Planning water resources management systems using a fuzzy-boundary interval-stochastic programming method

Advances in Water Resources 33 (2010) 1105–1117

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Advances in Water Resources j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / a d v wa t r e s

Planning water resources management systems using a fuzzy-boundary interval-stochastic programming method Y.P. Li a,⁎, G.H. Huang a, S.L. Nie b a b

Research Academy of Energy and Environmental Studies, North China Electric Power University, Beijing 102206, China College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology, Beijing 100022, China

a r t i c l e

i n f o

Article history: Received 22 July 2009 Received in revised form 21 June 2010 Accepted 27 June 2010 Available online 3 July 2010 Keywords: Fuzzy programming Interval optimization Planning Stochastic analysis Uncertainty Water resources

a b s t r a c t In this study, a fuzzy-boundary interval-stochastic programming (FBISP) method is developed for planning water resources management systems under uncertainty. The developed FBISP method can deal with uncertainties expressed as probability distributions and fuzzy-boundary intervals. With the aid of an interactive algorithm woven with a vertex analysis, solutions for FBISP model under associated α-cut levels can be generated by solving a set of deterministic submodels. The related probability and possibility information can also be reflected in the solutions for the objective function value and decision variables. The developed FBISP is also applied to water resources management and planning within a multi-reservoir system. Various policy scenarios that are associated with different levels of economic consequences when the pre-regulated water-allocation targets are violated are analyzed. The results obtained are useful for generating a range of decision alternatives under various system conditions, and thus helping decision makers to identify desired water resources management policies under uncertainty. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction The constantly increasing demand for water in terms of both sufficient quantity and satisfied quality has forced planners to contemplate comprehensive, complex and ambitious plans for water resources management systems. In recent decades, water shortage, flood event, unreliable water supply, and poor water quality have led to a variety of adverse impacts on social-economic development and human life. One of the major reasons for these disasters is the lack of efficient, equitable and sustainable water-resources management as well as effective policy instructions from decision makers. Consequently, effective planning of water resources management is important for developing regional and/or national socio-economic sustainability. Previously, a large number of methods were developed for allocating and managing water resources in more efficient and sustainable ways [2,5,6,17,23,24,26,32,37,42]. For example, Slowinski [43] proposed an interactive fuzzy multiobjective linear programming method and applied it to water supply planning; Huang [19] proposed an interval-parameter programming (IPP) method for dealing with uncertainties expressed as interval numbers in a water resources management system. Bender and Simonovic [3] proposed a fuzzy compromise approach to water resources planning under imprecision uncertainty. Jairaj and Vedula [22] optimized a multi-reservoir system

⁎ Corresponding author. Tel.: +86 10 5197 1215; fax: +86 10 5197 1255. E-mail address: [email protected] (Y.P. Li). 0309-1708/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.advwatres.2010.06.015

through using fuzzy mathematical programming (FMP) technique, where uncertainties existing in reservoir inflows were treated as fuzzy sets. Faye et al. [15] proposed a fuzzy approach for the shortterm of water resource systems under uncertainty. Lee and Chang [28] proposed an interactive fuzzy approach for planning a stream water resources management system that involved vague and imprecise information. Summarily, FMP is effective in dealing with decision problems under fuzzy goal and constraints and handling ambiguous coefficients in the objective function and constraints; however, it has difficulties in tackling uncertainties expressed as probabilistic distributions in a non-fuzzy decision space [21,35]. IPP can handle uncertain parameters that are expressed as intervals with known lower and upper bounds, but unknown membership or distribution functions; nevertheless, when some right-hand-side parameters have wide intervals, the IPP model may not have a feasible solution [18]. As a result, a number of researchers tackled uncertainties in water resources management problems through stochastic programming approaches [16,20,30,31,36,39,41,44–46,49]. For example, Stedinger and Loucks [44] proposed a stochastic dynamic programming model to calculate single reservoir operation, where optimal reservoir operating policies which are subject to reliability constraints could be derived. Pereira and Pinto [39] proposed a stochastic optimization approach for the planning of a multi-reservoir hydroelectric system under uncertainty, through associating a given probability to each of a range of inputs that occurred at different stages of an optimization horizon. Feiring and Sastri [16] advanced a stochastic programming model for planning of a water resources management system with a dual-purpose of generating electricity and supplying water for

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Y.P. Li et al. / Advances in Water Resources 33 (2010) 1105–1117

agricultural irrigation. Huang and Loucks [20] developed an inexact two-stage stochastic programming method for water resources management, which could tackle uncertainties expressed as both probability distributions and intervals and account for economic penalties due to infeasibility. Watkins Jr et al. [46] proposed a scenario-based multistage stochastic programming model for planning water supplies from highland lakes. By explicitly considering a number of inflow scenarios, the stochastic model could help determine a contract for water delivery in the coming year. Li et al. [30] proposed an inexact multistage stochastic programming method for water resources management under uncertainty; this method could deal with uncertainties expressed as random variables and interval values through constructing a set of scenarios that were representative for the universe of possible outcomes. In general, multistage stochastic programming (MSP) approach permitted modified decisions in each time stage based on the real-time realizations of uncertain system conditions [1,4,30]. The uncertain information in a MSP was often modeled through a multilayer scenario tree [13]. However, MSP had a difficulty in dealing with uncertain parameters when their probabilistic distributions were not available; moreover, even if these distributions were known, reflection of them in large-scale stochastic models could be extremely challenging [4,14]. In fact, in practical water resources management systems, uncertainties exist in many impact factors and system components such as available resources, water demands/supplies, related cost/benefit coefficients, sustainability requirements, and policy regulations. Some uncertainties can be quantified as probabilities while the others may exist as fuzzy membership functions and/or discrete intervals. For example, stream inflows may be expressed as several random variables with known probabilities; also, the economic data of benefit and cost may not be available as deterministic values, and they may be acquired as interval values. It may often be difficult for a planner to promise a deterministic water-allocation target to user when the available water resources are uncertain. Moreover, the system may possess multiple tributaries and multiple reservoirs where uncertainties can lead to interactive and dynamic complexities in terms of water allocation and flood diversion over a multistage context. The system may face both problems of insufficient capacity to retain the surplus water during the high-flow season as well as to satisfy the demand during the low-flow season (e.g. a serious water shortage could occur when flows are continuously low; while flooding disaster could happen when flows are continuously high exceeding the retention capacities of tributaries and/ or reservoirs). An effective approach that cannot only address the above uncertainties and complexities but also hedge against both drought and flooding is desired. Therefore, the objective of this study aims to develop a fuzzyboundary interval-stochastic programming (FBISP) method in response to the above challenges. The FBISP will incorporate techniques of interval-parameter programming (IPP), fuzzy programming (FP), and multistage stochastic programming (MSP) to deal with uncertainties expressed as probability distributions and fuzzy-boundary intervals (i.e. the lower and upper bounds of some intervals may rarely be acquired as deterministic values, and they may be fuzzy in nature). A case study will then be provided for demonstrating how the developed method will support the planning for water resources management within a multi-tributary, multi-reservoir and multiperiod context. The results obtained can help water resources managers identify desired alternatives against water shortage and flood control with a maximized economic objective. The paper will be organized as follows: Section 2 describes the development process of the FBISP; Section 3 provides a case study of water resources management planning; Section 4 presents result analysis and discussion; Section 5 draws some conclusions and extensions; the detailed solution method for solving FBISP is introduced in Appendix A.

2. Methodology The problem, whose coefficients in the objective and constraints are ambiguous and can be expressed as possibility distributions, can be formulated as a fuzzy programming (FP) model as follows: n

Max ˜ f = ∑ ˜ cj x j

ð1aÞ

j=1

subject to: n

∑ a˜ij xj ≤ ˜ bi ;

j=1

xj ≥0;

i = 1; 2; ⋯; m

j = 1; 2; ⋯; n

ð1bÞ ð1cÞ

˜ are fuzzy cj , ˜ aij and b where xj (j = 1, 2, …, n) are decision variables; ˜ i coefficients of the objective and constraints. The possibility distributions of fuzzy parameters can
be characterized  as fuzzy sets. For aij =  a ij ; aij1 ; aij2 ; aij can be presented as example, fuzzy parameter ˜ a triangular fuzzy set when aij1 = aij2, or a trapezoidal fuzzy set when aij1 b aij2. A fuzzy set (Ã) in X can be defined as a set of ordered pairs of à = {x, μÃ(x)|x ∈ X}, where μÃ(x)is named the membership function or grade of membership [50]. The μÃ(x) value ranges from 0 to 1, where 1 represents full membership and 0 denotes non-membership. The closer μÃ(x) is to 1, the more likely it is that an element x belongs to Ã; conversely, the closer μÃ(x) is to 0, the less likely it is that x belongs to à [27,48,50]. Application of the extension principle to fuzzy sets can be viewed as its extension to α-cuts when the membership functions are continuous [10,29]. An α-cut can be defined as the set of elements that belong to fuzzy set à at least to the degree of α, and this degree is also called the degree of confidence (or the degree of plausibility) [11,12]. The vertex method based on α-cut analysis is useful for dealing with fuzzy sets [8–10]. Through using the α-cut concept, each fuzzy variable characterized by a convex membership function can be converted into a group of intervals with various α-cut levels. Then, intervals with the same α-cut level from all fuzzy variables can be processed through interval analysis, resulting in an interval function associated with an α-cut level. The detailed definitions related to fuzzy vertex analysis can be found in a number of literatures [7,9,25,29,34]. In an interval-parameter programming (IPP) model, uncertain parameters are expressed as intervals without any distributional information that is always required in fuzzy and stochastic programming. The IPP allows the interval information to be directly communicated into the optimization process and resulting solution [18]. An IPP model can be written as follows: Max f

