ITSFP) approach has been devel-oped through incorporating an interactive fuzzy resolution (IFR) method within an inexact two-stagestochastic programming (ITSP) framework. ITSFP can not only tackle dual uncertainties presented asfuzzy boundary intervals that exist in the objective function and the left- and right-hand sides ofconstraints, but also permit in-depth analyses of various policy scenarios that are associated withdifferent levels of economic penalties when the promised policy targets are violated. A managementproblem in terms of water resources allocation has been studied to illustrate applicability of theproposed approach. The results indicate that a set of solutions under different feasibility degrees hasbeen generated for planning the water resources allocation. They can help the decision makers (DMs) toconduct in-depth analyses of tradeoffs between economic efficiency and constraint-violation risk, as wellas enable them to identify, in an interactive way, a desired compromise between satisfaction degree ofthe goal and feasibility of the constraints (i.e., risk of constraint violation).

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Over the past decades, the conflict-laden issues of water resourcesallocation among competing municipal, industrial and agriculturalinterests have been of increasing concerns (Huang and Chang, 2003;Wang et al., 2003). The competition among water users has beenintensified due to growing population shifts, shrinking water avail-abilities, varyingnatural conditions, anddeteriorating qualityofwaterresources (Li andHuang, 2008). The increasedwaterdemands and theinadequate water supplies have exacerbated the shortage of waterresources; this has been considered as amajor obstacle to sustainablewater resourcesmanagement.When the essential demand cannot besatisfied due to insufficient resources, losses can hardly be avoided,resulting in a variety of adverse impacts on socio-economic develop-ment (Lu et al., 2010). Therefore, a sound strategy for water resourcesallocation is desired to help reduce such losses. However, the waterresources management systems are complicated with a variety ofuncertainties and their interactions which may intensify the conflict-laden issuesofwaterallocation (Li et al., 2006). These complexities canbecome further compounded not only by interactions among theuncertain parameters but also by additional economic implications.

: þ1 306 585 4855.(S. Wang), gordon.huang@

All rights reserved.

To address the above concerns, a number of optimizationtechniques were developed (Slowinski, 1986; Wu et al., 1997;Huang, 1998; Jairaj and Vedula, 2000; Seifi and Hipel, 2001; Luoet al., 2003; Maqsood et al., 2005; Li et al., 2007; Wang et al.,2010). Among them, two-stage stochastic programming (TSP) iseffective in dealing with problems where an analysis of policyscenarios is desired and uncertainties can be expressed as proba-bilistic distributions. In TSP, an initial decision must be made beforethe realization of random variables (first-stage decision), and thena corrective action can be taken after random events have takenplace (second-stage decision). This implies that a second-stagedecision can be used to minimize “penalties” that may appear dueto any infeasibility (Birge and Louveaux, 1988, 1997). TSP waswidely studied over the past decades (Wang and Adams, 1986;Birge and Louveaux, 1988; Eiger and Shamir, 1991; Ruszczynskiand Swietanowski, 1997; Huang and Loucks, 2000; Maqsoodet al., 2005). For example, Wang and Adams (1986) proposeda two-stage optimization framework for planning reservoir oper-ations, where hydrologic uncertainty and seasonality of reservoirinflows were modeled as periodic Markov processes. Eiger andShamir (1991) developed a model for an optimal multi-periodoperation within a multi-reservoir system. Ferrero et al. (1998)examined a long-term hydrothermal scheduling of multi-reser-voir systems using a two-stage dynamic programming approach. Ingeneral, TSP can deal with uncertainties expressed as probabilisticdistributions and account for economic penalties with recourse

S. Wang, G.H. Huang / Journal of Environmental Management 92 (2011) 1986e1995 1987

against any infeasibility. In many real-world problems, however,the quality and quantity of uncertain information is often notsatisfactory enough to be presented as probabilistic distribution.Even if such distributions for uncertain parameters are available,reflecting them in large-scale optimization models can beextremely challenging (Huang and Loucks, 2000).

In comparison, interval mathematical programming (IMP) iseffective in tacklinguncertainties expressed as intervalswithknownlower and upper bounds but unknown distribution functions(Huang et al., 1992). IMP proves to be an effective approach to dealwithuncertainties in the left- and right-hand sides of the constraintsas well as in the objective function. Thus, one potential approach forbetter reflecting uncertainties is to incorporate IMP within a TSPframework. Huang and Loucks (2000) proposed an inexact two-stage stochastic programming (ITSP) approach for water resourcesmanagement. ITSP could not only tackle uncertainties expressed asprobabilistic distributions and intervals, but also analyze a variety ofpolicy scenarios that are associatedwithdifferent levels of economicpenalties when the promised policy targets are violated. However,a remarkable limitation of ITSP was its incapability in handlinguncertainties presented as possibilistic distributions (i.e., fuzzy sets)or reflecting the risk of violating the constraints. In real-worldproblems, uncertainties may be estimated as intervals; at the sametime, the lower and upper bounds of many intervals can rarely beacquired as deterministic values. Instead, theymayoften be given assubjective information that can be only expressed as fuzzy sets; thisleads to dual uncertainties. Thus, ITSP is incapable of addressingsuch a complexity. Moreover, ITSP lacks the information about therisk of violating the constraints, which is desired by decisionmakers(DMs) in order to conduct in-depth analyses of tradeoffs betweeneconomic efficiency and system risk. Recently, Jiménez et al. (2007)proposed an interactive fuzzy resolution (IFR) method for solvinglinear programming problems with fuzzy parameters. IFR permitsinteractive participation of DMs in all steps of decision process,expressing their preferences in linguistic terms. A set of solutionscould be obtained from IFR under different feasibility degrees of theconstraints (i.e., risks of constraint violation), which aremeaningfulto support in-depth analyses of tradeoffs between economic effi-ciency and constraint-violation risk. IFR enables DMs to identifya compromised solution between two key factors in conflict: feasi-bility of the constraints and satisfaction degree of the goal.

However, IFR can merely reflect uncertainties expressed as fuzzysets, and can hardly handle uncertainties presented in other formats.Onepotential approach for tackling this is to integrate IFR into an ITSPframework; this would lead to an interactive two-stage stochasticfuzzy programming (ITSFP) approach. Therefore, the objective of thisstudy is to develop such an ITSFP approach for water resourcesmanagement. ITSFP can not only tackle dual uncertainties presentedas fuzzy boundary intervals that exist in the objective function andthe left- and right-hand sides of constraints, but also permit in-depthanalyses of various policy scenarios that are associatedwith differentlevels of economic penalties when the promised policy targets areviolated. A management problem in terms of water resources allo-cation will be studied. The results can not only help DMs to conductin-depth analyses of tradeoffs between economic efficiency andconstraint-violation risk, but also enable them to identify, in aninteractive way, a desired compromise between satisfaction degreeof the goal and feasibility of the constraints.

