identifying optimal water resources allocation strategies through an interactive multi-stage...

Identifying Optimal Water Resources AllocationStrategies through an Interactive Multi-StageStochastic Fuzzy Programming Approach

S. Wang & G. H. Huang

Received: 12 November 2010 /Accepted: 1 February 2012 /Published online: 25 February 2012# Springer Science+Business Media B.V. 2012

Abstract In this study, an interactive multi-stage stochastic fuzzy programming (IMSFP)approach has been developed through incorporating an interactive fuzzy resolution (IFR)method within an inexact multi-stage stochastic programming framework. IMSFP can dealwith dual uncertainties expressed as fuzzy boundary intervals that exist in the objectivefunction and the left- and right-hand sides of constraints. Moreover, IMSFP is capable ofreflecting dynamics of uncertainties and the related decision processes through constructinga set of representative scenarios within a multi-stage context. A management problem interms of water resources allocation has been studied to illustrate applicability of theproposed approach. The results indicate that a set of solutions under different feasibilitydegrees (i.e., risk of constraint violation) has been generated for planning the water resourcesallocation. They can not only help quantify the relationship between the objective-functionvalue and the risk of violating the constraints, but also enable decision makers (DMs) toidentify, in an interactive way, a desired compromise between two factors in conflict:satisfaction degree of the goal and feasibility degree of constraints. Besides, a number ofdecision alternatives have been generated under different policies for water resourcesmanagement, which permits in-depth analyses of various policy scenarios that are associatedwith different levels of economic penalties when the promised water-allocation targets areviolated, and thus help DMs to identify desired water-allocation schemes under uncertainty.

Keywords Fuzzy boundary interval . Interactive . Multi-stage . Policy analysis . Stochasticprogramming . Uncertainty . Water resources

Water Resour Manage (2012) 26:2015–2038DOI 10.1007/s11269-012-9996-1

S. Wang : G. H. Huang (*)Faculty of Engineering and Applied Science, University of Regina,Regina, Saskatchewan S4S 0A2, Canadae-mail: [email protected]

S. Wange-mail: [email protected]

1 Introduction

In water resources management problems, the conflict-laden issues of regional waterallocation among multiple competing users have been of substantial concerns due to therapid population growth and economic development (Huang and Chang 2003; Wang et al.2003). The increasing water demands under limited water availability may cause watershortage, which has been considered as a major obstacle to sustainable water resourcesmanagement. Losses can hardly be avoided when the essential demands are not satisfied dueto insufficient resources, resulting in a variety of adverse impacts on socio-economicdevelopment and human life (Lu et al. 2010). From a long-term point of view, the increasedwater demands can be satisfied by developing new water resources; however, the significanteconomic and environmental costs associated with the new development may make thisapproach inapplicable (Maqsood et al. 2005). Consequently, systems analysis techniques canbe used for allocating and managing water resources in more efficient and environmentallybenign ways. Currently, statistical analyses of hydrologic data play an important role in waterresources management. However, uncertainties that exist in a variety of system components aswell as their interrelationships may intensify the dynamic complexities arising from the spatialand temporal variations of the relationships between water demand and supply. Moreover,uncertainties may exist in multiple levels such as randomness and/or fuzziness in the lower andupper bounds of an estimated range (Li et al. 2009a). These complexities can hardly beaddressed through conventional optimization methods.

As a result, a large number of inexact optimization methods were developed for dealingwith uncertainties and the associated complexities in the water resources management(Slowinski 1986; Wu et al. 1997; Huang 1998; Jairaj and Vedula 2000; Seifi and Hipel2001; Luo et al. 2003; Maqsood et al. 2005; Li et al. 2007; Wang and Huang 2011). Amongthese methods, multi-stage stochastic programming (MSP) with recourse is effective indealing with decision-making problems where an analysis of policy scenarios is desiredand the uncertain data are random with known probability distributions. The fundamentalidea behind MSP is the concept of recourse, which possesses the advantage of permittingcorrective actions after a random event has taken place. Over the past decades, MSP wasdeveloped and applied to a number of areas. For example, Pereira and Pinto (1991) proposeda multi-stage stochastic optimization approach and applied it to the planning of a hydro-electric energy system, where the large-scale problem could be decomposed into represen-tative scenarios. Watkins et al. (2000) proposed a scenario-based stochastic programmingmodel for planning water supplies from highland lakes, where dynamics and uncertainties ofwater availability could be reflected through modeling the decision process and constructinga set of scenarios. Generally, MSP is capable of tackling uncertainties expressed as proba-bility distributions, and permitting corrective actions against any infeasibility after randomevents have occurred as well as decomposing a large-scale problem into a set of scenarios.However, in many real-world problems, the quality of uncertain information is often notsatisfactory enough to be presented as probability density functions (PDFs). The distributioninformation for uncertain parameters is difficult to be estimated when the number ofobserved data is insufficient; besides, it is extremely challenging to solve a large-scaleMSP model with all uncertain parameters being expressed as PDFs, even if their probabilitydistributions are available (Huang and Loucks 2000).

In comparison, interval mathematical programming (IMP) is effective in tackling uncer-tainties expressed as intervals with known lower and upper bounds but unknown distributionfunctions (Huang et al. 1992). IMP can deal with uncertainties in the left- and right-handsides of constraints as well as in the objective function. Thus, one potential approach for

2016 S. Wang, G.H. Huang

better reflecting uncertainties is to incorporate IMP into the MSP framework. Li et al. (2006)proposed an interval-parameter multi-stage stochastic programming (IMSP) model for waterresources management under uncertainty. IMSP could not only deal with uncertaintiesexpressed as probability distributions and intervals, but also reflect the dynamics of wateravailability through constructing a set of scenarios that are representative for all possibleoutcomes over the planning horizon. Nevertheless, the remarkable limitations of IMSP are itsincapability in handling uncertainties presented as possibility distributions and reflecting the riskof violating the constraints. In real-world problems, uncertainties can be estimated as intervals;at the same time, the lower and upper bounds of many intervals may rarely be acquired asdeterministic values. Instead, they may often be given as subjective information that can beexpressed as fuzzy sets; this leads to dual uncertainties (i.e., fuzzy boundary intervals). Such acomplexity cannot be addressed through IMSP.Moreover, IMSP lacks the information about therisk of violating the constraints, which is desired by decision makers (DMs) in order toconduct in-depth analyses of tradeoffs between economic efficiency and system risk.

Recently, Jiménez et al. (2007) proposed an interactive fuzzy resolution (IFR) method forsolving linear programming problems with fuzzy parameters. IFR permitted interactiveparticipation of DMs in all steps of decision process through expressing their preferencesin linguistic terms. A set of solutions could be obtained from IFR under different feasibilitydegrees for constraints (i.e., risk of constraint violation), which were meaningful in support-ing in-depth analyses of tradeoffs between economic efficiency and constraint-violation risk.IFR enabled DMs to identify a balanced solution between two key factors in conflict:feasibility degree of constraints and satisfaction degree of the goal.

One potential approach for better accounting for multiple uncertainties expressed as intervalsand probability/possibility distributions is to integrate the IFR into the IMSP framework, leadingto an interactive multi-stage stochastic fuzzy programming (IMSFP) approach. Therefore, theobjective of this study is to develop such an IMSFP approach for supporting water resourcesmanagement under uncertainty. IMSFP can only tackle dual uncertainties expressed as fuzzyboundary intervals that exist in the objective function and the left- and right-hand sides ofconstraints, but also reflect dynamics of uncertainties and the related decision processes throughconstructing a set of scenarios. A management problem in terms of water resources allocationwill be studied to illustrate applicability of the proposed approach. The results can only helpDMs to conduct in-depth analyses of tradeoffs between economic efficiency and constraint-violation risk, but also enable them to identify, in an interactive way, a desired compromisebetween satisfaction degree of the goal and feasibility degree of constraints.