F

n

F F

= ∑ cj xj

ð2aÞ

j=1

subject to: n

F F

F

∑ aij xj ≤bi ; i = 1; 2; ⋯; m

ð2bÞ

j=1

F

xj ≥0;

∀j

ð2cÞ

± ± where c± j , aij and bi form sets of interval values with deterministic lower and upper bounds; the ‘−’ and ‘+’ superscripts represent the lower and upper bounds of parameters/variables, respectively. An interactive solution algorithm was proposed by Huang et al. [18] to solve the above problem through analyses of the interrelationships between the parameters and the variables and between the objective function and the constraints. Therefore, when the objective is to be

Y.P. Li et al. / Advances in Water Resources 33 (2010) 1105–1117

maximized, a submodel corresponding to f+ can be firstly formulated ± as follows (assume that b± i N 0, and f N 0): Max f

þ

k1

þ þ

= ∑ cj x j + j=1

n

þ −

∑

j = k1 + 1

cj x j

ð3aÞ


 þ þ − þ jaij j Sign aij xj ≤ bi ; ∀i

ð3bÞ

subject to: k1
 − − þ ∑ jaij j Sign aij xj +

j=1 þ

xj ≥0;

n

∑

j = k1 +1

j = 1; 2; ⋯; k1

ð3cÞ

−

xj ≥0; j = k1 + 1; k1 + 2; ⋯; n

ð3dÞ

where x+ j (j=1, 2, …, k1) are lower bounds of interval variables with positive coefficients in the objective function; x− j (j=k1 +1, k1 +2, …, n) are upper bounds of interval variables with negative coefficients. Solutions − + of x+ jopt (j=1, 2, …, k1), xjopt (j=k1 +1, k1 +2, …, n), and fopt can be obtained from submodels (3a)–(3d). Thus, the submodel corresponding to f− can be formulated as follows: Max f

−

k1

− −

= ∑ cj xj + j=1

n

∑

j = k1 +1

− þ

± ± where a± rj , bi and c′j are a set of interval numbers with deterministic F ˜ F F ˜ lower and upper bounds; a˜ ij , br and cj are a set of intervals with ˜ F ˜F would be fuzzy lower and upper bounds; assume that b± i , br and f greater than zero. Models (6a)–(6d) can deal with uncertainties (presented as intervals and fuzzy sets) existing in coefficients of the objective function as well as left- and right-hand sides of the constraints. However, it has difficulties in tackling uncertainties expressed as random variables in a non-fuzzy decision space [21]; moreover, it is lack of linkage to economic consequences of violated policies preregulated by authorities through taking recourse actions in order to correct any infeasibilities. MSP cannot only handle uncertainties expressed as random variables in non-fuzzy sets but also provide an effective linkage between the pre-regulated policies and the associated economic implications caused by improper policies. In MSP, the initial action is called the first-stage decision, and the corrective one is named the recourse decision. The first-stage decisions have to be made before further information of initial system uncertainties is revealed, whereas the recourse decisions are allowed to adapt to this information [14]. Generally, a multistage stochastic linear programming model can be formulated as follows:

ð4aÞ

cj x j

1107

T

Max f = ∑

n1

n2

Kt

!

∑ cjt xjt − ∑ ∑ ptk djtk yjtk

ð7aÞ

r = 1; 2; ⋯; m1 ; t = 1; 2; ⋯; T

ð7bÞ

ˆ itk ; i = 1; 2; ⋯; T; k = 1; 2; ⋯; Kt ∑ aijt xjt + ∑ a′ijt yjtk ≤ w

ð7cÞ

xjt ≥0;

ð7dÞ

t =1

j=1

j=l k=1

subject to:
 þ − ∑ jaij j Sign aij xj + k1

þ

j=1

n

∑

j = k1 +1


 − þ − jaij j Sign aij xj ≤bi ; −

subject to: ∀i

ð4bÞ

n1

∑ arjt xjt ≤brt ;

j=1 −

þ

0≤xj ≤xjopt ; þ

j = 1; 2; ⋯; k1

−

xj ≥xjopt ; j = k1 + 1; k1 + 2; ⋯; n:

ð4cÞ ð4dÞ

Through solving the two submodels, interval solutions for the ± decision variables (x± jopt) and the objective function value (fopt) can be obtained as follows: h i F − þ þ − xjopt = xjopt ; xjopt ; xjopt ≥xjopt ; j = 1; 2; ⋯; n

ð5aÞ

h i F − þ þ − fopt = fopt ; fopt ; fopt ≥fopt

ð5bÞ

n

j=1


   c˜j + c′j Þxj

n

ð6aÞ

ð6bÞ

j=1 n

F F F ˜ ∑ aij xj ≤ bi ; i = 1; 2; ⋯; m2

ð6cÞ

j=1 F

xj ≥0;

∀j

j=1

j=1

j = 1; 2; ⋯; n1 ; t = 1; 2; ⋯; T

yjtk ≥0;

j = 1; 2; ⋯; n2 ; t = 1; 2; ⋯; T; 1; 2; ⋯; Kt

ð7eÞ

where ptk is probability of occurrence for scenario k in period t, with Kt

k=1

subject to: F F F ˜ ∑ arj xj ≤ br ; r = 1; 2; ⋯; m1

n2

ptk N 0 and ∑ ptk = 1; cjt are coefficients of first-stage variables (xjt)

The IPP can tackle uncertainties expressed as intervals with known lower and upper bounds. However, in many real-world problems, the lower and upper bounds of some interval parameters can rarely be acquired as deterministic values [32,47]. Instead, they may often be given as subjective information that can only be expressed as fuzzy sets; this leads to dual uncertainties, such dual uncertainties cannot be addressed through the conventional IPP and FP methods. As a result, techniques of IPP and FP will be coupled in a general framework to handle such complexities; this leads to a hybrid fuzzy-boundary interval programming model as follows:  Max f˜ = ∑

n1

ð6dÞ

in the objective function; djtk are coefficients of recourse variables (yjtk) in the objective function; arjt and aijt are coefficients of xjt in ′ are coefficients of yjtk in constraint i; ˆ witk is constraints r and i; aijt random variable of constraint i, which is associated with probability level ptk; Kt is number of scenarios in period t, with the total being T

K = ∑ Kt . In models (7a)–(7e), the decision variables are divided t =1

into two subsets: those that must be determined before the realizations of random variables are disclosed (i.e. xjt), and those (recourse variables) that can be determined after the realized random-variable values are available (i.e. yjtk). MSP can tackle uncertainties with probabilistic specifications for random variables. However, in many real-world problems, it is often associated with difficulties in acquiring probability distribution for a random event when available data are not enough; even if the probability distributions are available, it could be difficult to reflect them in large-scale stochastic models [4,20,30]. On the other hand, many system parameters are highly uncertain and their interrelationships can be extremely complicated. For example, uncertainties may exist in both left- and right-hand sides of the constraints as well as coefficients of the objective function; some uncertainties may be expressed as random variables; at the same time, some random events can only be quantified as discrete intervals with fuzzy boundaries, leading to multiple uncertainties presented as different formats in the system's components. Therefore, when multiple uncertainties presented as complex formats, one potential approach

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Y.P. Li et al. / Advances in Water Resources 33 (2010) 1105–1117

for handling such complexities is to incorporate techniques of IPP, FP and MSP; this will lead to a fuzzy-boundary interval stochastic linear programming (FBISP) model as follows: n

T

1  Max f˜ = ∑ ∑

j=1 t=1




Kt
 n2  F T  ′F F ˜  ′ c˜ jt + cjt xjt − ∑ ∑ ∑ ptk djtk + djtk yjtk ð8aÞ j=1 t=1 k=1

subject to: n

1 F F F ˜ ∑ a˜ rt xjt ≤ brt ;

j=1

r = 1; 2; ⋯; m1 ; t = 1; 2; ⋯; T

n1 n2
 ˜ F F F F ′F ˆ ∑ aijt xjt + ∑ a˜ ijtk yjtk ≥ witk ;

j=1

ð8bÞ

ð8cÞ

j=1

i = 1; 2; ⋯; m2 ; t = 1; 2; ⋯; T; k = 1; 2; ⋯; Kt F

xjt ≥0; F

yjtk ≥0;

j = 1; 2; ⋯; n1 ; t = 1; 2; ⋯; T

ð8dÞ

j = 1; 2; ⋯; n2 ; t = 1; 2; ⋯; T; k = 1; 2; ⋯; Kt

ð8eÞ

˜ F ˆ where w itk are a set of discrete random variables and, at the same time, some random events can only be quantified as dual intervals with fuzzy boundaries. In this study, two solution methods (i.e. based on risk prone and risk adverse, respectively) are proposed for facilitating computations of the FBISP model with the aid of an interactive algorithm woven with a fuzzy vertex analysis. In the first solution method (i.e. risk þ prone), a set of submodels corresponding to f˜ (i.e. best-case submodels) can be first solved and, then a set of submodels − corresponding to f˜ can be formulated based on the solutions from the upper-bound submodel. In the second solution method (i.e. risk adverse), reversing the solution process to first solve the lower-bound submodels (i.e. worst-case submodels) and then solve the upperbound submodels. The detailed solution method for solving models (8a)–(8e) is presented in Appendix A. Fig. 1 shows the schematic of the FBISP model as well as its solution processes. It is indicated that FBISP is based on optimization techniques namely MSP, FP and IPP. Each technique has a unique contribution in enhancing the model's capability in dealing with uncertainties presented as multiple formats, such that robustness of the optimization effort can be enhanced.