2. Model development

2.1. Inexact two-stage stochastic programming

Consider a problem in which a water manager is responsible forallocating water to multiple users. The objective of the water

resources allocation problem is to maximize the system benefit.Based on local water management policies, a prescribed quantity ofwater is promised to each user. If the promised water is delivered, itwill result in net benefits to the local economy; otherwise, userswill have to either obtain water from more expensive sources orcurtail their development plans, resulting in economic penalties. Insuch a problem, the water flow levels are uncertain (expressed asrandomvariables), while a decision of water-allocation target (first-stage decision) must be made before the realization of randomvariables, and then a recourse action can be taken after thedisclosure of random variables (second-stage decision). Therefore,this problem under consideration can be formulated as a two-stagestochastic programming (TSP) model as follows:

Maximize f ¼Xmi¼1

NBiTi � E

"Xmi¼1

CiSiQ

#(1a)

subject to:

Xmi¼1

�Ti � SiQ

�ð1þ dÞ � Q (1b)

(Water availability constraints)

SiQ � Ti � Timax; ci (1c)

(Water-allocation target constraints)

SiQ � 0; ci (1d)

(Non-negativity and technical constraints)where f¼ system benefit ($); NBi¼ net benefit to user i per m3 ofwater allocated ($/m3) (first-stage revenue parameters);Ti¼ allocation target for water that is promised to user i (m3) (first-stage decision variables); E[$]¼ expected value of a random vari-able; Ci¼ loss to user i perm3 of water not delivered, Ci>NBi ($/m3)(second-stage cost parameters); SiQ¼ shortage of water to user iwhen the seasonal flow is Q (m3) (second-stage decision variables);Q¼ total amount of seasonal flow (m3) (random variables); d¼ rateof water loss during transportation; Timax¼maximum allowableallocation amount for user i (m3);m¼ total number of water users;i¼water user, i¼ 1, 2, 3, where i¼ 1 for the municipality, i¼ 2 forthe industrial user, and i¼ 3 for the agricultural sector.

To solve the above problem through the linear programmingmethod, the distribution of Q must be approximated by a set ofdiscrete values. Letting Q take values qjwith probabilities pj (j¼ 1, 2,., n), we have (Huang and Loucks, 2000):

E

"Xmi¼1

CiSiQ

#¼Xmi¼1

Ci

0@Xnj¼1

pjSij

1A: (2)

Thus, model (1) can be reformulated as follows:

Maximize f ¼Xmi¼1

NBiTi �Xmi¼1

Xnj¼1

pjCiSij (3a)

subject to:

Xmi¼1

�Ti � Sij

�ð1þ dÞ � qj; cj; (3b)

Sij � Ti � Timax; ci; j; (3c)

Sij � 0; ci; j; (3d)

where Sij denotes the amount by which the water-allocation target(Ti) is not met when the seasonal flow is qj with probability pj.

S. Wang, G.H. Huang / Journal of Environmental Management 92 (2011) 1986e19951988

Obviously, model (3) can reflect uncertainties in water availability(qj) expressed as probabilistic distributions. However, uncertaintiesin other parameters such as benefits (NBi), penalties (Ci) and water-allocation targets (Ti) also need to be addressed. For example, it mayoften be difficult for a planner to promise a deterministic water-allocation target (Ti) to a water user when available water flows areuncertain. However, the quality and quantity of information thatcan be obtained for these parameters are often not satisfactoryenough to be presented as probabilistic distributions. Even if suchdistributions are available, reflecting them in large-scale optimi-zation models can be extremely challenging (Huang and Loucks,2000). In response to the above concerns, interval parameters areintroduced into the TSP framework to communicate uncertaintiesin NBi, Ci and Ti into the optimization process. This leads to aninexact two-stage stochastic programming (ITSP) model as follows:

Maximize f� ¼Xmi¼1

NB�i T�i �

Xmi¼1

Xnj¼1

pjC�i S

�ij (4a)

subject to:

Xmi¼1

�T�i � S�ij

��1þ d�

�� q�j ; cj; (4b)

S�ij � T�i � T�imax; ci; (4c)

S�ij � 0; ci; j; (4d)

where NBi�, Ti�, Ci�, Sij�, d�, qj� and T�imax are interval parameters/

variables. An interval is defined as a number with known upper andlower bounds but unknown distribution information (Huang et al.,1992). For example, letting Ti

� and Tiþ be lower and upper bounds

of Ti�, respectively, we have Ti�¼ [Ti�, Tiþ]. When Ti

�¼ Tiþ, then Ti

�

becomes a deterministic number.

2.2. Interactive fuzzy resolution method

In many real-world problems, when the subjective judgment ofDMs is influential in the decision-making process, the fuzzy settheory is recognized as an effective tool to express their opinions,while ITSP can hardly reflect the ambiguous or vague informationfrom subjective estimations. For example, DMs may estimate thatthe most possible value for the net benefit to awater user is $90 perm3 of water allocated, and there is no possibility for it to be lowerthan $80 or more than $100 per m3 of water; such a subjectiveestimation can be expressed as a fuzzy set. Therefore, an interactivefuzzy resolution (IFR) method is introduced to tackle uncertaintiespresented as fuzzy sets. Firstly, consider the following linearprogramming model with fuzzy parameters:

Maximize f ¼Xnj¼1

~CjXj (5a)

subject to:

Xnj¼1

~AijXj � ~Bi; i ¼ 1;2;.;m; (5b)

Xj � 0; j ¼ 1;2;.;n; (5c)

where ~Cj˛fRg1�n; ~Aij˛fRgm�n; ~Bi˛fRgm�1; and fRg denotesa set of fuzzy parameters involved in the objective function andconstraints; Xj˛ {R}n�1, and fRg denotes a set of crisp decisionvariables.