2 Model Development

2.1 Inexact Multi-Stage Stochastic Programming

Consider a hypothetical problem wherein a water resources manager is responsible forallocating water to multiple users over a multi-period planning horizon. The objective ofthis problem is to maximize the system benefit through identifying optimized water-allocation schemes over the planning horizon. Based on local water management policies,a prescribed amount of water is promised to each user. These users need to know how muchwater they can expect so as to make appropriate decisions on their activities and investments.If the promised water is delivered, it will lead to net benefits to the local economy; otherwise,the users will have to either obtain water from more expensive sources or curtail theirdevelopment plans, resulting in economic penalties or negative consequences (Maqsood et al.

IMSFP for Water Resources Management 2017

2005). The water supplies during the planning horizon are random variables, and the relevantwater-allocation plan will be of dynamic features. Therefore, the relevant decisions must bemade at each time stage under varied probability levels.

The problem under consideration can be formulated as a multi-stage stochastic programming(MSP) model with recourse. Figure 1 presents a multi-stage scenario tree, where the nodesrepresent decisions while the arcs indicate the realizations of uncertain variables (Li et al. 2008).In MSP, a first-stage decision must be made without the knowledge of random variables, andthen a corrective action (i.e., recourse action) is undertaken to minimize the penalties that mayappear due to any infeasibility after a random event has taken place. Therefore, a multi-stagestochastic programming (MSP) model can be formulated as follows:

Maximize f ¼XI

i¼1

XTt¼1

NBitTit �XI

i¼1

XTt¼1

XKt

k¼1

ptkCitSitk ð1aÞ

subject to:

XI

i¼1

Tit � Sitkð Þ 1þ dð Þ � qht þ " t�1ð Þk ; 8t; h; k ¼ 1; 2; ∙∙∙;Kt ð1bÞ

"ðt�1Þk þPIi¼1

ðTiðt�1Þ � Siðt�1ÞkÞð1þ dÞ � qhðt�1Þ þ "ðt�2Þk ;

8t � 1; h; k ¼ 1; 2; :::; Kt�1ð1cÞ

Titmax � Tit � Sitk � 0; 8i; t; k ð1dÞwhere f is the system benefit over the planning horizon ($); i is the water user, i01, 2, ∙∙∙, I; tis the planning time period, t01, 2, ∙∙∙, T; NBit is the net benefit to user i per m3 of water

Stage 1

Period 1

Stage 2

Period 2

Stage 3

Stage T-1

Period T-1

Stage T

Fig. 1 Structure of a multi-stage scenario tree


allocated in period t ($/m3); Tit is the allocation target for water that is promised to user i inperiod t (m3) (first-stage decision variable); ptk is the probability of occurrence for scenario kin period t, with ptk>0 and

PKtk¼1 ptk ¼ 1; Kt is the number of scenarios in period t, with the

total number of scenarios being K ¼ PTt¼1 Kt; Cit is the loss to user i per m3 of water not

delivered in period t ($/m3), Cit > NBit; Sitk is the amount by which the water-allocationtarget (Tit) is not met when the stream flow is qht with probability of ptk in period t (m3)(recourse variable); h is the flow level (h01, 2, ∙∙∙, H) with h01 representing the lowest flowand h0H representing the highest one; δ is the rate of water loss during transportation; qht isthe total amount of stream flow in period t (m3) (random variable); εtk is the surplus water inthe reservoir when water is delivered in period t under scenario k (m3); Titmax is themaximum allowable allocation amount for user i in period t (m3).

Obviously, model (1) can reflect uncertainties in water availability (qht) expressed asrandom variables. However, uncertainties in other parameters such as benefits (NBit),penalties (Cit) and water-allocation targets (Tit) also need to be addressed. For example, itmay be difficult for a planner to promise a deterministic water-allocation target (Tit) to a userwhen available stream flows are uncertain. In response to the above concerns, intervalparameters can be introduced into the MSP framework to communicate uncertainties inNBit, Cit and Tit into the optimization process. This leads to an inexact multi-stage stochasticprogramming (IMSP) model as follows:

Maximize f � ¼XI

i¼1

XTt¼1

NB�it T

�it �

XI

i¼1

XTt¼1

XKt

k¼1

ptkC�it S

�itk ð2aÞ

subject to:

XI

i¼1

ðT�it � S�itkÞð1þ d�Þ � q�ht þ "�ðt�1Þk ; 8t; h; k ¼ 1; 2; :::;Kt ð2bÞ

"�ðt�1Þk þPIi¼1

ðT�iðt�1Þ � S�iðt�1ÞkÞð1þ d�Þ � q�hðt�1Þ þ "�ðt�2Þk ;

8t � 1; h; k ¼ 1; 2; :::; Kt�1ð2cÞ

T�itmax � T�

it � S�itk � 0; 8i; t; k ð2dÞwhere NB�

it , T�it , C

�it , S

�itk , δ

± and T�itmax are interval parameters/variables. An interval is

defined as a number with known upper and lower bounds but unknown distributioninformation. For example, letting T�

it and Tþit be lower and upper bounds of T�

it , respectively,we have T�

it ¼ T�it ; T

þit

� �. When T�

it ¼ Tþit ; T

�it becomes a deterministic number.

2.2 Interactive Fuzzy Resolution Method

When the subjective judgments of DMs need to be reflected in the decision-making process,fuzzy set theory is recognized as an effective means of expressing their opinions. Forexample, DMs may estimate that the most possible value for the net benefit to a water useris $90 per m3 of water allocated, and there is no possibility for it to be lower than $80 ormore than $100; such a subjective estimation on the net benefit can be expressed as a fuzzyset. Therefore, an interactive fuzzy resolution (IFR) method is introduced to deal with


uncertainties expressed as fuzzy sets. Firstly, consider the following linear programmingmodel with fuzzy parameters:

Maximize f ¼Xnj¼1

eCjXj ð3aÞ

subject to: Xnj¼1

eAijXj � eBi; i ¼ 1; 2; ∙∙∙;m ð3bÞ

Xj � 0; j ¼ 1; 2; ∙∙∙; n ð3cÞwhere eCj 2 <f g1�n; eAij 2 <f gm�n; eBi 2 <f gm�1, with <f g denoting a set of fuzzy param-eters involved in the objective function and constraints; Xj 2 Rf gn�1, with Rf g denoting aset of decision variables. A fuzzy set ðeAÞ in X is characterized by a membership functionðμeAÞ, where X represents a space of points (objects), with an element of X denoted by x(Zadeh 1965). The μeAðxÞ represents the membership grade of x in eA: μeAðxÞ ! ½0; 1�, whereμeAðxÞ can be viewed as the plausibility degree of eA taking value x. Zadeh (1978) defined apossibility distribution associated with eA as numerically equal to μeA. A convex fuzzy set (eA)is defined on a real line (R) with a continuous membership function (μeA) that can bedescribed as follows (Heilpern 1992; Jiménez et al. 2007):

μeA ðxÞ ¼0 for x � a1;feA ðxÞ for a1 � x � a2;1 for a2 � x � a3;geA ðxÞ for a3 � x � a4;0 for a4 � x:

8>>>><>>>>:

ð4Þ

where feA and geA represent a continuous and monotonically increasing function on the left-hand side of eA and a continuous and monotonically decreasing function on the right-handside of eA, respectively. If feA and geA are linear functions, a trapezoidal fuzzy set can bedenoted by eA ¼ a1; a2; a3; a4ð Þ. If a2 ¼ a3, the fuzzy set becomes triangular. The α-level setof a fuzzy set ðeAÞ can be defined as follows (Fang et al. 1999):

eAa ¼ fx 2 RjμeA � ag ð5Þ

where α ∈ [0, 1]. Since μeA is upper semi-continuous, the α-level set of eA forms a closed andbounded interval eAa ¼ ½ f �1eA ðaÞ; g�1eA ðaÞ� (Heilpern 1992), where

f �1eA ðaÞ ¼ inf x : μeA ðxÞ � a� � ð6aÞ

g�1eA ðaÞ ¼ sup x : μeA ðxÞ � a� � ð6bÞ

The expected interval of a fuzzy set ðeAÞ, denoted by EIðeAÞ, can be defined as follows(Jiménez et al. 2007):

EI eA� �¼ E

eA1 ;E

eA2

� ¼

ð10f �1eA ðaÞda;

ð10g�1eA ðaÞda

� ð7Þ


The expected value of a fuzzy set ðeAÞ, denoted by EV ðeAÞ, is the half point of its expectedinterval (Heilpern 1992):

EV eA� �¼ EeA1 þ EeA2

2ð8Þ

According to formulae (7) and (8), if a fuzzy set (eA) is trapezoidal or triangular, its expectedinterval and expected value can be calculated as follows:

EI eA� �¼ 1

2ða1 þ a2Þ; 12 ða3 þ a4Þ

� ; EV eA� �

¼ 1

4ða1 þ a2 þ a3 þ a4Þ

� ð9Þ

Two key factors, feasibility and optimality, need to be taken into account when compar-ing fuzzy sets due to the uncertain nature of modeling parameters. Following Jiménez et al.

(2007), for any pair of fuzzy sets eA and eB, the degree in which eA is smaller than eB can bedefined as follows:

μMeA; eB� �

¼0 if EeB2 � EeA1 < 0;

EeB2�EeA1EeB2 �EeA1 �ðEeB1 �EeA2 Þ if 0 2 EeB1 � EeA2 ;EeB2 � EeA1h i

;

1 if EeB1 � EeA2 > 0:

8>>><>>>: ð10Þ

where EeA1 ;EeA2h iand EeB1 ;EeB2h i

are the expected intervals of eA and eB. When μM ðeA; eBÞ � a,

where 0≤α≤1, it implies that eA is smaller than or equal to eB at least in degree α, and can be

represented by eA�aeB. In terms of the feasibility of a decision vector when theconstraints involve fuzzy sets, Jiménez et al. (2007) proposed that a decision vector(X ∈ Rn) was feasible in degree α (or α-feasible) if

mini¼1;:::;m

μM ðeAiX ; eBiÞn o

¼ a ð11Þ

where eAi ¼ eAi1; eAi2; :::; eAin

� �; α is the feasibility degree of a decision vector, and 1 – α

provides a measure of infeasibility risk. According to formula (10), this is equivalent to:

½ð1� aÞEeAi1 þ aEeAi

2 �X � aEeBi1 þ ð1� aÞEeBi

2 ð12Þ

The set of decision vectors that are α-feasible can be denoted by @ðaÞ. It is evident that:

a1 < a2 ) @ða1Þ � @ða2Þ: ð13Þ

Regarding the optimality for the objective function with fuzzy coefficients, a decision vector(X 0 ∈ Rn) is an acceptable solution if it can be verified that (Jiménez et al. 2007):

μM ðeCX ; eCX 0Þ � 1

2ð14Þ


where eC ¼ ðeC1; eC2; :::; eCnÞ; X 0 is a choice (with the objective of maximized f) at least indegree 1/2 higher as opposed to the other feasible decision vectors. According to formula (10),this is equivalent to:

EeCX 0

2 þ EeCX 0

1

2� EeCX2 þ EeCX1

2ð15Þ

Based on formulae (12) and (15), a decision vector [X 0(α)∈ Rn] is the α-acceptable solution ofmodel (3) if it is a solution to the followingα-parametric linear programming problem (Jiménezet al. 2007):

Maximize f ¼Xnj¼1

EV ðeCjÞXj ð16aÞ

subject to:

Xnj¼1

½ð1� aÞEeAij

1 þ aEeAij

2 �Xj � aEeBi1 þ ð1� aÞEeBi

2 ; i ¼ 1; 2; :::; m; a 2 ½0; 1� ð16bÞ

Xj � 0; j ¼ 1; 2; . . . ; n ð16cÞwhere EV ðeCÞ represents the expected value of the fuzzy vector ðeCÞ. Jiménez et al. (2007)proposed an interactive procedure to solve the α-parametric linear programming problem.The interactive fuzzy resolution (IFR) method allows DMs to consider two factors in aninteractive way when making a decision: feasibility degree of constraints and satisfactiondegree of the goal. A strong desire to obtain an optimal objective-function value will lead toa lower degree of feasibility for constraints (i.e., a higher risk of constraint violation);contrarily, a willingness to accept a lower satisfaction degree of the goal will guarantee ahigher feasibility degree of constraints. Therefore, DMs have to identify a compromisebetween two objectives in conflict: to improve the system benefit and to enhance theconstraint feasibility. The best way to reflect DMs’ preferences is to express them throughnatural language, establishing a semantic correspondence for different degrees of feasibility(Zadeh 1975). In order to distinguish between different preference levels sufficiently,Jiménez et al. (2007) established 11 scales, including unacceptable solution (α00),practically unacceptable solution (α00.1), almost unacceptable solution (α00.2), veryunacceptable solution (α00.3), quite unacceptable solution (α00.4), neither acceptablenor unacceptable solution (α00.5), quite acceptable solution (α00.6), very acceptablesolution (α00.7), almost acceptable solution (α00.8), practically acceptable solution(α00.9), and completely acceptable solution (α01). In real-world problems, DMs maynot be willing to admit a high risk of constraint violation. If α0 is the minimumconstraint feasibility that DMs are willing to accept, then the feasibility interval of α would bereduced to a0 � a � 1; thus, according to the semantic scale, the discrete values of α can beachieved as follows (Jiménez et al. 2007).

M ¼ ak ¼ a0 þ 0:1kjk ¼ 0; 1; :::;1� a0

0:1

� ½0; 1� ð17Þ

The first step of the IFR method is to obtain the space O ¼ fx0ðakÞ; ak 2 Mg of the αk-

acceptable solutions and the corresponding objective-function values ef 0ðakÞ ¼ eCX 0ðakÞ


through solving the α-parametric linear programming model (16) under each αk. After

obtaining the results with different ef 0ðakÞ, DMs would specify a goal and its tolerance

threshold. The goal can be expressed by means of a fuzzy set ðeGÞ. The second step of the

IFR method is to obtain the satisfaction degree of the fuzzy goal ðeGÞ for each α-acceptable

solution [i.e., membership grade of ef 0ðakÞ to eG] through an index proposed by Yager (1979).The third step of the IFR method is to identify a compromise between feasibility degree ofconstraints and satisfaction degree of the goal in the αk-acceptable solutions. In order toobtain a recommendation for a final alternative, a fuzzy decision can be defined as anintersection of fuzzy constraint and fuzzy goal (Bellman and Zadeh 1970). The solution withthe highest membership grade for fuzzy decision will be the final alternative for the fuzzylinear programming problem [i.e., model (3)].