curtailed, resulting in penalties to the local economy [35]. On the other hand, if the allocation targets are regulated too low, the corresponding policy may result in less water shortage and thus lower penalty but, at the same time, high opportunity loss (more waste of resources) would occur when the water availability is enough. Particularly, when the flow levels are continuously high over multiple periods while the allocation targets are relatively low, more surpluses in the reservoirs would be generated. When the surplus exceeds the reservoirs' storage capacity, spill would occur, which might potentially lead to a flooding event. Under such a condition, a water diversion project has to be undertaken to avoid flooding losses; this may result in a raised diversion cost. Summarily, the challenges for the study problem include: (i) how to identify the desired water-allocation target with a minimized risk of economic penalty and opportunity loss, (ii) how to obtain an optimized water-allocation plan with a maximized system benefit over a multi-period planning horizon, and (iii) how to generate an optimized flood-diversion scheme with sound timing and sizing considerations. Therefore, based on the developed FBISP method, the study problem can be formulated as follows:

Max f

T

F

F

K1t

T

F

= ∑ NBt Xt − ∑ t =1

K2t

F

F

ð9aÞ

∑ ∑ ptk1 ptk2 PEt Ytk1 k2

t = 1 k1 = 1 k2 = 1

T

−∑

K1t K2t
 F F F F ∑ ∑ ptk1 ptk2 FCt Ztk1 k2 + VCt Wtk1 k2

t = 1 k1 = 1 k2 = 1

subject to: 1. Constraints of water-mass balance ˜ F F F F F ˆ Rtk1 = Stk1 + Q tk1 −E1t −Sðt + 1Þk1 ;

F

a F

E1t = A1 e1t

F

F

F Stk + SF ðt + 1Þk1 1



ð9bÞ

! 0 F

+ A1 e1t

ð9cÞ

 ˜ F F F F Qˆ tk1 + Rtk1 −E2t −Sðt + 1Þk1 k2 ;

ð9dÞ

2

Rtk1 k2 = Stk1 k2 +

t

∀t; k1 = 1; 2;⋯; K1

t

t

∀t; k1 = 1; 2;⋯; K1 ; k2 = 1; 2;⋯; K2 3. Case study The following water resources management problem will be used to demonstrate the applicability of the developed FBISP method. An authority is charged with delivering water to a municipality to meet demands for regional socio-economic development. In the study region, there are two tributaries and two reservoirs that supply water to the municipality (as shown in Fig. 2). Water availabilities and demands are highly uncertain, and the relevant allocation plan involves in multiple periods (i.e. with dynamic feature). The relevant allocation decisions have to be made at each time stage under multiple uncertainties. The authority desires to identify an optimal allocation target with a maximized economic benefit over the planning horizon. Variations in water-allocation targets could not only reflect different policies for managing the water resources but also lead to different economic implications (e.g., probabilistic penalty, opportunity loss, and diversion cost caused by improper policies) under uncertainty. For example, when the allocation targets are regulated too high, more water must be supplied to the municipality at each time stage; however, due to water availabilities during the planning horizon are highly uncertain, the targeted flows may not be delivered when stream inflow level is low. When the promised water cannot be satisfied, either the water must be obtained from alternative and more expensive sources or the demand must be

F

a F

E2t = A2 e2t

!

StkF1 k2 + SF ðt + 1Þk1 k2

0 F

ð9eÞ

+ A2 e2t

2

2. Constraint of available water F

F

F

F

t

t

Xt −Ytk1 k2 + Wtk1 k2 = Rtk1 k2 ; ∀t; k1 = 1; 2;⋯; K1 ; k2 = 1; 2;⋯; K2 ð9fÞ 3. Constraints of reservoir capacity F

F

t

Stk1 ≤RSC1 ; ∀t; k1 = 1; 2;⋯; K1 F

F

ð9gÞ t

t

Stk1 k2 ≤RSC2 ; ∀t; k1 = 1; 2;⋯; K1 ; k2 = 1; 2;⋯; K2

ð9hÞ

4. Constraints of reserved storage requirement F

F

t

Stk1 ≥RSV1 ; ∀t; k1 = 1; 2;⋯; K1 F

F

ð9iÞ

t

t

Stk1 k2 ≥RSV2 ; ∀t; k1 = 1; 2;⋯; K1 ; k2 = 1; 2;⋯; K2

ð9jÞ

Y.P. Li et al. / Advances in Water Resources 33 (2010) 1105–1117

1109

Fig. 1. The framework of the FBISP model.

7. Non-negative and technical constraints

5. Constraints of water allocation target F

min max D˜ ≤X ≤ D˜ ; ∀t t t

ð9kÞ

6. Constraints of surplus-flow diversion ZtkF1 k2

F

f = 0; if otherwise

ð9lÞ t

∀t; k1 = 1; 2;⋯; K1 ; k2 = 1; 2;⋯; K2 F

F

Wtk1 k2 ≤Mk1 k2 Ztk1 k2 ;

t

F

Wtk1 k2 ≥ 0;

= 1; if flood diversion ; is undertaken

t

F

t

∀t; k1 = 1; 2;⋯; K1 ; k2 = 1; 2;⋯; K2

ð9mÞ

t

t

Xt ≥ Ytk1 k2 ≥ 0; ∀t;k1 = 1; 2;⋯; K1 ; k2 = 1; 2;⋯; K2 t

t

∀t; k1 = 1; 2;⋯; K1 ; k2 = 1; 2;⋯; K2

where: f± = net system benefit over the planning horizon ($); t = time period, and t = 1, 2, …, T; A01 = storage-area coefficient for reservoir 1 (m2); A02 = storage-area coefficient for reservoir 2 (m2);

Fig. 2. The schematic of the study system (symbols “T1”, “T2”, “R1” and “R2” denote “tributary 1”, “tributary 2”, “reservoir 1” and “reservoir 2”, respectively).

ð9nÞ

ð9oÞ

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Y.P. Li et al. / Advances in Water Resources 33 (2010) 1105–1117

Aa1 = area (per unit of active storage volume) above A01 (m− 2); Aa2 = area (per unit of active storage volume) above A02 (m− 2); ˜ Dmin = minimum amount of water demand for the municipality in t period t (m3); ˜ Dmax = maximum water demand for the municipality in period t t (m3); e± 1t = average evaporation rate for reservoir 1 in period t (m); e± 2t = average evaporation rate for reservoir 2 in period t (m); 3 E± 1t = evaporation loss of reservoir 1 in period t (m ); ± E2t = evaporation loss of reservoir 2 in period t (m3); 6 FC± t = fixed cost for surplus-flow diversion in period t ($10 ); t K1 = number of flow scenarios for tributary 1 in period t; Kt2 = number of flow scenarios for tributary 2 in period t; Mtk1k2 = variable upper bounds for surplus-flow diversion in period t under scenarios k1 and k2, which is assumed to be sufficiently large; 3 NB± t = net benefit per unit of water allocated in period t ($/m ); ± PEt = penalty per unit of water not delivered in period t ($/m3), and PEt N NBt; ptk1 = probability of occurrence of scenario k1 (for stream 1) in t K1

period t, with ptk1 N 0 and ∑ ptk1 = 1; k1 = 1

ptk2 = probability of occurrence of scenario k2 (for stream 2) in t K2

period t, with ptk2 N 0 and ∑ ptk2 = 1; k2 = 1

˜  = inflow level into stream 1 in period t under scenario k1 (m3); ˆ Q tk1

3 ˜ ˆ Qtk2 = inflow level into stream 2 in period t under scenario k2 (m );

Rtk±1 = release flow from reservoir 1 in period t under scenario k1 (m3); R± tk1k2 = release flow from reservoir 2 in period t under scenarios k1 and k2 associated with joint probabilities of ptk1ptk2(m3); 3 RSC± 1 = storage capacity of reservoir 1 (m ); ± RSC2 = storage capacity of reservoir 2 (m3); RSV± 1 = reserved storage level for reservoir 1, which could prevent storage in reservoir 1 from dropping below a given level at any time (m3); 3 RSV± 2 = reserved storage level for reservoir 2 (m ), which could prevent storage in reservoir 2 from dropping below a given level at any time (m3); ± = storage level in reservoir 1 in period t under scenario k1 (m3); Stk 1 ± Stk1k2 = storage level in reservoir 2 in period t under scenarios k1 and k2 (m3); 3 VC± t = variable cost for surplus flow diversion in period t ($/m ); ± Wtk1k2 = amount of surplus flow to be diverted in period t under scenarios k1 and k2 (m3); X± t = water allocation target that is promised to the municipality in period t (m3); ± = shortage level by which the water-allocation target is not Ytk 1k2 met under scenarios k1 and k2, which is associated with joint probabilities of ptk1ptk2 (m3); Z± tk1k2 = binary variables that are used for identifying whether a surplus-flow-diversion action needs to be undertaken in period t under scenarios k1 and k2. In the above model, integer programming is introduced into the FBISP for dealing with flood diversion problem under stochastic conditions. It is assume that, if the system requires additional capacity for flood diversion at the beginning of a particular time period, the construction project has to be completed at the start of this period. Fixed-charge cost functions are used to reflect the economies of scale in the diversion cost. The fixed-charge cost is linked to the flood-

diversion action, while the variable cost is linked to the amount of flood diverted [1,33]. Moreover, random flows can be reflected through constructing a set of scenarios that are representative for the universe of water-availability conditions. Fig. 3 shows the structure of the scenario tree for the study system, with a one-toone correspondence between the previous random variable and one of the nodes (states of the system) in each time stage. 258 scenarios will be generated for the two tributaries associated with different joint probabilities over the planning horizon (i.e., 6 scenarios in period 1, 36 scenarios in period 2, and 216 scenarios in period 3). Consequently, water allocation and surplus-flow-diversion schemes under all of flow scenarios can be generated. A decision can be made at each stage in a real-time manner based on both the progressively acquired information about the actual realizations of the random variables and the decisions in the previous stages; this allows corrective actions to be taken dynamically in reference to the preregulated policies, and can thus help minimize the penalties and/or diversion costs. There are three assumptions for the above modeling formulation. The random variables (i.e. random flows) are assumed to take on discrete distributions, such that the study problem can be solved through linear programming method; second, the two random variables are assumed to be mutually independent, such that the probabilistic shortages and surplus correspond to joint probabilities; third, evaporation is assumed to be a linear function of average reservoir storage in each time period. However, when water resources management problems are complicated by the need to take adequate account of persistence in hydrological records, conditional probabilities may need to be handled for quantifying water availability, particularly over a multi-reservoir context. This may lead to nonlinearity in system responses and raise major problems for the linear assumption in the above model. Table 1 provides the flow levels of the two tributaries and water resources demands from the municipality, as well as the associated probabilities and α-cut levels during the three periods. Obviously, the water availabilities will fluctuate dynamically due to the varying river inflows. Table 2 provides the water resources demand from the municipality. Shortage in water supply may be generated if the targeted water is not delivered; on the other hand, high stream inflows may lead to a raised surplus and thus mandate a decision of water diversion. Table 3 presents the economic data of water allocation and surplus-flow

Fig. 3. Scenario tree for the study system.