A fuzzy set ð~AÞ in X is characterized by a membership functionðm~A

Þ, where X represents a space of points (objects), with anelement of X denoted by x (Zadeh, 1965); m~A

ðxÞ represents themembership grade of x in ~A : m~A

ðxÞ/½0;1�, where m~AðxÞ can be

viewed as the plausibility degree of ~A taking value x. Zadeh (1978)defined the possibility associated with ~A as m~A

.A convex fuzzy set ð~AÞ is a set defined on the real line (R) with

a continuous membership function ðm~AÞ that can be described as

follows (Heilpern, 1992; Jiménez et al., 2007):

m~AðxÞ ¼

8>>>>><>>>>>:

0 for x � a1;f ~AðxÞ for a1 � x � a2;1 for a2 � x � a3;g~AðxÞ for a3 � x � a4;0 for a4 � x;

(6)

where functions f ~A and g~A represent a continuous and mono-tonically increasing function on the left-hand side of ~A anda continuous and monotonically decreasing function on the right-hand side of ~A, respectively. If f ~A and g~A are linear functions,a trapezoidal fuzzy number can be denoted by ~A ¼ ða1; a2; a3; a4Þ. Ifa2¼ a3, the fuzzy set becomes triangular. The a-level set of a fuzzyset ð~AÞ can be defined as follows (Fang et al., 1999):

~Aa ¼nx˛Rjm~A

� ao;

where a˛ [0,1]. Since m~Ais upper semi-continuous, the a-level set

of ~A forms a closed and bound interval ~Aa ¼ ½f ~A�1ðaÞ; g~A

�1ðaÞ�(Heilpern, 1992), where

f ~A�1ðaÞ ¼ inf

nx : m~A

ðxÞ � ao; (7a)

g~A�1ðaÞ ¼ sup

nx : m~A

ðxÞ � ao: (7b)

The expected interval of a fuzzy set ð~AÞ, denoted by EIð~AÞ, can bedefined as follows (Jiménez et al., 2007):

EI�~A�

¼�E1

~A; E2~A�¼24Z 1

0f ~A�1ðaÞda;

Z 1

0g~A�1ðaÞda

35: (8)

The expected value of a fuzzy set ð~AÞ, denoted by EVð~AÞ, is thehalf point of its expected interval (Heilpern, 1992):

EV�~A�

¼ E1~A þ E2

~A

2: (9)

According to formulae (8) and (9), if a fuzzy set ð~AÞ is trapezoidalor triangular, its expected interval and expected value can becalculated as follows:

EI�~A�

¼�12ða1 þ a2Þ ;

12ða3 þ a4Þ

�;

EV�~A�

¼�14ða1 þ a2 þ a3 þ a4Þ

�: (10)

The uncertain and/or imprecise nature ofmany parameters leadsto concerns of system feasibility and optimality. It is thereforenecessary to answer the following two questions (Jiménez et al.,2007):

(1) How to define the feasibility of a decision vector, when theconstraints involve fuzzy sets?

(2) How to define the optimality for an objective function withfuzzy coefficients?

S. Wang, G.H. Huang / Journal of Environmental Management 92 (2011) 1986e1995 1989

Several focuses have been developed to answer these questions(Lai and Hwang, 1994; Rommelfanger and Slowinski, 1998). Avariety of methods for comparing or ranking fuzzy sets have beenreported (Wang and Kerre, 1996). Jiménez et al. (2007) conducteda fuzzy relationship analysis to compare fuzzy sets; such a methodwas computationally efficient.

Following Jiménez et al. (2007), for any pair of fuzzy sets ~A and ~B,the degree in which ~A is smaller than ~B can be defined as follows:

mM

�~A;~B�¼

8>>>>><>>>>>:0 if E2

~B�E1~A<0;

E2~B�E1

~A

E2~B�E1

~A��E1

~B�E2~A if 0˛

�E1

~B�E2~A;E2

~B�E1~A�;

1 if E1~B�E2

~A>0:

(11)

where ½E1~A;E2

~A� and ½E1~B;E2

~B� are the expected intervals of ~A and ~B.When mMð~A;~BÞ � a, where 0� a� 1, it implies that ~A is smaller thanor equal to ~B at least in a degree a, and can be represented by ~A�a

~B.For thefirst question, according to Jiménez et al. (2007), a decision

vector (X˛ Rn) will be feasible in degree a (or a-feasible) if

mini¼1;.;m

nmM

�~AiX; ~Bi

�o¼ a; (12)

where ~Ai ¼ ð~Ai1;~Ai2;.; ~AinÞ; a is the feasibility of a decision vector,

and 1� a provides a measure of the risk of infeasibility. Accordingto formula (11), this is equivalent to:�ð1� aÞE~Ai

1 þ aE~Ai2

�X � aE

~Bi1 þ ð1� aÞE~Bi

2 : (13)

The set of decision vectors that are a-feasible can be denoted byaðaÞ. It is evident that:

a1 < a20aða1ÞIaða2Þ: (14)

To address the second question, vector X0˛ Rn is an acceptablesolution if it can be verified that (Jiménez et al., 2007):

mM

�~CX; ~CX0

�� 1

2; (15)

where ~C ¼ ð~C1;~C2;.; ~CnÞ; X0 is a better choice (with a maximized

objective) at least in degree 1/2 as opposed to the other feasibledecision vectors. According to formula (11), it is equivalent to:

E~CX0

2 þ E~CX0

12

� E~CX2 þ E

~CX1

2: (16)

Based on formulae (13) and (16), vector X0(a)˛ Rn is an a-acceptable solution of initial fuzzy model (5) if it is an optimalsolution to the following a-parametric linear programmingproblem (Jiménez et al., 2007):

Maximize f ¼Xnj¼1

EV�~Cj

�Xj (17a)

subject to:Xnj¼1

�ð1� aÞE

~Aij

1 þ aE~Aij

2

�Xj � aE

~Bi1 þ ð1� aÞE~Bi

2 ;

i ¼ 1;2;.;m; a˛½0;1�;(17b)

Xj � 0; j ¼ 1;2;.;n; (17c)

where EVð~CÞ represents the expected value of the fuzzy vector ð~CÞ.Jiménezet al. (2007)proposedan interactiveprocedure to solve the

a-parametric linear programming problem. The interactive fuzzy

resolution (IFR) method allows DMs to consider two factors in aninteractiveway: feasibility of the constraints and satisfaction degree ofthegoal. A strongdesire to obtain abetterobjective-functionvaluewilllead to a lowerdegree of constraint feasibility; contrarily, awillingnessto accept a lower satisfactiondegree of the goalwill guarantee ahigherfeasibility of the constraints. Therefore, the DMs have to identifya balanced solution between the two objectives in conflict: to improvethe objective-function value or to improve the constraint feasibility.