2.3 Interactive Multi-Stage Stochastic Fuzzy Programming

Although the IFR method is capable of solving linear programming problems with fuzzyparameters, it becomes ineffective in reflecting uncertainties that exist in decision variables.Likewise, IMSP is able to deal with uncertainties expressed as intervals and probabilitydistributions, but it has difficulties in reflecting DMs’ subjective estimations on uncertainparameters in the decision-making process. In real-world problems, however, the lower andupper bounds of many interval parameters may often be given as subjective information thatcan be expressed as fuzzy sets; this leads to dual uncertainties (i.e., fuzzy boundaryintervals). As shown in Fig. 2, ea� and eaþ are lower and upper bounds of an interval ðea�Þ,and can be quantified as fuzzy sets with triangular membership functions. Assume that thereis no intersection between the fuzzy sets of the two bounds. In fact, intervals are used toexpress uncertainties without distribution information. If the fuzzy sets of an interval’s lowerand upper bounds intersect, then the so-called “interval” can actually be described as fuzzymembership functions, such that the interval representation becomes unnecessary (Nie et al.2007). To address dual uncertainties expressed as fuzzy boundary intervals, the IFR methodcan be integrated into the IMSP framework. This leads to an interactive multi-stagestochastic fuzzy programming (IMSFP) model as follows:

Maximize ef � ¼XI

i¼1

XTt¼1

NeB�it T

�it �

XI

i¼1

XTt¼1

XKt

k¼1

ptk eC�it S

�itk ð18aÞ

0

1

a−� a+� ±

μ

Fig. 2 Fuzzy boundary interval


subject to:

PIi¼1

ðT�it � S�itkÞð1þ ð1� aÞEed �

1 þ aEed �

2 Þ � aEeq �ht

1 þ ð1� aÞEeq �ht

2 þ "�ðt�1Þk ;

8t; h; k ¼ 1; 2; :::; Kt

ð18bÞ


ðT�iðt�1Þ � S�iðt�1ÞkÞð1þ ð1� aÞEed �

1 þ aEed �

2 Þ � aEeq�hðt�1Þ

1

þð1� aÞEeq �hðt�1Þ

2 þ "�ðt�2Þk ; 8t � 1; h; k ¼ 1; 2; :::; Kt�1

ð18cÞ

T�itmax � T�

it � S�itk � 0; 8i; t; k ð18dÞ

where α is the feasibility degree of a decision vector, and 1 – α provides a measure of therisk of infeasibility for a decision vector; T�

it , S�itk and T�

itmax are interval parameters/variables; NeB�

it , eC�it , ed�, eq�ht are fuzzy boundary intervals.

According to Huang and Loucks (2000), if T�it are considered as uncertain inputs, the

existing methods for solving inexact linear programming problems cannot be used directly.Therefore, an optimized set of target values (T�

it ) will be identified by having yit beingdecision variables. The optimized set corresponds to the maximized system benefit underuncertain water-allocation targets. Accordingly, let T�

it ¼ T�it þΔTityit, whereΔTit ¼ Tþ

it � T�it

and yit 2 ½0; 1�; yit are decision variables that are used for identifying an optimized set of targetvalues (T�

it ) in order to support the related policy analyses. Thus, by introducing decisionvariables (yit), model (18) can be reformulated to:


i¼1

XTt¼1

NeB�it ðT�

it þΔTityitÞ �XI

i¼1

XTt¼1

XKt

k¼1

ptk eC�it S

�itk ð19aÞ

subject to:

PIi¼1

ðT�it þΔTityit � S�itkÞð1þ ð1� aÞEed �

1 þ aEed �

2 Þ � aEeq �ht


2

þ"�ðt�1Þk ; 8t; h; k ¼ 1; 2; :::; Kt

ð19bÞ

"�t�1ð Þk þPIi¼1

ðT�iðt�1Þ þΔTiðt�1Þyiðt�1Þ � S�iðt�1ÞkÞð1þ ð1� aÞEed�1 þ aEed�2 Þ

� aEeq �hðt�1Þ

1 þ ð1� aÞEeq �hðt�1Þ

2 þ "�ðt�2Þk ; 8t � 1; h; k ¼ 1; 2; :::; Kt�1

ð19cÞ

T�itmax � T�

it þΔTityit � S�itk � 0; 8i; t; k ð19dÞ

When T�it are known, model (19) can be transformed into two submodels, which

correspond to the lower and upper bounds of the objective-function value (Huang et al.1992). The resulting solution can provide intervals for the objective-function value anddecision variables, and can be easily interpreted for generating decision alternatives. Since


the objective is to maximize the system benefit, the submodel corresponding to the upper

bound of the objective-function value (ef þ) can be first formulated as follows:

Maximize ef þ ¼XI

i¼1

XTt¼1

NeBþit ðT�

it þΔTityitÞ �XI

i¼1

XTt¼1

XKt

k¼1

ptk eC�it S

�itk ð20aÞ

subject to:

PIi¼1

ðT�it þΔTityit � S�itkÞð1þ ð1� aÞEed þ

1 þ aEed þ

2 Þ � aEeq þht

1 þ ð1� aÞEeq þht

2

þ"þðt�1Þk ; 8t; h; k ¼ 1; 2; :::;Kt

ð20bÞ

"þðt�1Þk þPIi¼1

ðT�iðt�1Þ þΔTiðt�1Þyiðt�1Þ � S�iðt�1ÞkÞð1þ ð1� aÞEed þ

1 þ aEed þ

2 Þ

� aEeq þhðt�1Þ

1 þ ð1� aÞEeq þhðt�1Þ

2 þ "þðt�2Þk ; 8t � 1; h; k ¼ 1; 2; :::; Kt�1

ð20cÞ

Tþitmax � T�

it þΔTityit � S�itk � 0; 8i; t; k ð20dÞ

where S�itk and yit are decision variables; Solutions of S�itkopt, yitopt and ef þopt can be obtainedthrough submodel (20). The optimized water-allocation targets can be determined bycalculating T�

itopt ¼ T�it þΔTityitopt.Then, the submodel corresponding to the lower bound

of the objective-function value ðef �Þ can be formulated as follows:


i¼1

XTt¼1

NeB�it ðT�

it þΔTityitoptÞ �XI

i¼1

XTt¼1

XKt

k¼1

ptk eCþit S

þitk ð21aÞ

subject to:

PIi¼1

ðT�it þΔTityitopt � SþitkÞð1þ ð1� aÞEed �

1 þ aEed �

2 Þ � aEeq �ht


2

þ"�ðt�1Þk ; 8t; h; k ¼ 1; 2; :::;Ktð21bÞ


ðT�iðt�1Þ þΔTiðt�1Þyiðt�1Þopt � Sþiðt�1ÞkÞð1þ ð1� aÞEed �

1 þ aEed �

2 Þ

� aEeq �hðt�1Þ

1 þ ð1� aÞEeq �hðt�1Þ

2 þ "�ðt�2Þk ; 8t � 1; h; k ¼ 1; 2; :::; Kt�1

ð21cÞ

T�it þΔTityitopt � Sþitk ; 8i; t; k ð21dÞ

Sþitk � S�itkopt; 8i; t; k ð21eÞ


where Sþitk are decision variables. Solutions of Sþitkopt and ef �opt can be obtained throughsubmodel (21). By combining the solutions from submodels (20) and (21), the finalsolutions of model (19) under the optimized water-allocation targets are:

S�itkopt ¼ ½S�itkopt; Sþitkopt�; 8i; t; k ð22aÞ

ef �opt ¼ ½ef �opt; ef þopt� ð22bÞThus, the optimized water-allocation scheme over the planning horizon is:

A�itkopt ¼ T�

itopt � S�itkopt; 8i; t; k ð23ÞThe solution algorithm of the IMSFP approach can be summarized as follows:

Step 1: Formulate the IMSFP model;Step 2: Reformulate the IMSFP model by introducing T�

it ¼ T�it þΔTityit, where

ΔTit ¼ Tþit � T�

it and yit ∈ [0,1];Step 3: Transform the IMSFP model into two submodels, where the submodel

corresponding to ef þ is first desired since the objective is to maximize ef �;Step 4: Obtain the αk-acceptable solutions through solving the ef þ submodel under

each αk;Step 5: Compute the satisfaction degree of the fuzzy goal for each α-acceptable

solution;Step 6: Identify a balance between feasibility of constraints and satisfaction degree

of the goal for each α-acceptable solution;Step 7: Formulate and solve the ef � submodel by following the same interactive

procedure as that in ef þ submodel;Step 8: Reach a desired compromise between feasibility of constraints and satisfac-

tion degree of the goal by considering the solutions from two submodels;Step 9: Obtain the αk-feasibility optimal solution:

ef �opt ¼ ½ef �opt; ef þopt�; S�ijopt ¼ ½S�itkopt; Sþitkopt�; 8i; t; kStep 10: Obtain the optimal water-allocation scheme over the planning horizon:

A�itkopt ¼ T�

itopt � S�itkopt; 8i; t; k:Step 11: Stop.

3 Case Study

3.1 Overview of the Study System

Water resources management is associated with a variety of complexities such asuncertainties in economic and technical data, dynamic variations in system components,randomness in water availability, and policy implications in water allocation (Li et al.2009a). These complexities could become further compounded not only by interactions


among uncertain system components but also through their economic effects. Uncer-tainties often arise from a number of factors such as the random characteristics ofnatural processes (e.g., precipitation and climate change), stream conditions (e.g., streamflow and water-quality requirement), human-induced imprecision in acquiring modelingparameters (e.g., lack of available data and biased judgments), and vagueness of systemobjectives and constraints (Li et al. 2009b). Moreover, uncertainties may exist inmultiple levels. For example, in many real-world problems, the lower and upper boundsof some interval parameters such as the rate of water loss during a runoff process couldrarely be acquired as a deterministic value. Instead, they may often be given assubjective information that could be expressed as fuzzy sets; this leads to dual uncer-tainties (i.e., fuzziness in the lower and upper bounds of an interval). These complex-ities could not be addressed through conventional optimization methods. Therefore, it isdesired that advanced optimization methods be developed for supporting sustainablewater resources management.

The following water resources management problem will be used to demonstrateapplicability of the developed approach. A water manager is responsible for allocatingwater from an unregulated reservoir to three users: a municipality, an industry unit andan agricultural sector (Fig. 3). The time horizon under consideration is 15 years, whichis divided into three planning periods. The water supplies are random variables withknown probability distributions, and thus the relevant water-allocation plan would be ofdynamic feature over the planning periods. All water users need to know how muchwater they could expect so as to make appropriate decisions on their activities andinvestments over the planning horizon. If the promised water is delivered, a net benefitto the local economy will be generated for each unit of water allocated; otherwise,either the water must be obtained from higher-priced alternatives or the regionaldevelopment plans must be curtailed, resulting in economic penalties (Maqsood et al.2005). Table 1 shows the available water resources (expressed as fuzzy boundaryintervals) and the associated probabilities of occurrence in three planning periods, aswell as the rate of water loss (expressed as fuzzy boundary intervals). Table 2 showsthe water-allocation targets prescribed by local water management policies and the

Reservoir

Municipality

Industry

Agriculture

Fig. 3 Schematic of water allocation system


maximum allowable allocations to municipal, industrial and agricultural sectors. Table 3provides the related economic data (presented as fuzzy boundary intervals), which aremainly from governmental reports and public surveys. Due to the spatial and temporalvariations of the relationships between water demand and supply, the desired water-allocation patterns may vary dynamically. These complexities could be further intensi-fied by uncertainties that exist in many system components.

The hypothetical problem under consideration is how to allocate the water to multipleusers to maximize the system benefit over the planning horizon. In this study, therandom variables (stream flows) with known probabilities could be addressed throughconstructing three scenario trees with each having a branching structure of 1-3-3-3. Thus,a three-period (four-stage) scenario tree could be generated for each of the three waterusers. All of the scenario trees have the same structure with one initial node at time 0and three succeeding ones in period 1; each node in period 1 has three succeeding nodesin period 2, and so on for each node in period 3. Consequently, there would be 9scenarios in period 1, 27 in period 2, and 81 in period 3 for the stream flows associatedwith different joint probabilities over the planning horizon. Since the dual uncertaintiespresented as fuzzy boundary intervals exist in a variety of system components, and alinkage between pre-regulated policies and the associated corrective actions against anyinfeasibility is desired, IMSFP is considered to be a feasible approach for tackling thiswater-allocation problem.

Table 1 Stream flows (in 106 m3) and the rates of water loss

Flow level Probability (%) Stream flow

t01 t02 t03

Low (L) 0.2 [(3.7, 4.0, 4.3),(4.7, 5.0, 5.3)]

[(4.7, 5.0, 5.3),(5.7, 6.0, 6.3)]

[(3.2, 3.5, 3.8),(4.2, 4.5, 4.8)]

Medium (M) 0.6 [(7.0, 8.0, 9.0),(10.0, 11.0, 12.0)]

[(8.0, 9.0, 10.0),(11.0, 12.0, 13.0)]

[(6.5, 7.5, 8.5),(9.5, 10.5, 11.5)]

High (H) 0.2 [(14.0, 15.0, 16.0),(18.0, 19.0, 20.0)]

[(16.0, 17.0, 18.0),(20.0, 21.0, 22.0)]

[(13.0, 14.0, 15.0),(17.0, 18.0, 19.0)]

Water loss ðed�Þ [(0.10, 0.15, 0.20), (0.30, 0.35, 0.40)]

Table 2 Water-allocation targets(in 106 m3) Time period

t01 t02 t03

Water-allocation target:

T�1t (to municipal) [2.0, 3.0] [2.5, 3.5] [3.0, 4.0]

T�2t (to industrial) [2.5, 4.0] [3.5, 5.5] [4.0, 6.0]

T�3t (to agricultural) [3.5, 5.5] [4.0, 6.0] [4.0, 6.0]

Maximum allowable allocation:

T�1tmax(to municipal) 7.0 7.0 7.0

T�2tmax(to industrial) 7.0 7.0 7.0

T�3tmax(to agricultural) 7.0 7.0 7.0


3.2 Result Analysis

Table 4 shows the system benefits expressed as fuzzy boundary intervals under a set offeasibility degrees (α). In this study, 0.4 was the minimum constraint feasibility degree thatDMs were willing to accept (suppose that DMs would not admit a high risk in violating theconstraints); therefore, the discrete values of α under consideration were 0.4, 0.5, 0.6, 0.7,0.8, 0.9 and 1. Figure 4 shows the expected values of system benefits expressed as intervalsunder different α levels. It is indicated that the minimum and maximum expected systembenefits would be $[909.92, 2075.25]×106 under α01 and $[1073.23, 2170.40]×106 underα00.4. The results indicate that the expected system benefits would gradually decreasewhen α increases (higher constraint feasibility is achieved with a cost of reduced systembenefit), and thus reflect a tradeoff between system benefit and constraint-violation risk. Inreal-world problems, a lower feasibility degree for constraints would correspond to a higherobjective-function value; contrarily, a lower satisfaction degree of the objective wouldguarantee a higher feasibility degree for constraints. Therefore, DMs would have to identifya balance between two factors in conflict: feasibility of constraints and satisfaction degree ofthe goal. After obtaining the system benefits (expressed as fuzzy boundary intervals) under