Y.P. Li et al. / Advances in Water Resources 33 (2010) 1105–1117

1111

Table 1 Tributary flows (106 m3).

Low inflow of tributary 1 (probability = 0.2)

Medium inflow of tributary 1 (probability = 0.6)

High inflow of tributary 1 (probability = 0.2)

Low inflow of tributary 2 (probability = 0.4)

High inflow of tributary 2 (probability = 0.6)

α-cut level

t=1

t=2

t=3

α = 0.2 α = 0.5 α = 0.8 α=1 α = 0.2 α = 0.5 α = 0.8 α=1 α = 0.2 α = 0.5 α = 0.8 α=1 α = 0.2 α = 0.5 α = 0.8 α=1 α = 0.2 α = 0.5 α = 0.8 α=1

[[59.39, 62.81], [68.92, 73.08]] [[60.15, 62.05], [69.7, 72.3]] [[60.91, 61.29], [70.48, 71.52]] [61.1, 71.0] [[96.2, 102.6], [111.14, 118.66]] [[97.4, 101.4], [112.55, 117.25]] [[98.6, 100.2], [113.96, 115.84]] [99.4, 114.9] [[149.7, 160.9], [171.96, 182.84]] [[151.8, 158.8], [174, 180.8]] [[153.9, 156.7], [176.04, 178.76]] [155.3, 177.4] [[32.06, 34.94], [39.06, 43.54]] [[32.6, 34.4], [39.9, 42.7]] [[33.14, 33.86], [40.74, 41.86]] [33.5, 41.3] [[67.82, 72.78], [78.68, 84.12]] [[68.75, 71.85], [79.7, 79.7]] [[69.68, 70.92], [80.72, 82.08]] [70.3, 81.4]

[[65.68, 70.32], [75.42, 80.38]] [[66.55, 69.45], [76.35, 79.45]] [[67.42, 68.58], [77.28, 78.52]] [68.0, 77.9] [[100.12, 106.68], [113.74, 119.06]] [[101.35, 105.45], [114.8, 118]] [[102.58, 104.22], [115.86, 116.94]] [103.4, 116.4] [[157.04, 168.56], [176.54, 188.06]] [[159.2, 166.4], [178.7, 185.9]] [[161.36, 164.24], [180.86, 183.74]] [162.8, 182.3] [[33.78, 36.82], [40.42, 43.78]] [[34.35, 36.25], [41.05, 43.15]] [[34.92, 35.68], [41.68, 42.52]] [35.3, 42.1] [[73.32, 78.28], [83.32, 88.88]] [[74.25, 77.35], [84.55, 88.65]] [[75.18, 76.42], [87.42, 87.42]] [75.8, 86.1]

[[55.16, 59.64], [62.4, 68.2]] [[56, 58.8], [64.95, 67.45]] [[56.84, 57.96], [65.7, 66.7]] [57.4, 66.2] [[90.96, 97.04], [103.14, 109.86]] [[92.1, 95.9], [104.4, 108.6]] [[93.24, 94.76], [105.66, 107.34]] [94.0, 106.5] [[144.88, 155.92], [166.26, 177.14]] [[146.95, 153.85], [168.3, 175.1]] [[149.02, 155.92], [166.26, 177.14]] [150.4, 171.7] [[29.44, 32.16], [35.36, 38.24]] [[29.95, 31.65], [35.9, 37.7]] [[30.46, 31.14], [36.44, 37.16]] [30.8, 36.8] [[65.06, 69.54], [74.86, 78.54]] [[65.9, 68.7], [75.55, 77.85]] [[66.74, 67.86], [76.24, 77.16]] [67.3, 76.7]

In this study, two solution methods were proposed for solving models (9a)–9o), which are based on risk prone and risk adverse respectively. The resulting system benefits would be $[7562.1, 16,527.6] × 10 6 from solution method (1a–1c) and $[7967.0, 15,964.5] × 106 from solution method (2a–2c) when α = 1 (i.e. with the highest possibility degree). The system benefit under α = 1 presents in a single interval; this is because the triangular membership functions for fuzzy sets are used in this study. The interval solution can be easily interpreted for generating decision alternatives, where upper-bound system benefit is associated with more advantageous conditions (i.e. associated with upper-bound inflows, upper-bound benefit coefficients, lower-bound cost coefficients, upper-bound reservoir capacities, lower-bound reserved storage requirements) while the lower-bound one corresponds to the demanding conditions. Solution method (1) (i.e. risk prone) provides a wider interval for objective function value (i.e. a wider interval associated with a higher uncertainty level); solution method (2) (i.e. risk adverse) provides a narrower interval for objective function value but, it may lead to some information loss (i.e. the system is incapable of achieving the highest benefit under advantageous conditions). Tables 4, 5 and A1 present the results for shortage and allocation obtained from the two solution methods when α = 1. Since random inflows were conceptualized into a multilayer scenario tree, 258 flow

scenarios were generated for the two tributaries associated with different joint probabilities over the planning horizon (as shown in Fig. 3). Shortage in water supply would be generated if the pre-regulated targets were not satisfied (i.e. shortage = targeted value − available inflow). Under such a situation, the actual water allocation would be the difference between the pre-regulated target and the probabilistic shortage (i.e. allocation = target − shortage). For example, from solution method (1), the optimized targets would be [141.6, 148.4]× 106 m3 in period 1, [164.4, 189.7] × 10 6 m3 in period 2, and [157.5, 178.8] × 106 m3 in period 3. Under the worst-shortage condition (i.e. when inflows of the two tributaries are both low during the entire planning horizon), the shortages would be [40.1, 67.0]× 106 m3 in period 1, [73.2, 73.2] × 106 m3 in period 2, and [79.3, 79.3] × 106 m3 in period 3. The actual allocations would be [74.6, 108.3] × 106 m3 in period 1, [91.2, 116.5]× 106 m3 in period 2, and [78.2, 99.5]× 106 m3 in period 3; the total of allocated water would be 244.0 to 324.3 × 106 m3 in the three periods. However, the total water demand over the planning horizon would be [457.8, 516.9] × 106 m3 (under α = 1), indicating a serious shortage in water supply when inflows are low. Thus, the municipality would have to obtain water from other sources to satisfy its essential demands. The solutions for water shortage and allocation under the other scenarios can be similarly interpreted based on the results presented in Tables 4, 5 and A1. Different solution methods imply different risk attitudes of decision makers considering system uncertainties, leading to varied solutions for pre-regulated target and probabilistic shortage. For example, the optimized targets from solution method (2a–2c) would be [141.6, 144.8] × 106 m3 in period 1, [155.2, 176.6] × 106 m3 in period 2, and [145.6, 170.3] × 106 m3 in period 3. When inflows of the two tributaries are both low during the entire planning horizon, the shortages would be [50.0, 67.0] × 106 m3, [55.7, 55.7] × 106 m3, and [61.2, 61.2] × 106 m3 in periods 1, 2 and 3, respectively. The actual

Table 2 Water demands (106 m3).

Table 3 Technical and economic data.

diversion. Besides, the storage capacities of reservoirs 1 and 2 are [27.0, 37.0] × 106 and [50.0, 63.0] × 106 m3, respectively; the initial storages in reservoirs 1 and 2 are [19.5, 21.9] × 106 and [27.3, 30.1] × 106 m3, respectively; the reserved storage levels for reservoirs 1 and 2 are [20.0, 24.0] × 106 and [32.5, 39.0] × 106 m3, respectively.