The best way to reflect the DMs’ preferences is to use naturallanguage, establishing a semantic correspondence for differentdegrees of feasibility (Zadeh, 1975); therefore, Jiménez et al. (2007)established 11 scales, which allowed a sufficient distinction betweendifferent levels without being excessive. The 11 scales are as follows:

0.0 e Unacceptable solution0.1 e Practically unacceptable solution0.2 e Almost unacceptable solution0.3 e Very unacceptable solution0.4 e Quite unacceptable solution0.5 e Neither acceptable nor unacceptable solution0.6 e Quite acceptable solution0.7 e Very acceptable solution0.8 e Almost acceptable solution0.9 e Practically acceptable solution1.0 e Completely acceptable solution

Obviously, other scales can be used depending on the wishes ofDMs. The 11 scales correspond to different risk levels of violating themodeling constraints. The higher the scale value, the lower the risklevel. In real-world problems, the DMs may not be willing to admita high risk of constraint violation. If a0 is the minimum constraintfeasibility that theDMsarewilling to accept, the feasibility interval ofa will be reduced to a0� a� 1. According to the semantic scale, thediscrete values of a can be achieved as follows (Jiménez et al., 2007):

M ¼ak ¼ a0 þ 0:1kjk ¼ 0;1;.;

1� a00:1

�3½0;1�: (18)

In thefirst stepof the IFRmethod, the spaceO¼ {X0(ak),ak˛M} ofthe ak-acceptable solutions and the corresponding possibilisticdistributions of the objective-function values ~f

0ðakÞ ¼ ~CX0ðakÞ canbe obtained through solving model (17) under each ak. In order toobtain a decision vector that complies with the expectations of theDMs, two conflicting factors should be evaluated: feasibility of theconstraints and acceptability of the objective-function value. Afterobtaining the results under different ~f

0ðakÞ, the DMs are asked tospecify a goal ðGÞ and its tolerance threshold ðGÞ. The goal isexpressedbymeansof a fuzzy set ð~GÞwhosemembership function is:

m~GðzÞ ¼

1 if z�G;

l˛½0;1�; increasing onG� z�G;0 if z�G:

(19)

Fig. 1 shows the possibilistic distributions of the objective-function values under different feasibility degrees and the fuzzygoal provided by the DMs.

In the second step of the IFR method, the satisfaction degree ofthe fuzzy goal ð~GÞ for each a-acceptable solution (i.e., membership

grade of ~f0ðakÞ to ~G) can be analyzed through an index proposed by

Yager (1979):

K ~G

�~f0ðaÞ

¼

ZSm~f

0ðaÞ

ðzÞm~GðzÞ dzZ

Sm~f

0ðaÞ

ðzÞ dz; (20)

Fig. 1. Possibilistic distributions of the objective-function values and the fuzzy goalprovided by the decision makers.

S. Wang, G.H. Huang / Journal of Environmental Management 92 (2011) 1986e19951990

whereRdenotes the classical integral; S represents the support set

of the fuzzy set ð~f 0ðakÞÞ. The denominator is the area under themembership function ðm~f 0ðaÞÞ; in the numerator, the possibility of

occurrence ðm~f 0ðaÞðzÞÞ for each z is weighted by its satisfaction

degree ðm~GðzÞÞ of the goal ð~GÞ (see Fig. 1). This is an extension of the

widely accepted center of gravity defuzzification method, using thegoal function ðm~G

Þ as a weighting factor (Jiménez et al., 2007).

In the third step of the IFR method, in an attempt to identifya balance between feasibility of the constraints and satisfactiondegree of the goal in the space of ak-acceptable solutions (O),Jiménez et al. (2007) proposed to build two fuzzy sets: ~F and ~Swiththe following membership functions: m~F

ðX0ðakÞÞ ¼ ak and

m~SðX0ðakÞÞ ¼ K ~G

ð~f 0ðakÞÞ, respectively. To obtain a final recom-

mendation, a fuzzy decision is defined by aggregating the twoaforementioned fuzzy sets (i.e., ~D ¼ ~FX~S) (Bellman and Zadeh,1970):

m~D

�X0ðakÞ

�¼ ak*K ~G

�~f0ðakÞ

; (21)

where * denotes a minimum t-normwhich is used to construct theintersection of two fuzzy sets. The relation among ~F, ~S and ~D isdepicted in Fig. 2. The solution with the highest membership gradewill be the final decision for the fuzzy linear programming problem(i.e., model (5)):

m~D

�X*�

¼ maxak˛M

ak*K ~G

�~f0ðakÞ

�: (22)

2.3. Interactive two-stage stochastic fuzzy programming

Although the IFR method is capable of solving linear program-ming problems with fuzzy parameters, it becomes ineffective inreflecting uncertainties that exist in decision variables. Likewise,ITSP is able to handle uncertainties expressed as intervals and

Fig. 2. Relation between the fuzzy constraint ð~FÞ, the fuzzy goal ð~SÞ and the fuzzydecision ð~DÞ.

probabilistic distributions, but it encounters difficulties in reflectingsubjective information in decision making. In real-world problems,however, the lower and upper bounds of many interval parametersmay often be given as subjective information that can only beexpressed as fuzzy sets; this leads to dual uncertainties (i.e., fuzzyboundary intervals). As shown in Fig. 3, ~a

�and ~a

þare lower and

upper bounds of an interval ð~a�Þ, and can be quantified as fuzzy setswith triangular membership functions. Assume that there is nointersection between the fuzzy sets of the two bounds. In fact,intervals are used to express uncertainties without distributioninformation. If the fuzzy sets of an interval’s lower andupper boundsintersect, then the so-called “interval” is actually described by fuzzymembership functions, such that the interval representationbecomes unnecessary (Nie et al., 2007). To tackle dual uncertaintiesexpressed as fuzzy boundary intervals, the IFR method is integratedinto the ITSP framework. This leads to an interactive two-stagestochastic fuzzy programming (ITSFP) problem as follows:

Maximize ~f� ¼

Xmi¼1

NeB�i T�i �Xmi¼1

Xnj¼1

pj~C�i S

�ij (23a)

subject to:

Xmi¼1

�T�i � S�ij

� 1þ ð1� aÞE~d

�

1 þ aE~d�

2

!

� aE~q�j

1 þ ð1� aÞE~q�j

2 ; cj; (23b)

S�ij � T�i � T�

imax; ci; (23c)

S�ij � 0; ci; j; (23d)

where T�i , S�ij and T�imax are interval parameters/variables; NeB�i , ~C�

i ,~d�, ~q

�j are fuzzy boundary intervals; a is the feasibility degree of

a decision vector, and 1� a provides a measure about the risk ofinfeasibility for a decision vector.