Table 3 Net benefits and penalties (in $/m3)

Time period

t01 t02 t03

Net benefit when water demand is satisfied:

Municipal (i01) [(85, 90, 95), (105, 110, 115)] [(90, 95, 100), (110, 115, 120)] [(90, 95, 100), (110, 115, 120)]

Industrial (i02) [(42, 45, 48), (52, 55, 58)] [(52, 55, 58), (67, 70, 73)] [(62, 65, 68), (82, 85, 88)]

Agricultural (i03) [(29, 30, 31), (34, 35, 36)] [(34, 35, 36), (49, 50, 51)] [(34, 35, 36), (49, 50, 51)]

Penalty when water is not delivered:

Municipal (i01) [(195, 200, 205), (245, 250, 255)] [(195, 200, 205), (245, 250, 255)] [(195, 200, 205), (245, 250, 255)]

Industrial (i02) [(57, 60, 63), (82, 85, 88)] [(67, 70, 73), (92, 95, 98)] [(97, 100, 103), (127, 130, 133)]

Agricultural (i03) [(49, 50, 51), (69, 70, 71)] [(54, 55, 56), (74, 75, 76)] [(49, 50, 51), (69, 70, 71)]

Table 4 The α-acceptable system benefits obtained from the IMSFP model

Feasibility degree System benefit ($106)

0.4 [(987.97, 1073.23, 1158.49), (2078.75, 2170.40, 2262.04)]

0.5 [(962.22, 1046.75, 1131.26), (2063.44, 2154.58, 2245.72)]

0.6 [(931.87, 1015.78, 1099.69), (2047. 71, 2138.52, 2229.32)]

0.7 [(910.49, 993.66, 1076.83), (2032.82, 2122.57, 2212.31)]

0.8 [(880.36, 962.88, 1045.41), (2017.22, 2106.68, 2196.14)]

0.9 [(858.95, 940.74, 1022.53), (2002.53, 2090.91, 2179.29)]

1 [(828.75, 909.92, 991.09), (1987.12, 2075.25, 2163.39)]

The level of 0.4 is the minimum constraint feasibility degree which DMs can accept (suppose that DMs willnot admit a high risk of constraint violation); the discrete values of α are: M0{0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1}


different levels of α, DMs would be asked to specify a goal and its tolerance threshold. Thegoals of lower- and upper-bound system benefits would then be, respectively, expressed by

means of a fuzzy set ðeGÞ whose membership function is shown as follows:

μeGðzÞ ¼1 if z � 1158;z�829

1158�829 if 829 � z � 1158;0 if z � 829:

8<: ð24aÞ

μeGðzÞ ¼1 if z � 2262;z�1987

2262�1987 if 1987 � z � 2262;0 if z � 1987:

8<: ð24bÞ

Table 5 presents the satisfaction degrees of fuzzy goals under each α-acceptable systembenefit. In an attempt to identify a balance between feasibility degree of constraints andsatisfaction degree of the goal, a fuzzy decision was employed to aggregate the feasibilityand satisfaction degrees under each α-acceptable solution. In order to obtain a recommen-dation for a final decision, the mean and deviation of the lower- and upper-bound member-ship grades in the fuzzy decision were calculated under each α-acceptable solution. Theoptimal solution would be the one with the highest mean of membership grades in the fuzzydecision. As shown in Fig. 5, the 0.7-feasibility solution with the highest mean (0.347) andlowest deviation (0.004) would be the best choice.

Figures 6, 7 and 8 present the 0.7-feasibility water-allocation patterns under all possiblescenarios of stream flow over the planning horizon. Most of non-zero decision variables areintervals. These interval solutions reflect potential system-condition variations caused by

uncertain modeling inputs such as NeB�it , eC�

it , ed� and q�ht as well as the complexities of theirinteractions. Deficits would occur if available water is insufficient to satisfy the promisedwater-allocation targets over the planning horizon. Under such a situation, the actual waterallocation would be the difference between the water-allocation target and the probabilisticshortage (i.e., water allocation0promised target—water shortage) under a given streamcondition with an associated probability level. In the case of insufficient water supply, thewater allocation should first be guaranteed to the municipal user, secondly to the industry,

1073.23 1046.75 1015.78 993.66 962.88 940.74 909.92

2170.4 2154.58 2138.52 2122.57 2106.68 2090.91 2075.25

500

1000

1500

2000

2500

0.4 0.5 0.6 0.7 0.8 0.9 1

Exp

ecte

d s

yste

m b

enef

it (

$106

)

Feasibility degree (αα )

Lower bound Upper bound

Fig. 4 Expected values of system benefits under different α levels


and lastly to the agriculture. This is because the municipal user brings about the highestbenefit when its water demand is satisfied; meanwhile, it is subject to the highest penalty ifthe promised water is not delivered. In comparison, the industrial and agricultural userscorrespond to lower benefits and lower penalties.

The optimized water-allocation targets for the municipal, industrial and agriculturalsectors could be obtained by letting T�

itopt ¼ T�it þΔTityitopt. These optimized targets could

help DMs to maximize the system benefit under uncertainty. The results of y11opt ¼ y12opt ¼y13opt ¼ 1 indicate that the optimized water-allocation targets would be 3.0×106 m3, 3.5×106 m3 and 4.0×106 m3 for the municipal user in periods 1, 2 and 3, respectively, whichcorrespond to their upper-bound targets (i.e., Tþ

11, Tþ12 and T

þ13). This is because the municipal

user could bring about the highest benefit when its demand is satisfied; thus, DMs wouldhave to promise as much as possible for this user to achieve a maximized system benefit.The results of y31opt ¼ 0, y32opt ¼ 0 and y33opt ¼ 1 indicate that the targets for the agriculturalsector would be 3.5×106 m3, 4.0×106 m3 and 6.0×106 m3 in periods 1, 2 and 3, respectively;

Table 5 Membership grades of fuzzy decisions in the α-acceptable solutions

Feasibilitydegree

Satisfaction degreeof fuzzy goal

Membership gradeof fuzzy decision

Mean ofmembershipgrade forfuzzy decision

Deviation ofmembershipgrade for fuzzydecisionLower-bound

solutionUpper-boundsolution

Lower-boundsolution

Upper-boundsolution

0.4 0.74 0.67 0.296 0.268 0.282 0.014

0.5 0.66 0.61 0.330 0.305 0.318 0.013

0.6 0.57 0.55 0.342 0.330 0.336 0.006

0.7 0.50 0.49 0.350 0.343 0.347 0.004

0.8 0.41 0.44 0.328 0.352 0.340 0.012

0.9 0.34 0.38 0.306 0.342 0.324 0.018

1 0.25 0.32 0.250 0.320 0.285 0.035

0.2

0.25

0.3

0.35

0.4

0.4 0.5 0.6 0.7 0.8 0.9 1

Mea

n a

nd

dev

iati

on

of

mem

ber

ship

gra

de

Feasibility degree (αα)

Fig. 5 Means and deviations of membership grades for fuzzy decisions under different α levels


the results of y31opt ¼ 0 and y32opt ¼ 0 indicate that the targets would reach their lowerbounds (i.e., T�

31 and T�32) in periods 1 and 2, since the agricultural sector is associated with

the lowest benefit. In comparison, the benefit from the industry is lower than that from themunicipal user and higher than that from the agriculture. The optimized targets for theindustrial sector would be 2.5×106 m3, 5.5×106 m3 and 6.0×106 m3 in periods 1, 2 and 3,respectively; the results of y22opt ¼ 1 and y23opt ¼ 1 indicate that the targets would reachtheir upper bounds (i.e., Tþ

22 and Tþ23), representing a situation that DMs have an optimistic

attitude towards water supply to the industry in periods 2 and 3. These decisions represent acompromise between water shortage and surplus under uncertain water availability. A highertarget level would lead to a higher benefit associated with a higher risk of penalty caused bywater shortage when the stream flow is low; however, a lower target would result in a lowerrisk of water shortage and thus lower penalties but, at the same time, potential waste of waterresources when the stream flow is high.