4. Results analysis

Minimum demand level

Maximum demand level

α-cut

t=1

t=2

t=3

0.2 0.5 0.8 1.0 0.2 0.5 0.8 1.0

[134.32, 148.88] [137.05, 146.15] [139.78, 143.42] 141.6 [167.3, 183.3] [170.3, 178.3] [173.3, 177.3] 175.3

[147.28, 163.12] [150.25, 160.15] [153.22, 157.18] 155.2 [181.3, 198.1] [184.45, 194.95] [187.6, 191.8] 189.7

[139.04, 152.16] [141.5, 149.7] [143.96, 147.24] 145.6 [171.42, 185.98] [174.15, 183.25] [176.98, 180.52] 178.7

Net benefit when water demand is satisfied ($/m3) Penalty when pre-regulated water is not delivered ($/m3) Fixed cost for surplus-flow diversion ($106) Variable cost for surplus-flow diversion ($/m3)

t=1

t=2

t=3

[27.6, 33.2]

[32.3, 38.9]

[37.6, 45.3]

[110.0, 130.0] [120.0, 150.0] [130.0, 160.0] [20.0, 25.0]

[23.4, 29.2]

[27.2, 34.0]

[28.5, 34.0]

[32.8, 39.1]

[37.7, 45.2]

1112

Y.P. Li et al. / Advances in Water Resources 33 (2010) 1105–1117

Table 4 Solution of water allocation (106 m3) in period 1 (α = 1). Flow sequence

Solution method (1a–1c) Shortage

Allocation

Solution method (2a–2c) Shortage

Allocation

Best/worst Shortage

Allocation

L-L L-H M-L M-H H-L H-H

[40.1, 67.0] [0, 19.1] [9.6, 31.7] [0, 0] [0, 0] [0, 0]

[74.6, 108.3] [122.5, 148.4] [109.9, 138.8] [141.6, 148.4] [141.6, 148.4] [141.6, 148.4

[50.5, 67.0] [10.4, 19.1] [9.6, 31.7] [0, 0] [0, 0] [0, 0]

[74.6, 94.3] [122.5, 134.4] [109.9, 135.2] [141.6, 144.8] [141.6, 144.8] [141.6, 144.8

(40.1, 67.0) (0, 19.1) (9.6, 31.7) (0, 0) (0, 0) (0, 0)

(108.3, (148.4, (138.8, (148.4, (148.6, (148.4,

74.6) 122.5) 109.9) 141.6) 141.6) 141.6)

Note: Symbol of L-L denotes the available inflows of tributaries 1 and 2 are both low in period 1.

allocations would be [74.6, 94.3] × 106 m3 in period 1, [99.5, 120.9] × 106 m3 in period 2, and [84.4, 109.1] × 106 m3 in period 3. Generally, the results (from the two solution methods) indicate that there would be no shortage when the inflow levels of the two tributaries are medium and high; moreover, the shortage levels may be low under advantageous conditions, while the shortage levels may be raised under demanding conditions. In addition, the shortage and allocation from the best/worst case (under α = 1) are also provided in Tables 4, 5 and A1, while the system benefit from best/worst case are $(7967.0, 16,527.6) × 106. The results from the best/worst case are useful for judging the system's capability to realize the desired goal; however, they could not necessarily construct a set of stable intervals for objective function value and decision variables. In this study, other several α-cut levels (i.e. 0.2, 0.5 and 0.8) were also examined to help investigate the relationship among the targets/ deficits, system benefits, and fuzzy-interval values (i.e. water

demands and availabilities). Tables 6 and 7 present the system benefits under α-cut levels of 0.2, 0.5 and 0.8. For each lower- and − upper-bound submodel (i.e. corresponding to f˜ and f˜þ ), eight cases were generated based on different combinations of the fuzzyboundary intervals. The results indicate that different combinative considerations on the uncertain inputs (under different α-cut levels) would lead to varied solutions for objective function values and decision variables. For example, when α = 0.2, the minimum and maximum values (from solution method 1) would be 5466.5 and 8060.8 million dollars, respectively; they would form the lower0.2 6 bound system benefit (i.e. f− opt = $[5466.5, 8060.8] × 10 ); similarly, the upper-bound system benefit would respectively be 15,925.6 and 0.2 6 17,047.0 million dollars (i.e. f+ opt = $[15,925.6, 17,047.0] × 10 ). Through integration of the lower- and upper-bound values, the system benefit (from solution method 1) would be $[[5466.5, 8060.8], [15,925.6, 17,047.0]] × 106 when α = 0.2. Similarly, the system

Table 5 Solution of water allocation (106 m3) in period 2 (α = 1). Flow sequence

Solution method (1a–1c) Shortage

Allocation

Solution method (2a–2c) Shortage

Allocation

Shortage

Allocation

LL-LL LM-LL LH-LL ML-LL MM-LL MH-LL HL-LL HM-LL HH-LL LL-LH LM-LH LH-LH ML-LH MM-LH MH-LH HL-LH HM-LH HH-LH LL-HL LM-HL LH-HL ML-HL MM-HL MH-HL HL-HL HM-HL HH-HL LL-HH LM-HH LH-HH ML-HH MM-HH MH-HH HL-HH HM-HH HH-HH

[73.2, 73.2] [34.7, 34.7] [0, 0] [59.8, 61.9] [30.2, 30.2] [0, 0] [55.3, 61.9] [0, 18.3] [0, 0] [29.2, 29.2] [0, 0] [0, 0] [15.8, 21.4] [0, 0] [0, 0] [0, 10.4] [0, 0] [0, 0] [73.2,73.2] [34.7, 34.7] [0, 0] [29.3, 50.9] [0, 18.5] [0, 0] [25.7, 50.9] [0, 18.5] [0, 0] [29.2, 29.2] [0, 0] [0, 0] [0, 10.4] [0, 0] [0, 0] [0, 10.4] [0, 0] [0, 0]

[91.2, 116.5] [129.7, 155.0] [164.4, 189.7] [102.5, 129.9] [134.2, 159.5] [164.4, 189.7] [102.5, 134.4] [146.1, 189.7] [164.4, 189.7] [135.2, 160.5] [164.4, 189.7] [164.4, 189.7] [143.0, 173.9] [164.4, 189.7] 164.4, 189.7] [154.0, 189.7] [164.4, 189.7] [164.0, 189.7] [91.2, 116.5] [129.7, 155.0] [164.4, 189.7] [113.5, 160.4] [145.9, 164.4] [164.4, 189.7] [113.5, 164.0] [145.9, 189.7] [164.4, 189.7] [135.2, 160.5] [164.4, 189.7] [164.4, 189.7] [154.0, 189.7] [164.4, 189.7] [164.4, 189.7] [154.0, 189.7] [164.4, 189.7] [164.4, 189.7]

[55.7, 55.7] [20.2, 23.3] [0, 0] [52.7, 52.7] [17.2, 20.3] [0, 0] [33.7, 41.7] [0, 9.3] [0, 0] [11.7, 15.2] [0, 0] [0, 0] [8.7, 12.2] [0, 0] [0, 0] [0, 1.2] [0, 0] [0, 0] [55.7, 55.7] [20.2, 23.3] [0, 0] [22.2, 41.7] [0, 9.3] [0, 0] [22.2, 41.7] [0, 9.3] [0, 0] [11.7, 15.2] [0, 0] [0, 0] [0, 1.2] [0, 0] [0, 0] [0, 1.2] [0, 0] [0, 0]

[99.5, 120.9] [131.9, 156.4] [155.2, 176.6] [102.5, 123.9] [134.9, 159.4] [155.2, 176.6] [113.5, 142.9] [145.9, 176.6] [155.2, 176.6] [140.0, 164.9] [155.2, 176.6] [155.2, 176.6] [143.0, 167.9] [155.2, 176.6] [155.2, 176.6] [154.0, 176.6] [155.2, 176.6] [155.2, 176.6] [99.5, 120.9] [131.9, 156.4] [155.2, 176.6] [113.5, 154.4] [145.9, 176.6] [155.2, 176.6] [113.5, 154.4] [145.9, 176.6] [155.2, 176.6] [140.0, 164.9] [155.2, 176.6] [155.2, 176.6] [154.0, 176.6] [155.2, 176.6] [155.2, 176.6] [154.0, 176.6] [155.2, 176.6] [155.2, 176.6]

(73.2, 55.7) (34.7, 23.3) (0, 0) (59.8, 52.7) (30.2, 20.3) (0, 0) (55.3, 41.7) (0, 9.3) (0, 0) (29.2, 15.2) (0, 0) (0, 0) (15.8, 12.2) (0, 0) (0, 0) (0, 1.2) (0, 0) (0, 0) (73.2, 55.7) (34.7, 23.3) (0, 0) (29.3, 41.7) (0, 9.3) (0, 0) (25.7, 41.7) (0, 9.3) (0, 0) (29.2, 15.2) (0, 0) (0, 0) (0, 1.2) (0, 0) (0, 0) (0, 1.2) (0, 0) (0, 0)

(116.5, (155.0, (189.7, (129.9, (159.5, (189.7, (134.4, (189.7, (189.7, (160.5, (189.7, (189.7, (173.9, (189.7, (189.7, (189.7, (189.7, (189.7, (116.5, (155.0, (189.7, (160.4, (189.7, (189.7, (164.0, (189.7, (189.7, (160.5, (189.7, (189.7, (189.7, (189.7, (189.7, (189.7, (189.7, (189.7,

Note: Symbol of LL-LL denotes the available inflows of tributaries 1 and 2 are both low in periods 1 and 2.

Best/worst

99.5) 131.9) 155.2) 102.5) 134.9) 155.2) 113.5) 145.9) 155.2) 140.0) 155.2) 155.2) 143.0) 155.2) 155.2) 154.0) 155.2) 155.2) 99.5) 131.9) 155.2) 113.5) 145.9) 155.2) 113.5) 145.9) 155.2) 140.0) 155.2) 155.2) 154.0) 155.2) 155.2) 154.0) 155.2) 155.2)

Y.P. Li et al. / Advances in Water Resources 33 (2010) 1105–1117

1113

Table 6 Lower-bound system benefits under α-cut levels of 0.2, 0.5, and 0.8 ($106). Case

Fuzzy combination

Solution method (1a–1c) α = 0.2

Q− ,  tk1

Q− )  tk2

1

, D max , ( D min t t

2

,Dmax , Q− , Q− ) ( D min t t  tk1  tk2 max − , D , Q , Q− ) (Dmin t ―t  tk1  tk2 max − ,D , Q , Q− ) (Dmin t t  tk1 ― tk2

3 4

− , D max , Q− ( D min tk1 , Q tk2 ) t t − − max ( D min , D ,Q , Q tk1 tk2 ) t t