According to Huang and Loucks (2000), if Ti� are considered asuncertain inputs, the existing methods for solving inexact linearprogramming problems cannot be used directly. Therefore, anoptimized set of target values (Ti�) will be identified by having yibeing decision variables. The optimized set corresponds to a maxi-mized system benefit under uncertain water-allocation targets.Accordingly, let Ti�¼ Ti

�þDTiyi, where DTi¼ Tiþ� Ti

� and yi˛ [0,1],and yi are decision variables that are used for identifying an opti-mized set of target values (Ti�) in order to support the related policyanalyses. Thus, by introducing decision variables (yi), model (23)can be reformulated to:

Maximize ~f� ¼

Xmi¼1

NeB�i �T�i þDTiyi

��Xmi¼1

Xnj¼1

pj~C�i S

�ij (24a)

Fig. 3. An interval with fuzzy boundaries.

S. Wang, G.H. Huang / Journal of Environmental Management 92 (2011) 1986e1995 1991

subject to:

Xm �T�i þ DTiyi � S�ij

� 1þ ð1� aÞE~d

�

1 þ aE~d�

2

!

i¼1

� aE~q�j

1 þ ð1� aÞE~q�j

2 ; cj; (24b)

S�ij � T�i þ DTiyi � T�imax; ci; (24c)

S�ij � 0; ci; j: (24d)

When Ti� are known, model (24) can be transformed into two

deterministic submodels, which correspond to the lower and upperbounds of the objective-function value. This transformation processis based on an interactive algorithm, which is different from thebest/worst case analysis (Huang, 1996). The resulting solutionprovides intervals for the objective function and decision variables,which can be easily interpreted for generating decision alterna-tives. Since the objective is to maximize the system benefit, thesubmodel corresponding to the upper bound of the objective-function value ð~fþÞ can be first formulated as follows:

Maximize ~fþ ¼

Xmi¼1

NeBþi �T�i þDTiyi��Xmi¼1

Xnj¼1

pj~C�i S

�ij (25a)

subject to:

Xmi¼1

�T�i þ DTiyi � S�ij

� 1þ ð1� aÞE~d

�

1 þ aE~d�

2

!

� aE~qþj

1 þ ð1� aÞE~qþj

2 ; cj; (25b)

S�ij � T�i þ DTiyi � Tþimax; ci; (25c)

S�ij � 0; ci; j; (25d)

where Sij� and yi are decision variables; Let Sij

� and yi are thesolutions of model (25). Then, the optimized water-allocationtargets can be determined by calculating Ti

�¼ Ti�þDTiyi. Accord-

ing to Huang (1996), the submodel corresponding to the lowerbound of the objective-function value ð~f�Þ can be formulated asfollows:

Maximize ~f� ¼

Xmi¼1

NeB�i �T�i þDTiyi��Xmi¼1

Xnj¼1

pj~Cþi S

þij (26a)

subject to:

Xmi¼1

�T�i þ DTiyi � Sþij

� 1þ ð1� aÞE~d

þ

1 þ aE~dþ

2

!

� aE~q�j

1 þ ð1� aÞE~q�j

2 ; cj; (26b)

Sþij � T�i þ DTiyi � T�imax; ci; (26c)

Sþij � S�ijopt; ci; j; (26d)

where Sijþ are decision variables. Submodels (25) and (26) are

deterministic linear programming problems, the solutions of model(24) under the optimized water-allocation targets are:

S�ijopt ¼hS�ijopt; Sþijopt

i; ci; j; (27a)

~f�opt ¼

�~f�opt;

~fþopt

�; (27b)

where S�ijopt and~fþopt are the solutions of submodel (25), and Sþijopt

and ~f�opt are those of submodel (26). Thus, the optimized water-

allocation scheme is:

A�ijopt ¼ T�iopt � S�ijopt; ci; j: (28)

The detailed solution process can be summarized as follows:

Step 1: Formulate the ITSFP model.Step 2: Reformulate the ITSFP model by introducing T�

i ¼ T�i

þDTiyi, where DTi ¼ Tþi � T�i and yi˛½0;1�.

Step 3: Transform the ITSFP model into two submodels, wherethe submodel corresponding to ~f

þis first desired since the

objective is to maximize ~f�.

Step 4: Obtain the ak-acceptable solutions through solving the ~fþ

submodel under each ak.Step 5: Compute the satisfaction degree of the fuzzy goal foreach a-acceptable solution.Step 6: Identify a balance (i.e., membership grade of the fuzzydecision) between feasibility of the constraints and satisfactiondegree of the goal for each a-acceptable solution.Step 7: Formulate and solve the ~f

�submodel by following the

same interactive procedure as that in ~fþsubmodel.

Step 8: Reach a desired compromise between feasibility of theconstraints and satisfaction degree of the goal by consideringthe solutions from two submodels.Step 9: Obtain the ak-feasibility optimal solution:

~f�opt ¼

�~f�opt;

~fþopt

�; S�ijopt ¼

hS�ijopt; S

þijopt

i; ci; j:

Step 10: Obtain the optimal water-allocation scheme:

A�ijopt ¼ T�iopt � S�ijopt; ci; j:

Step 11: Stop.

3. Case study

3.1. Overview of the study system

The following water resources management problem will beused to demonstrate applicability of the developed ITSFP approach.A water manager is responsible for allocating water from anunregulated reservoir to three users: a municipality, an industryconcern and an agricultural sector. These water users need to knowhow much water they can expect so as to make appropriate deci-sions on their various activities and investments. If the promisedwater is delivered, a net benefit to the local economy will begenerated for each unit of water allocated; otherwise, either thewater must be obtained from higher-priced alternatives or thedemandmust be curtailed by reduced industrial and/or agriculturalproductions, resulting in a reduced system benefit (Huang andLoucks, 2000). Table 1 shows the maximum allowable water allo-cation, the water-allocation target, the net benefit to user i per m3

of water allocated and the loss to user i per m3 of water notdelivered. Moreover, Table 1 presents the fuzzy boundary intervalsfor the rate of water loss during transportation and the seasonalflows under different probability levels.

The problem under consideration is how to effectively allocatethe limited water supply to multiple users in order to achieve

Table 1Related economic data ($/m3) and seasonal flows (in 106 m3) under different probability levels.