Figure 6 shows the optimized water-allocation patterns under 9 scenarios in the firstperiod. The solutions of S�111opt ¼ S�112opt ¼ S�113opt ¼ 0 indicate that there would be no watershortage for the municipal user under low, medium and high flow levels; accordingly, the

0

1

2

3

4

L M H L M H L M H

Wat

er a

lloca

tio

n (

106m

3)

Municipal Industrial Agricultural

Optimized water allocation (lower bound)Optimized water allocation (upper bound)Optimized water allocation target

Fig. 6 Optimized water-allocation patterns in period 1

0

2

4

6

LL LM LH ML

MM

MH HL

HM

HH LL LM LH ML

MM

MH HL

HM

HH LL LM LH ML

MM

MH HL

HM

HH

Wat

er a

lloca

tio

n (

106

m3)

AgriculturalIndustrialMunicipal




total amount of water allocated to the municipal user would equal the corresponding target of3.0×106 m3. The solutions of S�211opt ¼ 2:5� 106 m3, S�212opt ¼ 0:79; 2:50½ � � 106 m3 andS�213opt ¼ 0 indicate that, for the industrial user, zero allocation of water would occur undera low flow level since the shortage ðS�31opt ¼ 2:5� 106 m3Þ would equal the promised target

0

2

4

6LL

L

LLM

LLH

LML

LMM

LMH

LHL

LHM

LHH

MLL

MLM

MLH

MM

L

MM

M

MM

H

MH

L

MH

M

MH

H

HLL

HLM

HLH

HM

L

HM

M

HM

H

HH

L

HH

M

HH

HWat

er a

lloca

tio

n (

106

m3)

Wat

er a

lloca

tio

n (

106

m3)

Wat

er a

lloca

tio

n (

106

m3)

Scenario of stream flow

Municipal sector


0

2

4

6

8

LLL

LLM

LLH

LML

LMM

LMH

LHL

LHM

LHH

MLL

MLM

MLH

MM

L

MM

M

MM

H

MH

L

MH

M

MH

H

HLL

HLM

HLH

HM

L

HM

M

HM

H

HH

L

HH

M

HH

H


Industrial sector


0

2

4

6

8

LLL

LLM

LLH

LML

LMM

LMH

LHL

LHM

LHH

MLL

MLM

MLH

MM

L

MM

M

MM

H

MH

L

MH

M

MH

H

HLL

HLM

HLH

HM

L

HM

M

HM

H

HH

L

HH

M

HH

H


Agricultural sector




ðT�21opt ¼ 2:5� 106 m3Þ. Under a medium flow level (with a probability of 60%), the

situation is more ambiguous for the industrial user. There would be a water shortage of0.79×106 m3 under advantageous conditions [e.g., when the other users do not consume thefull amounts of the targeted demands and/or the medium flow ðeq�21Þ approaches its upperbound]; however, under demanding conditions, the shortage would become as high as 2.5×106 m3. When the stream flow is high (with a probability of 20%), the promised target of2.5×106 m3 would be satisfied for the industrial user. Likewise, the solutions ofS�311opt ¼ 3:5� 106 m3, aS�312opt ¼ 3:5� 106 m3 and S�313opt ¼ 1:31; 3:50½ � � 106 m3 indi-cate that, for the agricultural sector, zero allocation would occur under a low ormedium flow level, since the water shortage would equal the promised targetðT�

31opt ¼ 3:5� 106 m3Þ. When the stream flow is high, the water shortage would beas low as 1.31×106 m3 under advantageous conditions, and become as high as 3.5×106 m3 under demanding ones.

Figure 7 presents the optimized water-allocation patterns under 27 scenarios in period 2.It is indicated that there would be no water shortage for the municipal user under allscenarios; accordingly, the total amount of water allocated to the municipal user would bethe promised target of 3.5×106 m3. The solutions of S�221opt ¼ 5:5� 106 m3, S�222opt ¼2:12; 3:74½ � � 106 m3 and S�223opt ¼ 0 indicate that there would respectively be 5.5×

106 m3, [2.12, 3.74]×106 m3 and 0 of water shortage for the industrial user under low,medium and high flow levels in period 2 if the stream flow is low in the first period;accordingly, the total amounts of water allocated to this user would respectively be 0, [1.76,3.38]×106 m3 and 5.5×106 m3 under three scenarios (with joint probabilities of 4, 12 and4%). If the stream flow becomes higher in the first period (resulting in surplus), there wouldbe more water allocated to the industry under low, medium and high flow levels in period 2.The solutions of S�223opt ¼ S�226opt ¼ S�229opt ¼ 0 imply that, there would be no water shortagefor the industrial user as long as the stream flow is high in period 2. In comparison, therewould be less water allocated to the agricultural sector in period 2. The solutions of S�321opt ¼S�324opt ¼ S�327opt ¼ 4� 106 m3 indicate that, there would be zero allocation of water for theagricultural sector if the stream flow is low in period 2, since the water shortage would equalthe promised target ðT�

23opt ¼ 4� 106 m3Þ.Figure 8 presents the optimized water-allocation patterns to the municipal, industrial and

agricultural sectors under 81 scenarios in period 3. It is indicated that there would be nowater shortage for the municipal user under all possible scenarios; accordingly, the totalamount of water allocated to the municipal user would be the target of 4×106 m3. When thestream flow is low over the entire planning horizon (under the worst scenario with a jointprobability of 0.8%), the water allocated to the industrial user would be [0.12, 0.76]×106 m3

in period 3; contrarily, when the stream flow is high during the entire planning horizon(under the best scenario with a joint probability of 0.8%), there would be no water shortagefor the industrial user in period 3, and thus the water allocated to this user would be the targetof 6×106 m3. The water shortage for the industrial user in period 3 would become less ifthere is some surplus in the reservoir due to the high-flow condition in period 2. Disregard-ing the flow levels in the previous two periods, there would be no water shortage for theindustrial user if the stream flow is high in period 3. In terms of the agricultural sector, whenthe stream flow is low over the entire planning horizon (under the worst scenario), zeroallocation would occur in period 3; contrarily, when the stream flow is high during the entireplanning horizon (under the best scenario), there would be no water shortage for theagricultural sector in period 3, and thus water allocated to this user would be the promisedtarget of 6×106 m3. In general, in the case of insufficient water supply, DMs would have to


give the top priority to the municipal user. Therefore, there would be no water shortage forthe municipal user under all possible scenarios over the entire planning horizon. In compar-ison, the optimized water-allocation plans for industrial and agricultural users would varyunder different stream flow scenarios.