5 6

α = 0.5

Solution method (2a–2c) α = 0.8

α = 0.2

α = 0.5

α = 0.8

7203.6

7184.5

7186.5

7706.1

7775.7

7844.1

7203.6

7184.5

7186.5

7706.1

7775.7

7844.1

5466.5

6173.2

6825.2

6511.0

7211.1

7628.0

5466.5

6173.2

6825.2

6511.0

7211.1

7628.0

8060.8

7853.5

7525.4

8522.5

8328.4

8077.8

8060.8

7853.5

7525.4

8522.5

8328.4

8077.8

7

− , D max ,Q − (Dmin tk1 , Q tk2 ) t t

7710.1

7501.3

7271.5

8049.4

7984.0

7878.9

8

(Dt

− ,Q − tk1 , Q tk2 )

7710.1

7501.3

7271.5

8049.4

7984.0

7878.9

min

max

,Dt

Table 7 Upper-bound system benefits under α-cut levels of 0.2, 0.5, and 0.8 ($106). Case

Fuzzy combination

Solution method (1a–1c)

Solution method (2a–2c)

α = 0.2

α = 0.5

α = 0.8

α = 0.2

α = 0.5

15,946.5

16,189.6

16,442.8

14,875.6

15,292.1

α = 0.8 15,752.2

16,033.4

16,189.6

16,479.4

14,875.6

15,292.1

15,752.2

5

, D max , Q− , Q− ) ( D min t t  tk1  tk2 max þ ,D , Q , Qþ ) ( D min t t  tk1  tk2 min þ , Q , Qþ ) (Dt , D max t  tk1  tk2 min max þ , Q ) (Dt ,Dt , Q þ  tk1  tk2 þ þ min max D t , Q tk1 , Q tk2 ) ( D t ,

16,696.3

16,664.4

16,571.7

15,780.5

15,793.8

6

D min ( , D max ,Q tk1 , Q tk2 ) t t

17,047.0

16,664.4

16,637.4

15,780.5

15,793.8

15,903.7

7

, D max ,Q tk1 , Q tk2 ) t þ þ min max (Dt ,Dt ,Q tk1 , Q tk2 )

16,696.3

16,664.4

16,571.7

15,780.5

15,792.5

15,903.7

17,047.0

16,861.9

16,637.4

15,780.5

15,792.5

15,903.7

1 2 3 4

þ

min

(Dt

8

þ

þ

þ

15,925.6

16,186.8

16,442.8

14,831.7

15,212.8

15,752.1

15,984.9

16,248.4

16,479.2

14,831.7

15,212.8

15,752.1 15,903.7

benefits (from solution method 1) would be $[[6173.2, 7853.5], [16,186.8, 16,861.9]] × 106 under α = 0.5 and $[[6825.2, 7271.5], [16,442.8, 16,637.4]] × 106 under α = 0.8. The solution for the objective-function value under each α-cut level provides four options of maximized system benefit corresponding to different conditions (i.e. different reliability levels). A higher α-cut level would result in a Pþ higher fP but a lower f opt , leading to a narrow interval. In detail, opt Pþ

under α = 0.2, the highest f opt may be achieved under advantageous

conditions; however, the system may encounter the lowest fP under opt

demanding conditions. When α = 0.8, the system would encounter Pþ

the lowest f opt under advantageous conditions; however, the highest would achieve under demanding conditions. fP opt The results also indicate that solution method (2a–2c) would identify a narrower interval for objective function under each case and each α-cut level, compared with the benefit from solution method

(1a–1c). The system benefits (from solution method 2a–2c) would be $[[6511.0, 8522.5], [14,831.7, 15,780.5]] × 10 6 under α = 0.2, $[[7211.1, 8328.4], [15,212.8, 15,793.8]] × 106 under α = 0.5, and $[[7628.0, 8077.8], [15,752.1, 15,903.7]] × 106 under α = 0.8. In general, solution method (1a–1c) could identify the highest system benefit under all of α-cut levels; however, it may be associated with high risk since it is based on an optimistic anticipation for system components. Solution method (2a–2c) could help achieve a narrower range for system benefit with a lower risk since it is based on a conservative anticipation for system components and constraints; however, the system might lose the opportunity of achieving the highest benefit value. Since uncertainties exist in the available flows and the water demands, variations in water-allocation targets could lead to different policies for water allocation and thus create different system benefit. Therefore, identification of an optimal allocation

Fig. 4. Relationship between optimal targets and system benefits (solution method 1).

1114

Y.P. Li et al. / Advances in Water Resources 33 (2010) 1105–1117

Fig. 5. Relationship between optimal targets and system benefits (solution method 2).

target (i.e. the first-stage variable) which could help maximize the system benefit and minimize the probabilistic penalty is very important. Figs. 4 and 5 show the relationship between the optimized targets and the system benefits obtained from solution methods (1a–1c) and (2a–2c). The optimized allocation targets (from solution method 1a–1c) would be (464.2, 463.0, 501.6, 533.2) × 106 m3 (α = 0.2), (456.0, 458.2, 504.8, 523.8) × 106 m3 (α = 0.5), and (451.4, 456.0, 511.8, 520.7) × 106 m3 (α = 0.8). The optimized allocation targets (from solution method 2a–2c) would be (464.9, 424.4, 464.2, 487.6) × 106 m3 (α = 0.2), (456.0, 428.8, 475.8, 483.6) × 10 6 m 3 (α = 0.5), and (447.8, 437.0, 485.4, 492.4) × 106 m3 (α = 0.8). They are four options for water allocation P targets, which correspond to lower-min ( Pf  ), lower-max ( f opt ), P þopt þ upper-min ( Pf opt ), and upper-max ( f opt ) system benefits, respectively.

Although this study is the first attempt for planning a waterresources management system through the developed FBISP method, research extensions still exist. In this study, the FBISP handles uncertainties through constructing a set of scenarios that are representative for the universe of water-availability conditions for two tributaries. With such a scenario-based approach, the resulting mathematical programming model could become too large to be applied to large-scale real-world problems. Moreover, the random variables (i.e. flows of two tributaries) are assumed to take on discrete distributions and to be mutually independent, such that the study problem can be solved through linear programming method. However, conditional probabilities need to be handled for quantifying water availability, particularly for a multi-stream and multi-reservoir system. This may lead to non-linearity in system responses and raise a main challenge for identifying global optimal solution.

5. Conclusions Acknowledgements In this study, a fuzzy-boundary interval-stochastic programming (FBISP) method has been developed for planning water resources management systems under uncertainty. The developed FBISP integrates interval-parameter programming, fuzzy vertex analysis, and multistage stochastic programming techniques. It can deal with uncertainties expressed as probability distributions and fuzzy-boundary intervals (i.e. the lower and upper bounds of some intervals may rarely be acquired as deterministic values, and they may be fuzzy in nature). The FBISP model can be solved based on an interactive algorithm and a vertex analysis approach, solutions under a number of α-cut levels can be generated. The related probability and possibility information can also be included in the solutions for the objective function value and decision variables. The developed method has also been applied to water resources management and planning within a multi-reservoir and multi-period context. It can reflect the dynamics in terms of decisions for water allocation and surplus-flow diversion, through transactions at discrete points of a complete scenario set based on a multilayer scenario tree. The results obtained demonstrate that variations in water-allocation targets could reflect different policies for managing water resources and thus result in varied economic implications under uncertainty. The results are useful for (i) identifying the desired water-allocation target with a minimized risk of economic penalty and opportunity loss, (ii) obtaining an optimized water-allocation plan with a maximized system benefit over a multi-period planning horizon, and (iii) generating an optimized surplus-diversion scheme with sound timing and sizing considerations to minimize flooding event risk.

This research was supported by the Natural Sciences Foundation of China (50979001 and 50775081), the Major State Basic Research Development Program of MOST (2006CB403307), and the Special Water Project of China (2009ZX07104-004). The authors are grateful to the editors and the anonymous reviewers for their insightful comments and suggestions. Appendix A. Solution method for FBISP In the FBISP model, assume that there is no intersection h between F ˜þ the fuzzy sets at the two bounds (e.g. let ˜ brt = ˜ b− rt ; brt  = hh i h ii − − þ þ þ − ˜ ˜ Xb rt ; brt ; Xb rt ; brt , where brt and brt are fuzzy lower and upper F − − bounds of ˜ brt ; Xb rt and brt are the lower- and upper-boundary of b˜− rt ; þ þ b and b are the lowerand upper-boundary of ˜ bþ X rt rt ). This is due to rt satisfy the definition of an interval value that its lower bound should not be larger than its upper bound [18]. Secondly, interval numbers are used to express uncertainties without distribution information. If the fuzzy sets of an interval's lower and upper bounds intersect, then the so-called “interval” is actually described by fuzzy membership functions, such that the interval representation becomes unnecessary [38]. Thirdly, if the fuzzy sets of lower and upper bounds intersect, the interactive algorithm for solving the interval-parameter programming problem cannot be used for solving such a FBISP model. Then, two solution methods (i.e.) are proposed for solving the FBISP model, which are based on risk prone and risk adverse, respectively. In the first solution method (i.e. risk prone), a set of submodels corresponding to ˜ f þ can be first formulated based on the

Y.P. Li et al. / Advances in Water Resources 33 (2010) 1105–1117

interactive algorithm; for each ˜ f þ submodel, take one end point from h i h i h i þ − þ d− bþ each of the fuzzy intervals (i.e.,  cþ rt ; brt ); jtk ; djtk , and  jt ; cjt ,  then, the obtained end points can be combined into an n-array, leading to 2n combinations for n fuzzy sets [7,9]. Through solving 2n + problems, a set of lower-bound objective-function values (f+ 1 , f2 , ⋅ ⋅ ⋅, + þ f2n ) can be obtained. In detail, for each α-cut level, a set of f˜ submodels can be formulated as follows (assume that the right-hand sides and objective are both greater than zero):

f

T

j1

Max ˜ fþ = ∑ ∑

t =1 j=1 Kt T

½½c ; c  + c x þ jt

(

þ jt

hh

j2

− ∑ ∑ ptk ∑ t =1 k=1

′þ jt

j=1

þ jt

n1

+

∑

j = j1 + 1

hh

þ

þ

c jt ; cjt 

i i − − ′− − d jtk ; djtk + djtk yjtk + 

i

i − + c′jtþ xjt

n2

∑

j = j2 +1

hh

subject to: j1

 þ þ þ − ∑ j a rjt jSign  a rjt ; jaþ jSign arjt xjt rjt j=1 n1 h

i h i − þ − − − − xjt ≤  + ∑ j a rjt jSign  a− b rt ; brt ; ∀r; t rjt ; jarjt jSign arjt