Activity User

Municipal (i¼ 1) Industrial (i¼ 2) Agricultural (i¼ 3)

Maximum allowable allocation ðT�imaxÞ 7.0 7.0 7.0Water-allocation target (Ti�) [1.5, 2.5] [2.0, 4.0] [3.5, 6.5]Net benefit when water demand is satisfied ðNeB�i Þ [(85, 90, 95), (105, 110, 115)] [(40, 45, 50), (60, 65, 75)] [(27, 28, 29), (31, 32, 33)]Reduction of net benefit when demand is not delivered ð~C�

i Þ [(215, 220, 225), (275, 280, 285)] [(55, 60, 65), (85, 90, 95)] [(45, 50, 55), (65, 70, 75)]

Flow level Probability (%) Seasonal flow ð~q�j ÞLow (j¼ 1) 0.2 [(3.3, 3.5, 3.7), (4.3, 4.5, 4.7)]Medium (j¼ 2) 0.6 [(7.0, 8.0, 9.0), (11.0, 12.0, 13.0)]High (j¼ 3) 0.2 [(14.0, 15.0, 16.0), (18.0, 19.0, 20.0)]

Water loss ð~d�Þ [(0.10, 0.15, 0.20), (0.30, 0.35, 0.40)]

S. Wang, G.H. Huang / Journal of Environmental Management 92 (2011) 1986e19951992

a maximum system benefit while incorporating water policies witha minimum risk of system disruption. In the study system, the first-stage decision variables represent the allocation target for waterthat is promised to user i (denoted as Ti

�); the second-stage deci-sion variables represent the shortage of water to user i under flowlevel j (denoted as Sij

�) after the probabilistic seasonal flows areknown. The objective is to maximize the system benefit througheffectively allocating water flows to three users; the constraintsinvolve all relationships among decision variables and water-allo-cation conditions. Since dual uncertainties exist in a variety ofsystem components, and a linkage between pre-regulated policiesand associated corrective actions against any infeasibility is desired,ITSFP is considered to be a feasible approach for tackling this water-allocation problem.

3.2. Result analysis

Model (23) can be solved through the solution method asdescribed in Section 2. Table 2 shows the solutions obtained from theITSFP model under a set of feasibility degrees (a). In this study,the level of 0.4 was the minimum constraint feasibility degree thatthe DMs were willing to accept (suppose that the DMs would notadmit high risks of constraint violation); therefore, the discretelevels of a were 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 and 1. The solutions undereach level of a indicate that most of non-zero decision variables (Sij

�)

are intervals and the objective-function value ð~f�Þ is expressed asa fuzzy boundary interval. These interval solutions reflect potentialsystem-condition variations caused by uncertain inputs of NeB�i , ~C�

i ,~d�and qj

�as well as complexities of their interactions. As shown in

Table 2, the minimum and maximum system benefits would be$[(217.51, 235.54, 253.57), (498.92, 523.20, 567.48)]� 106 undera¼ 1 and $[(248.92, 269.20, 289.48), (517.96, 544.32, 590.68)]� 106

under a¼ 0.4. The results reveal that the system benefits graduallydecrease when a increases (higher constraint feasibility is achieved

Table 2The a-acceptable solutions obtained from the ITSFP model (in 106 m3).

Feasibilitydegree, a

Shortage, S�ij ðaÞS�11 S�21 S�31 S�12 S�22 S�32

0.4 0 [3.14, 3.43] 3.50 0 0 [1.00, 2.93]0.5 0 [3.17, 3.46] 3.50 0 0 [1.11, 3.04]0.6 0 [3.19, 3.49] 3.50 0 0 [1.22, 3.16]0.7 0 [3.22, 3.52] 3.50 0 0 [1.32, 3.28]0.8 0 [3.25, 3.55] 3.50 0 0 [1.43,3.39]0.9 0 [3.27, 3.58] 3.50 0 [0, 0.004] [1.53, 3.50]1 0 [3.30, 3.61] 3.50 0 [0, 0.12] [1.64, 3.50]

Note: The level of 0.4 is the minimum constraint feasibility degree which the DMs can adiscrete levels of a are: M¼ {0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1}.

with a cost of reduced system benefit), and thus reflect the tradeoffbetween system benefit and constraint-violation risk. In real-worldproblems, a lower constraint feasibility would correspond toa higher objective value; contrarily, a lower satisfaction degree ofthe objectivewould guarantee a higher feasibility of the constraints.Therefore, the DMs had to identify a balance between feasibility ofthe constraints and satisfaction degree of the goal. After obtainingthe system benefits (expressed as fuzzy boundary intervals) underdifferent levels of a, the DMs would be asked to specify a goal ðGÞand its tolerance threshold ðGÞ. The goals of lower- and upper-bound system benefits would then be, respectively, expressed bymeans of a fuzzy set ð~GÞ whose membership function is shown asfollows:

m~GðzÞ ¼

8<:1 if z � 289;z� 218

289� 218if 218 � z � 289;

0 if z � 218:

(29a)

m~GðzÞ ¼

8<:1 if z � 591;z� 499

591� 499if 499 � z � 591;

0 if z � 499:

(29b)

Table 3 presents the satisfaction degree ðK ~Gðf�ðaÞÞÞ of the fuzzy

goal ð~GÞ under each a-acceptable system benefit ð~f�ðaÞÞ. In anattempt to identify a balance between constraint feasibility andsatisfaction degree of the goal, a fuzzy decision ð~DÞwas employed toaggregate feasibility and satisfaction degrees under each a-acceptable solution. In order to obtain a recommendation for a finaldecision, the mean ðm~D

ðS�ij ðaÞÞÞ and deviation ðbm ~DðS�ij ðaÞÞÞ of the

membership grade for the fuzzy decision were calculated under

Possibilistic distribution of system benefit, ~f�ðaÞ ($106)

S�13 S�23 S�330 0 0 [(248.92, 269.20, 289.48), (517.96, 544.32, 590.68)]0 0 0 [(244.12, 264.04, 283.96), (514.66, 540.66, 586.66)]0 0 0 [(238.93, 258.46, 277.99), (511. 47, 537.12, 582.77)0 0 0 [(233.74, 252.88, 272.02), (508.44, 533.76, 579.08)]0 0 0 [(228.94, 247.72, 266.50), (505.14, 530.10, 575.06)]0 0 0 [(223.94, 242.34, 260.75), (502.22, 526.86, 571.50)]0 0 0 [(217.51, 235.54, 253.57), (498.92, 523.20, 567.48)]

ccept (suppose that the DMs will not admit high risks of constraint violation); the

Table 3Membership grades of the fuzzy decision ð~DÞ in the a-acceptable solutions.