The solution of the objective function (i.e., ef �opt ¼ $ 910:49; 993:66; 1076:83ð Þ;½2032:82; 2122:57; 2212:31ð Þ� � 106) expressed as a fuzzy boundary interval providestwo extremes of the system benefit over the planning horizon. As the values of continuousvariables vary within their lower and upper bounds, the system benefit would correspond-ingly change between ef �opt and ef þopt.3.3 Policy Analysis

Variations in water-allocation targets could reflect different policies for water resourcesmanagement under uncertainty. Solutions under various scenarios of water-allocation targetscould be obtained by letting T�

it have different deterministic values. Table 6 providessolutions of the objective function under different scenarios of water-allocation targets.The solution when all T�

it reach their lower bounds indicates a situation when DMs areconservative regarding water availability, and thus promise the lower-bound target values toall users. Such a policy in regulating the water-allocation targets would lead to both lesswater shortage and less water allocation, but a higher risk of wasting available water, andthus a lower system benefit of $[(1009.42, 1078.82, 1263.50), (1734.52, 1806.59,1878.67)]×106. Conversely, the solution when all T�

it reach their upper bounds wouldbe applicable when DMs are optimistic of water availability. Due to the potentiallyhigher system benefit of $[(688.99, 772.60, 856.22), (1990.41, 2081.16, 2171.91)]×106, such a policy should be effective under advantageous conditions (e.g., when thestream flow is high) when all promised water demands are delivered; but it wouldbecome highly risky under demanding conditions (e.g., when the stream flow is low)due to over-promise of allocation targets. The solution when all T�

it equal their mid-

values T ðmidÞit ¼ ðT�

it þh

Tþit Þ=2; i ¼ 1; 2; 3; t ¼ 1; 2; 3� corresponds to a situation

when DMs stand between conservative and optimistic attitudes towards water avail-ability. As a result, the system benefit would be $[(869.84, 948.30, 1026.76),(1888.61, 1971.96, 2055.31)]×106. In general, if the water-allocation targets areregulated at too low levels, such a conservative policy would result in a lower riskof violating promised targets and thus lower penalties but, at the same time, more

Table 6 Solutions of the IMSFP model under different scenarios of water-allocation targets (in 106 m3)

User/Period T�it ¼ T�

it T�it ¼ Tþ

it T�it ¼ T ðmidÞ

it

t01 t02 t03 t01 t02 t03 t01 t02 t03

Municipal (i01) 2.0 2.5 3.0 3.0 3.5 4.0 2.5 3.0 3.5

Industrial (i02) 2.5 3.5 4.0 4.0 5.5 6.0 3.25 4.5 5.0

Agricultural (i03) 3.5 4.0 4.0 5.5 6.0 6.0 4.5 5.0 5.0

System benefit, ef �ð$106Þ [(1009.42, 1078.82,1263.50), (1734.52,1806.59, 1878.67)]

[(688.99, 772.60,856.22), (1990.41,2081.16, 2171.91)]

[(869.84, 948.30,1026.76), (1888.61,1971.96, 2055.31)]


waste of resources under advantageous conditions. Conversely, if the targets areregulated at too high levels, such an optimistic policy would lead to a higher systembenefit along with a higher risk of penalty when the promised water is not deliveredunder demanding conditions (e.g., when the stream flow is low). Therefore, differentpolicies in regulating water-allocation targets are associated with different levels of economicbenefit and system-failure risk.

3.4 Comparison with IMSP

The existing interval-parameter multi-stage stochastic linear programming (IMSP) approach(Li et al. 2006) could only tackle uncertainties expressed as intervals with known lower andupper bounds. However, in real-world problems, the lower and upper bounds of someintervals could rarely be acquired as deterministic values. Instead, they might often be givenas subjective information that could be expressed as fuzzy sets; this leads to dual uncertain-ties. IMSFP is effective in handling dual uncertainties expressed as fuzzy boundary intervalsthat exist in the objective function and the left- and right-hand sides of constraints. Thus, therobustness of optimization efforts could be enhanced (“robustness” means that IMSFPpossesses enhanced capacities in reflecting complexities of system uncertainties). On theother hand, the solutions from IMSP lack the information about constraint-violation risks asdefined by feasibility degrees (α) in the solutions of IMSFP. Through the concept ofsolutions with different feasibility degrees, IMSFP could provide plenty of information forDMs to determine their goal and its tolerance threshold in the decision process. Besides,IMSFP can not only permit interactive participation of DMs in the decision process throughexpressing their opinions in linguistic terms, but also enable DMs to identify a desiredcompromise between two objectives in conflict: to improve the objective-function value andto enhance the constraint feasibility.

The main limitation of the IMSFP approach is its high computational requirements dueto the utilization of the center of gravity defuzzification technique in IMSFP. Especiallywhen fuzzy membership functions are complex (e.g., shapes of membership functions areirregular), it would result in extremely high computational demands. Therefore, onepotential extension of this research is to develop more computationally efficient defuzzi-fication techniques. Besides, IMSFP deals with uncertainties through constructing scenar-ios that are representative for the universe of water-availability conditions; such ascenario-based stochastic programming model may become too large when all water-availability scenarios are taken into account. Consequently, a more advanced solutiontechnique is desired for further enhancing the developed IMSFP approach. Although thisstudy is the first attempt for planning a water resources management system throughIMSFP, the results suggest that this proposed approach is applicable to many otherenvironmental management problems that involve an analysis of policy scenarios associ-ated with uncertain parameters. Nevertheless, IMSFP has difficulties in dealing withlarge-scale problems that involve nonlinear complexity. Therefore, it will be necessaryto integrate IMSFP with other techniques such as nonlinear programming to enhance itsapplicability to practical situations.

4 Conclusions

In this study, an interactive multi-stage stochastic fuzzy programming (IMSFP) approach hasbeen developed through incorporating the interactive fuzzy resolution method within an


inexact multi-stage stochastic programming framework. IMSFP can not only directly incor-porate a variety of uncertainties expressed as intervals and probability/possibility distribu-tions within its optimization framework, but also reflect dynamics of uncertainties and therelated decision processes through generating a set of scenarios that are representative forpossible random outcomes within a multi-stage context. Since IMSFP possesses the abilityto take corrective actions against any infeasibility after uncertainties are disclosed, thepenalties or negative consequences caused by improper policies can be minimized.

The developed IMSFP has been applied to a hypothetical case study of water resourcesallocation for demonstrating its applicability. The results indicate that a set of solutions underdifferent feasibility degrees (α) have been obtained. The solutions under different α levelscan help quantify the relationship between the objective-function value and the risk ofviolating the constraints, which is meaningful for supporting in-depth analyses of tradeoffsbetween economic efficiency and constraint-violation risk, as well as for enabling DMs toidentify, in an interactive way, a desired compromise between satisfaction degree of the goaland feasibility degree of constraints. Furthermore, a number of decision alternatives havebeen generated under different policies for water resources management, which permitsanalyses of various policy scenarios that are associated with different levels of economicpenalties when the promised water-allocation targets are violated, and thus help DMs toidentify desired water-allocation schemes under uncertainty.

Acknowledgements The research was supported by the Major State Basic Research Development Programof MOST (2005CB724200 and 2006CB403307), and the Natural Science and Engineering Research Councilof Canada. The authors would like to express thanks to the editor and the anonymous reviewers for theirconstructive comments and suggestions.

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