½



j = j1 +1

ðA3  bÞ

g

j=1

i i − − ′− þ d jtk ; djtk +djtk yjtk 

)

½

ijtk

j=1 n2

h

∑

+

j1

 þ − − a− a− ∑ ½ j rjt jSign  rjt ; jarjt jSign arjt  xjt j=1 n1 h

i h i þ þ þ þ − þ þ + ∑ j a rjt jSign  a rjt ; jarjt jSign arjt xjt ≤  b rt ; brt ; j = j1+1

−

þ

∀r; t

j=1 n2

+




h

∑

j = j2 +1

þ a ′ijtk 

ðA1  cÞ

j = j1+1

þ − − þ xjt ; xjt ; yjtk ; yjtk

þ

∀t; j = 1; 2; ⋯; j1

ðA3  dÞ

∀t; j = j1 + 1; j1 + 2; ⋯; n1

ðA3  eÞ

−

yjtk ≥ yjtk opt ; ∀t; j = 1; 2; ⋯; j2 ; k = 1; 2;⋯; Kt þ

0 ≤ yjtk ≤ yjtk opt ;

ðA3  fÞ

∀t; j = j2 + 1; j2 + 2; ⋯; n2 ; k = 1; 2;⋯; Kt ðA3  gÞ

− − where x+ jt opt (j = 1, 2, …, j1), xjt opt (j = j1 + 1, j1 + 2, …, n1), yjtk opt (j = 1, 2, …, j 2 ) and y + jtk opt (j = j 2 + 1, j 2 + 2, …, n 2 ) are solutions corresponding to f þ . Through solving 2n deterministic problems, a −opt − set of values (f− , f , ⋯, f2n) can be obtained. The optimized lower1 2 bound interval for the objective-function value (under an α-cut level) can be identified as follows:


i þ − ′þ ; ja′ijtk jSign aijtk yjtk



i þ ; ja′− yjtk j a ′− jSign  a ′− jSign a′− ijtk ijtk ijtk ijtk

h − − ≥ w  itk ; witk ;

−

xjt ≥ xjt opt ;

−

j1 n1

 − − þ þ þ − ∑ jaijt j Sign aijt xjt + ∑ jaijt j Sign aijt xjt þ j a ′ijtk jSign

þ

0 ≤ xjt ≤ xjt opt ;

ðA1  bÞ

+ ∑



i þ þ − þ þ j a ′ijtk jSign  a ′ijtk ; ja′ijtk jSign a′ijtk yjtk

h þ þ ≥  w itk ; witk ; ∀i; t; k = 1; 2;⋯; Kt

subject to:

h

ðA3  cÞ

j = j1 +1



i ′− ′− þ − − + ∑ j a ′ijtk jSign  a ′ijtk Þ; ja jSign aijtk yjtk j2

j = j2 +1

j2


 n1 − þ − ∑ jaijt j Sign aijt xjt

j1
 þ þ − ∑ jaijt j Sign aijt xjt +

ðA1  aÞ

j=1

1115

∀i; t; k = 1; 2;⋯; Kt

h

≥0

ðA1  dÞ

f− − opt ; f opt

iα

−

−

−

−

−

−

= ½minðf1 ; f2 ; ⋯; f2n Þ; max ðf1 ; f2 ; ⋅⋅⋅; f2n Þ

α

ðA4Þ

−

where x+ jt (j = 1, 2, …, j1) are upper bounds of the first-stage decision variables (x± jt ) with positive coefficients in the objective function, and x− jt (j = j1 + 1, j 1 + 2, …, n1) are lower bounds with negative coefficients; y− jtk (k = 1, 2, …, Kt and j = 1, 2, …, j2) are lower bounds of the recourse decision variables (y± jtk) with positive coefficients in the objective function, and y+ jtk (k = 1, 2, …, Kt and j = j2 + 1, j2 + 2, …, n2) are upper bounds with negative coefficients. + + Through solving 2n submodels, a set of values (f+ 1 , f2 , ⋯, f2n) can be obtained. Let Pf þ be the minimum value of the upper bound (for the opt   Pþ þ objective-function value) with Pf þ = min f1þ ; f2þ ; ⋯; f2n , and f opt be opt  þ þ  Pþ þ . the maximum value of the upper bound with f opt = max f1 ; f2 ; ⋯; f2n Then, the optimized upper-bound interval for the objective function value (under an α-cut level) can be identified as follows: h

þ

þ

f ;f  opt opt

iα

h

iα þ þ þ þ þ þ = min f1 ; f2 ; ⋯; f2n ; max f1 ; f2 ; ⋯; f2n

ðA2Þ

Based on the solutions from the first set of submodels, submodels f − can be formulated as: corresponding to ˜

f

g

n1 i i hh i i − − ′− − − þ + c′jt− xjt c jt ; cjt + cjt xjt + ∑ c jt ; c−   jt j = j1 +1 ( ) j2 hh n2 i i hh i i T þ þ þ þ þ þ ′þ − − ∑ ∑ ptk ∑  d jtk ; djtk +d′jtk yjtk + ∑ d jtk ; djtk + djtk yjtk  T

− Max f˜ = ∑

t =1

j1

∑

hh

j=1 Kt

t =1 k=1

j=1

j = j2 +1

ðA3  aÞ

where f opt is the minimum value of the lower bound (for the  − − objective function value) with f − = minðf1− ; f2− ; ⋯; f2n Þ; f opt is the  opt − − − − Þ. maximum value of the lower bound with f opt = max ðf1 ; f2 ; ⋯; f2n Then, through integrating the computational results of the two sets of submodels, the solution for the objective function value (under an αcut level) can be obtained. Iteratively, the computational process can be repeated with the other α-cut levels. The above risk-prone solution method identifies the solutions for the first-stage and recourse decisions variables by first solving the best-case submodel (i.e. upper-bound objective function value when the problem is to be maximized). The resulting solution can provide intervals for the objective function value and decision variables, and can be easily interpreted for generating decision alternatives. However, this solution method may provide a wide ranging objective function value because significant (and costly) first-stage and recourse decisions are required under unfavorable conditions (represented by worst-case parameter values) [40]. Consequently, another solution method (based on risk adverse) is proposed for solving the FBISP model to reduce the interval width of the objective-function value, in which the worst-case submodel (i.e. corresponding to the lower-bound objective function) can be first solved to identify a more appropriate set of first-stage and recourse decision variables. Thus, we have:

f

T

j1

− Max f˜ = ∑ ∑

hh

t=1 j=1 T

Kt

−

−

c jt ; cjt  ( j2

− ∑ ∑ ptk ∑ t=1 k=1

j=1

i

i − + c′jt− xjt +

hh

þ

þ

i

d jtk ; djtk + 

n1

∑

hh

j = j1 + 1 ′þ djtk

i

þ

yjtk +

i i − + c′− xþ c− jt jt jt ; cjt  n2

∑

j = j2 +1

hh

i

g

i þ þ ′þ − d jtk ; djtk + djtk yjtk 

)

ðA5  aÞ

1116

Y.P. Li et al. / Advances in Water Resources 33 (2010) 1105–1117

Appendix B. Supplementary data

subject to: j1

 þ þ − þ ; ja a rjt jSign  aþ jSign arjt xjt ∑ j rjt rjt

j=1

½



h

n1

∑

+

j = j1 +1

Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.advwatres.2010.06.015.



 h i þ − − − − j a− a− b rt ; brt ; rjt jSign  rjt ; jarjt jSign arjt xjt ≤ 

∀r; t

ðA5  bÞ j1
 þ þ − ∑ jaijt j Sign aijt xjt +

j=1

n1
 − − þ ∑ jaijt j Sign aijt xjt

j = j1 +1



 þ ′− − − ′− ; jaijtk + ∑ j a ′ijtk jSign  a ′ijtk jSign aijtk yjtk

½

j2



j=1 n2

h

∑

+

j = j2 +1

ðA5  cÞ



i þ þ − ′ þ jSign a ′ þ yjtk j a ′ijtk jSign  a ′ijtk ; jaijtk ijtk

h i þ þ ≥  w itk ; witk ; ∀i; t; k = 1; 2;⋯; Kt −

þ

þ

−

xjt ; xjt ; yjtk ; yjtk ≥ 0

ðA5  dÞ

Solving 2n deterministic lower-bound submodels, a set of lower− − bound objective-function values (f− 1 , f2 , ⋯, f2n) can be obtained. Then, another set of submodels corresponding to ˜ f þ can be formulated:

f

T

Max ˜ fþ = ∑

t =1

j1

∑

j=1 Kt

T

hh

þ

þ

c jt ; cjt  ( j2

− ∑ ∑ ptk ∑ t=1 k=1

i

i ′þ þ + cjt xjt +

hh

−

−

i

n1

∑

hh

′

i −

þ

þ

c jt ; cjt 

j = j1 +1 −

i

n2

i ′þ − + cjt xjt

d jtk ; djtk + djtk yjtk + ∑  j = j +1

j=1

g

hh

i i − − ′− þ d jtk ; djtk + djtk yjtk 

)