Feasibilitydegree, a

Satisfaction degree of fuzzy goal,K ~G

ðf�ðaÞÞMembership grade of fuzzydecision, m~D

ðS�ij ðaÞÞMean of membership grade forfuzzy decision, m~D

ðS�ij ðaÞÞDeviation of membership gradefor fuzzy decision, bm ~D

ðS�ij ðaÞÞLower-boundsolution

Upper-boundsolution

Lower-boundsolution

Upper-boundsolution

0.4 0.72 0.57 0.288 0.228 0.258 0.0300.5 0.65 0.53 0.325 0.265 0.295 0.0300.6 0.57 0.49 0.342 0.294 0.318 0.0240.7 0.49 0.45 0.343 0.315 0.329 0.0140.8 0.42 0.41 0.336 0.328 0.332 0.0040.9 0.34 0.38 0.306 0.342 0.324 0.0181 0.25 0.34 0.250 0.340 0.295 0.045

S. Wang, G.H. Huang / Journal of Environmental Management 92 (2011) 1986e1995 1993

each a-acceptable solution. The optimal solution would be the onewith the highest mean of membership grades in the fuzzy decision.As shown in Fig. 4, the 0.8-feasibility optimal solution with thehighest mean of 0.332 and the lowest deviation of 0.004 would bethe best choice.

Table 4 presents the 0.8-feasibility solutions obtained from theITSFP model. The optimized water-allocation targets for threewater users could be obtained by letting Tiopt

� ¼ Ti�þDTiyiopt. The

results of y1opt¼ 1, y2opt¼ 1 and y3opt¼ 0 indicate that the opti-mized water-allocation targets would be 2.5�106 m3 for municipaluse, 4.0�106 m3 for industrial one and 3.5�106 m3 for agriculturalone, respectively. The optimized water-allocation targets couldhelp the DMs maximize the system benefit under uncertainty.Deficits would occur if available water is insufficient to satisfy thepromised water-allocation targets. Under such a situation, theactual water allocationwould be the difference between the water-allocation target and the probabilistic shortage (i.e., water alloca-tion¼promised target�water shortage) under a given streamflowcondition with an associated probability level. The solutions ofS�11opt ¼ 0, S�12opt ¼ 0 and S�13opt ¼ 0 indicate that there would beno water shortage for municipal use under low, medium and highflow levels; accordingly, the total amount of water allocated to themunicipal user would equal the corresponding optimized water-allocation target of 2.5�106 m3. Similarly, the solutions ofS�21opt ¼ ½3:25;3:55� � 106, S�22opt ¼ 0 and S�23opt ¼ 0 indicate that,for industrial use, there would be nowater shortage under mediumand high flow levels, but a shortage of [3.25, 3.55]� 106 m3 undera low flow level with a probability of 20%; accordingly, the totalamount of water allocated to the industrial user would be [0.45,0.75]� 106, 4.0�106 and 4.0�106 m3 under low,medium and highflow levels, respectively. The solutions of S�31opt ¼ 3:5� 106,

Fig. 4. Means and deviations of membership grades f

S�32opt ¼ ½1:43;3:39� � 106 and S�33opt ¼ 0 indicate that, for agri-cultural use, zero allocation of water would occur under a low flowlevel since the shortage ðS�31opt ¼ 3:5� 106 m3Þ would equal thepromised target ðT�

3opt ¼ 3:5� 106 m3Þ. Under a medium flowlevel, the situation is more ambiguous for agricultural use. Therewould be a shortage of 1.43�106 m3 under advantageous condi-tions (e.g., when the other users do not consume full amounts ofthe targeted demands and/or the medium flow ð~q�2 Þ approaches itsupper bound); however, under demanding conditions, the shortagemight become as high as 3.39�106 m3 with a probability of 60%.Likewise, the solutions of S�13opt ¼ 0, S�23opt ¼ 0 and S�33opt ¼ 0indicate that, under a high flow level, there would be no watershortage, and thus water would be fully allocated to the three userswith a probability of 20%. Fig. 5 presents the optimized water-allocation patterns and the associated allocation targets to the threeusers under different water flow levels. In the case of insufficientwater, the water allocation would firstly be guaranteed to themunicipality, secondly to the industry, and lastly to the agriculture.This is because the municipal user brings about the highest benefitwhen its water demand is satisfied; meanwhile, it is subject to thehighest penalty if the promised water is not delivered. In compar-ison, the industrial and agricultural users correspond to lowerbenefits and lower penalties.

The solution of the objective-function value (i.e.,~f�opt ¼ $½ð228:94;247:72;266:50Þ; ð505:14;530:10;575:06Þ� � 106),expressed asan intervalwith fuzzy lowerandupper bounds, providestwo extreme values of the system benefit. Such a solution (of fuzzyboundary interval) reflects potential system-condition variationscaused by uncertain modeling inputs as well as complexities of theirinteractions. Through adjusting the values of continuous variableswithin their lower and upper bounds, the system benefit would

or the fuzzy decision under different levels of a.

Table 4Solutions of the ITSFP model under optimized water-allocation targets (in 106 m3).

Activity/user Probability(%)

Municipal(i¼ 1)

Industrial(i¼ 2)

Agricultural(i¼ 3)

Target ðT�ioptÞ 2.5 4.0 3.5

Shortage ðS�ijoptÞ under a flow level ofLow (j¼ 1) 0.2 0 [3.25, 3.55] 3.5Medium (j¼ 2) 0.6 0 0 [1.43, 3.39]High (j¼ 3) 0.2 0 0 0

Allocation ðA�ijoptÞ under a flow level of

Low (j¼ 1) 0.2 2.5 [0.45, 0.75] 0Medium (j¼ 2) 0.6 2.5 4.0 [0.11, 2.07]High (j¼ 3) 0.2 2.5 4.0 3.5

System benefit ($106):~f�opt ¼ ½ð228:94;247:72;266:50Þ; ð505:14;530:10;575:06Þ�

Fig. 5. Optimized water-allocation patterns under low (L), medium (M) and high (H)flows.