2

ðA6  aÞ subject to: j1

 − þ − ∑ j a− a− xjt rjt jSign  rjt ; jarjt jSign arjt j=1 n1 h

i h þ − þ þ þ þ xjt ≤  + ∑ j a rjt jSign  aþ b rt ; brt ; rjt ; jarjt jSign arjt

½



j = j1 + 1

∀r; t ðA6  bÞ

j1
 − − þ ∑ jaijt j Sign aijt xjt +

j=1

j2

+ ∑

j=1 n2




½

þ j a ′ijtk jSign

h

∑

+

j = j2 +1

n1
 þ þ − ∑ jaijt j Sign aijt xjt

j = j1 +1


 − ′þ ′þ ; jaijtk jSign aijtk yjtk

þ a ′ijtk 





i − − − ′− þ j a ′ijtk jSign  a ′ijtk ; ja′ijtk jSign aijtk yjtk

ðA6  cÞ

h i − − ≥ w  itk ; witk ; ∀i; t; k = 1; 2;⋯; Kt þ

−

xjt ≥ xjt opt ;

∀t;j = 1; 2; ⋯; j1

−

þ

−

þ

0 ≤ xjt ≤ xjt opt ; 0 ≤ yjtk ≤ yjtk opt ; þ

−

yjtk ≥ yjtk opt ;

∀t; j = j1 + 1; j1 + 2; ⋯; n1 ∀t; j = 1; 2; ⋯; j2 ; k = 1; 2;⋯; Kt

∀t; j = j2 + 1; j2 + 2; ⋯; n2 ; k = 1; 2;⋯; Kt

ðA6  dÞ ðA6  eÞ ðA6  fÞ ðA6  gÞ

Through solving 2n deterministic problems, a set of upper-bound + + objective-function values (f+ 1 , f2 , ⋯, f2n) can be obtained. Then, through integrating the computational results of the two sets of submodels, the solution for the objective function value (based on risk adverse) can be obtained.

References [1] Ahmed S, King AJ, Parija G. A multi-stage stochastic integer programming approach for capacity expansion under uncertainty. J Glob Optim 2003;26:3–24. [2] Bashir MA, Tanakamaru H, Tada A. Spatial and temporal analysis of evapotranspiration using satellite remote sensing data: a case study in the Gezira Scheme, Sudan. J Environ Inform 2009;13(2):86–92. [3] Bender MJ, Simonovic SP. A fuzzy compromise approach to water resources planning under uncertainty. Fuzzy Sets Syst 2000;115(1):35–44. [4] Birge JR. Decomposition and partitioning methods for multistage stochastic linear programs. Oper Res 1985;33:989–1007. [5] Chang NB, Wen CG, Chen YL, Yong YC. A grey fuzzy multiobjective programming approach for the optimal planning of a reservoir watershed, part A: theoretical development. Water Res 1996;30:2329–40. [6] Chang NB, Wen CG, Chen YL, Yong YC. A grey fuzzy multiobjective programming approach for the optimal planning of a reservoir watershed. Part B: application. Water Res 1996;30:2341–52. [7] Chen HK, Hsu WK, Chiang WL. A comparison of vertex method with JHE method. Fuzzy Sets Syst 1998;95:201–14. [8] Civanlar MR, Trussel HJ. Constructing membership functions using statistical data. Fuzzy Sets Syst 1986;18:1–13. [9] Dong W, Shah HC. Vertex method for computing functions of fuzzy variables. Fuzzy Sets Syst 1987;24:65–78. [10] Dou C, Woldt W, Bogardi I, Dahab M. Numerical solute transport simulation using fuzzy sets approach. J Contam Hydrol 1997;27(1–2):107–26. [11] Dubois D, Prade H. Systems of linear fuzzy constraints. Fuzzy Sets Syst 1980;3: 37–48. [12] Dubois D, Prade H. Fuzzy sets and statistical data. Eur J Oper Res 1986;25:345–56. [13] Dupačová J, Consigli G, Wallace SW. Generating scenarios for multi-stage stochastic programming. Ann Oper Res 2000;100:25–53. [14] Dupačová J. Applications of stochastic programming: achievements and questions. Eur J Oper Res 2002;140:281–90. [15] Faye RM, Sawadogo S, Mora-Camino F. Flexible management of water resource systems. Appl Math Comput 2005;167(1):516–27. [16] Feiring BR, Sastri T. A stochastic programming model for water resource planning. Math Comput Modell 1998;27(3):1–7. [17] Hipel KW. Multiple objective decision making in water resource. Water Resour Bull 1992;28(1):3–13. [18] Huang GH, Baetz BW, Patry GG. A grey linear programming approach for municipal solid waste management planning under uncertainty. Civ Eng Syst 1992;9:319–35. [19] Huang GH. IPWM: an interval parameter water quality management model. Eng Optim 1996;26:79–103. [20] Huang GH, Loucks DP. An inexact two-stage stochastic programming model for water resources management under uncertainty. Civ Eng Environ Syst 2000;17: 95–118. [21] Inuiguchi M, Ramík J. Possibilistic linear programming: a brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem. Fuzzy Sets Syst 2000;111:3–28. [22] Jairaj PG, Vedula S. Multireservoir system optimization using fuzzy mathematical programming. Water Resour Manage 2000;14:457–72. [23] Karmakar S, Mujumdar PP. Grey fuzzy optimization model for water quality management of a river system. Adv Water Resour 2006;29:1088–105. [24] Karmakar S, Mujumdar PP. A two-phase grey fuzzy optimization approach for water quality management of a river system. Adv Water Resour 2007;30: 1218–35. [25] Kaufmann A, Gupta MM. Introduction a fuzzy arithmetic: theory and applications. New York: Van Nostrand Reinhold; 1991. [26] Kuby MJ, Fagan WF, ReVelle CS, Graf WL. A multiobjective optimization model for dam removal: an example trading off salmon passage with hydropower and water storage in the Willamette basin. Adv Water Resour 2005;28:845–55. [27] Lai YJ, Hwang CL. Fuzzy mathematical programming methods and applications. Berlin: Springer-Verlag; 1992. [28] Lee CS, Chang SP. Interactive fuzzy optimization for an economic and environmental balance in a river system. Water Res 2005;39:221–31. [29] Li, J.B. Development of an inexact environmental modeling system for the management of petroleum-contaminated sites. Ph.D. Dissertation, University of Regina, Regina, Saskatchewan, Canada, 2003. [30] Li YP, Huang GH, Nie SL. An interval-parameter multi-stage stochastic programming model for water resources management under uncertainty. Adv Water Resour 2006;29:776–89. [31] Li YP, Huang GH, Nie SL. Mixed interval-fuzzy tow-stage integer programming and its application to flood-diversion planning. Eng Optim 2007;39(2):163–83. [32] Li YP, Huang GH, Yang ZF, Nie SL. Interval-fuzzy multistage programming for water resources management under uncertainty. Resour Conserv Recycl 2008;52: 800–12. [33] Li YP, Huang GH, Nie SL, Liu L. An inexact multi-stage stochastic integer programming method for water resources management. J Environ Manage 2008;88:93–107.

Y.P. Li et al. / Advances in Water Resources 33 (2010) 1105–1117 [34] Li YP, Huang GH, Guo P, Yang ZF, Nie SL. A dual-interval vertex analysis method and its application to environmental decision making under uncertainty. Eur J Oper Res 2010;200:536–50. [35] Liu KFR, Liang HH, Yeh K, Chen CW. A qualitative decision support for environmental impact assessment using fuzzy logic. J Environ Inform 2009;13(2):93–103. [36] Loucks DP, Stedinger JR, Haith DA. Water resource systems planning and analysis. Englewood Cliffs, N.J.: Prentice-Hall; 1981. [37] Maqsood I, Huang GH, Yeomans JS. An interval-parameter fuzzy two-stage stochastic program for water resources management under uncertainty. Eur J Oper Res 2005;167:208–25. [38] Nie XH, Huang GH, Li YP, Liu L. IFRP: a hybrid interval-parameter fuzzy robust programming approach for municipal solid waste management planning under uncertainty. J Environ Manage 2007;84:1–11. [39] Pereira MVF, Pinto LMVG. Stochastic optimization of a multireservior hydroelectric system: a decomposition approach. Water Resour Res 1985;6:779–92. [40] Rosenberg DE. Shades of grey: a critical review of grey-number optimization. Eng Optim 2009;41(6):573–92. [41] Seifi A, Hipel KW. Interior-point method for reservoir operation with stochastic inflows. J Water Resour Plan Manage (ASCE) 2001;127(1):48–57.

1117

[42] Sivakumar C, Elango L. Application of solute transport modeling to study tsunami induced aquifer salinity in India. J Environ Inform 2010;15(1):33–41. [43] Slowinski R. A multicriteria fuzzy linear programming method for water supply system development planning. Fuzzy Sets Syst 1986;19:217–37. [44] Stedinger S, Loucks DP. Stochastic dynamic programming models for reservoir operation optimatization. Water Resour Res 1984;20:1499–505. [45] Trezos T, Yeh WWG. Use of stochastic dynamic programming for reservoir movement. Water Resour Res 1987;23:983–96. [46] Watkins Jr DW, McKinney DC, Lasdon LS, Nielsen SS, Martin QW. A scenario-based stochastic programming model for water supplies from the highland lakes. Inter Transit Oper Res 2000;7:211–30. [47] Yeomans JS. Applications of simulation-optimization methods in environmental policy planning under uncertainty. J Environ Inform 2008;12(2):174–86. [48] Zadeh LA. The concept of a linguistic variables and its application to approximate reasoning-1. Inf Sci 1975;8:199–249. [49] Zarghami M, Szidarovszky F. Stochastic-fuzzy multi criteria decision making for robust water resources management. Stoch Environ Res Risk Assess 2009;23:329–39. [50] Zimmermann HJ. Fuzzy set theory and its applications. 3rd Edition. Kluwer Academic Publishers; 1995.