S. Wang, G.H. Huang / Journal of Environmental Management 92 (2011) 1986e19951994

correspondingly change between ~f�opt and

~fþopt with a variety of reli-

ability levels.Solutions under other scenarios of water-allocation targets

could also be obtained by letting Ti� have different deterministic

values. Table 5 provides solutions under different scenarios ofwater-allocation targets. The solutionwhen all Ti� reach their lowerbounds indicates a situation when the DMs are conservativeregardingwater availability and thus promises merely lower-boundtarget values to all users. Such a policy would lead to both less

Table 5Solutions of the ITSFP model under different scenarios of water-allocation targets (in 10

Activity/user Ti�¼ Ti

� Ti�¼ Ti

þ

i¼ 1 i¼ 2 i¼ 3 i¼ 1

Target (Ti�) 1.5 2.0 3.5 2.5

Shortage (Sij�)Low (j¼ 1) 0 [0.25, 0.55] 3.5 0Medium (j¼ 2) 0 0 [0, 0.39] 0High (j¼ 3) 0 0 0 0

Allocation (Aij�)

Low (j¼ 1) 1.5 [1.45, 1.75] 0 2.5Medium (j¼ 2) 1.5 2.0 [3.11, 3.50] 2.5High (j¼ 3) 1.5 2.0 3.5 2.5

System benefit ($106)~f� ¼ ½ð231:94;247:72;263:50Þ;

ð351:75;369:00;396:25Þ�~f� ¼ ½ð14

ð49

water shortage and less water allocation, but a higher risk ofwasting available water and thus a lower system benefit of$[(231.94, 247.72, 263.50), (351.75, 369.00, 396.25)]� 106.Conversely, the solution when all Ti

� reach their upper boundswould be applicable when the DMs are optimistic of water avail-ability. Due to the potentially higher system benefit of $[(149.00,158.40,167.80), (490.14, 506.10, 542.06)]� 106, such a policy shouldbe effective under advantageous conditions (e.g., when water flowis high) when all promised water demands are delivered; but itwould become highly risky under demanding conditions (e.g.,when water flow is low) due to the over-promise of water-alloca-tion targets and the relevant penalties. The solution when all Ti�

equal their mid-values [Ti(mid)¼ (Ti�þ Tiþ)/2, i¼ 1, 2, 3] corre-

sponds to a situation when the DMs stand between conservativeand optimistic attitudes of water availability. As a result, the systembenefit would be $[(192.94, 205.72, 218.50), (442.14, 461.10,495.06)]� 106. The complexities associated with the water alloca-tion mainly arise from limited supply and increasing demand.Therefore, variations in water-allocation targets could reflectdifferent policies for water resources management under uncer-tainty. If thewater-allocation targets are regulated to a too low levelunder a conservative policy, it would result in a lower risk ofviolating the promised targets and thus lower penalties but, at thesame time, more waste of resources when the water flow is high.Conversely, if the water-allocation targets are regulated to a toohigh level, such an optimistic policy would lead to a higher systembenefit along with a higher risk of penalty when the promisedwater is not deliveredunderdemanding conditions (e.g.,whenwaterflow is low). Therefore, different policies in regulating the water-allocation targets are associated with different levels of economicbenefit and system-failure risk.

4. Discussion

The hypothetical problem could also be solved through aninexact two-stage stochastic programming (ITSP) method bysimplifying the intervals with fuzzy boundaries into the ones withdeterministic boundaries (Huang and Loucks, 2000). The obtainedsolution is a set of interval values, and is actually one of manyalternatives from the ITSFP approach. Compared with ITSP, theproposed ITSFP approach provides the following advantages: (a)ITSFP could handle dual uncertainties expressed as fuzzy boundaryintervals that exist in the objective function and the left- and right-hand sides of constraints, and thus enhance the robustness of theoptimization efforts.; (b) ITSFP provides DMs with the information

6 m3).

Ti�¼ Ti

(mid)

i¼ 2 i¼ 3 i¼ 1 i¼ 2 i¼ 3

4.0 6.5 2.0 3.0 5.0

[3.25, 3.55] 6.5 0 [1.75, 2.05] 5.00 [4.43, 6.39] 0 0 [1.43, 3.39]0 [0, 0.38] 0 0 0

[0.45, 0.75] 0 2.0 [0.95, 1.25] 04.0 [0.11, 2.07] 2.0 3.0 [1.61, 3.57]4.0 [6.12, 6.50] 2.0 3.0 5.0

9:00;158:40;167:80Þ;0:14;506:10;542:06Þ�

~f� ¼ ½ð192:94;205:72; 218:50Þ;

ð442:14;461:10;495:06Þ�

S. Wang, G.H. Huang / Journal of Environmental Management 92 (2011) 1986e1995 1995

about the risk of constraint violation in all steps of decision process;(c) ITSFP enables DMs to quantify the relationship between theobjective-function value and the risk of violating the constraints,which is meaningful for supporting in-depth analyses of tradeoffsbetween economic efficiency and constraint-violation risk; (d)Through its concept of solutions with different feasibility degrees,ITSFP could provide plenty of information for DMs to determinea goal and its tolerance threshold during the decision process; and(e) ITSFP enables DMs to identify, in an interactive way, a balancebetween the two objectives in conflict: to improve the objective-function value or to improve the constraint feasibility.

The main limitation of the ITSFP approach is its high computa-tional requirements. This is because the center of gravitydefuzzification technique in ITSFP may result in relatively highcomputational requirements. Especially, when the fuzzy member-ship functions are complex (e.g., shapes of membership functionsare irregular). Therefore, one potential extension of this research isto develop more computationally efficient defuzzification tech-niques. Although this study is the first attempt for planning a waterresources management system through the ITSFP method, theresults suggest that this proposed approach is applicable to manyother environmental management problems that involve an anal-ysis of policy scenarios associated with uncertain parameters. It canalso be incorporated within other optimization frameworks tohandle various real-world problems under uncertainty.

5. Conclusions

In this study, an interactive two-stage stochastic fuzzyprogramming (ITSFP) approach has been developed for waterresources management. ITSFP incorporates the interactive fuzzyresolution method within an inexact two-stage stochasticprogramming framework. It not only reflects dual uncertaintiesexpressed as fuzzy boundary intervals that exist in the objectivefunction and the left- and right-hand sides of the modelingconstraints, but also provides an effective linkage between the pre-regulated policies and the associated corrective actions against anyinfeasibility arising from random outcomes.

The developed approach has been applied to a case study ofwater resources allocation for demonstrating its applicability. Theresults indicate that a set of solutions under different feasibilitydegrees (a) have been obtained for supporting the decisions ofwater resources allocation. Moreover, a number of decision alter-natives have been generated under different policies for waterresourcesmanagement, which permits in-depth analyses of variousscenarios that are associated with different levels of economicpenalties when the promised policy targets are violated, and thushelp the DMs to identify desired water-allocation schemes.

Acknowledgements

The research was supported by the Major State Basic ResearchDevelopment ProgramofMOST (2005CB724200 and 2006CB403307),and the Natural Science and Engineering Research Council ofCanada.

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