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E-mail:journal-services@https://www.doczj.com/doc/246784139.html, Editorial
1The Role of the Systems Community in the National Science Foundation’s Environmental Observatories
Gregory W.Characklis,Patrick M.Reed,and Barbara S.Minsker Technical Papers
4Optimizing Operational Policies of a Korean Multireservoir System Using Sampling Stochastic Dynamic Programming with Ensemble Stream?ow
Prediction
Young-Oh Kim,Hyung-Il Eum,Eun-Goo Lee,and Ick Hwan Ko
15Water Resources Management:Stewardship and Services
Harald D.Frederiksen
23Dynamic Optimization Approach for Solving an Optimal Scheduling Problem in Water Distribution Systems
B.Ulanicki,J.Kahler,and H.See
33Vulnerability of Drinking Water Treatment Plants to Low Water Levels in the https://www.doczj.com/doc/246784139.html,wrence River
A.Carrière,
B.Barbeau,and J.-F.Cantin
39Probabilistic Estimation of Water Conservation Effectiveness
David E.Rosenberg
50Mathematical Programming Approaches for Modeling Water Rights Allocation
Lizhong Wang,Liping Fang,and Keith W.Hipel
60Water Sector Structure,Size,and Demographics
Neil S.Grigg
67Robust Least-Cost Design of Water Distribution Networks Using Redundancy and Integration-Based Methodologies
Artem V.Babayan,Dragan A.Savic,Godfrey A.Walters,
and Zoran S.Kapelan
78Simulation of Operations and Water Quality Performance of Reservoir Multilevel Intake Con?gurations
Rakesh K.Gelda and Steven W.Ef?er
Technical Notes
87Ant Colony Optimization Applied to Water Distribution System Design: Comparative Study of Five Algorithms
Aaron C.Zecchin,Holger R.Maier,Angus R.Simpson,Michael Leonard,
and John B.
Nixon
VOLUME133/NUMBER1JANUARY/FEBRUARY2007
Mathematical Programming Approaches for Modeling Water
Rights Allocation
Lizhong Wang1;Liping Fang2;and Keith W.Hipel3
Abstract:Three methods for modeling allocation of water rights under different types of water rights systems are presented.The priority-based maximal multiperiod network?ow?PMMNF?programming method searches for optimal allocations over the whole river basin and multiple periods,strictly preserving priority order by sequential programming.PMMNF is a?exible method that can be applied under prior,riparian and public water regimes with priorities determined by different criteria.The modi?ed riparian water rights allocation ?MRWRA?approach for water allocation under a riparian rights system is a special form of PMMNF,which assigns senior priorities to the basic water demands of all riparian landowners and junior priorities to their surplus demands.MRWRA recognizes that,an upstream riparian has the privilege to take water but should constrain its diversions for surplus demands to meet the basic needs of downstream riparian demands.The lexicographic minimax water shortage ratios method is a technique for water allocation under a public water rights regime adopting the lexicographic minimax fairness concept,which interprets fairness as minimizing weighted water shortage ratios of all water uses and the differences among them.The weights can be set based on the“equivalent weighted water shortage ratios”rule.The proposed methods may also be applied to international basins as long as water demands and associated priorities or weights are selected to re?ect relevant international water allocation principles and agreements.These approaches and their algorithms are evaluated using an illustrative application to the Amu Darya River Basin located within the Aral Sea Basin of Central Asia.
DOI:10.1061/?ASCE?0733-9496?2007?133:1?50?
CE Database subject headings:Water shortage;Water rights;Allocation;River basins;Optimization models;Mathematics.
Introduction
Driven by the rapid increases of water demands from population growth and other stresses such as industrialization,urbanization, pollution,and impacts of climate change,water scarcity is now a common occurrence especially in developing countries.It is pro-jected that by the middle of this century at worst7billion people in60countries,will face water scarcity,and at best2billion in48 countries?UN/WWAP2003?.The competition for water is now evident not only in terms of quantity but also quality.Many ne-gotiations begin with the parties basing their initial positions in terms of rights—the sense that a riparian is entitled to a certain allocation based on intracountry water rights regimes or interna-tional river basin agreements?Giordano and Wolf2001?.The basic underlying theme in water allocation relates to what are “fair”or“equitable”water rights.National and international laws,
for example,the Helsinki Rules,adopted by the International Law
Association?1967?and Convention on the Law of the Non-
Navigational Uses of International Watercourses,embraced by the
United Nations?UN1997?,assert that water should be allocated
in an equitable and reasonable manner and should not cause ap-
preciable harm to riparians.However,the principles or guidelines
of reasonable and equitable use are dif?cult to apply directly in
practice.Different riparian countries interpret these principles in
different ways.Measurable criteria and models need to be de-
signed and used to achieve fair apportionment of water in light of
water shortages?Seyam et al.2000;Van der Zaag et al.2002?.
From the point of view of sovereignty of a river basin,models
and algorithms can be grouped into two basic categories of intra-
country and intercountry water rights allocation,or they can also
be classi?ed as simulation and optimization models according to
modeling techniques.While it is hard to?nd quantitative models
or algorithms in the literature for intracountry water allocation
under riparian rights systems?Cech2002?and intercountry water
rights allocation?Seyam et al.2000;Van der Zaag et al.2002?,
conventional simulation?Wurbs2001?,minimum cost pure ?Fredericks et al.1998?,and generalized?Hsu and Cheng2002?network?ow,and mixed integer linear programming models ?Tu et al.2003?have been developed for prior water allocation. However,simulation models cannot provide,either spatially or
temporally,optimal allocations due to structural limitations.The
minimum cost network?ow models and linear programming for-
mulations also have a common shortcoming in that they lack
systematic and formal methods to set proper unit cost coef?cients
to ensure that water is allocated in the priority order when return
?ows,instream uses,or reservoir storage rights are included in
the programming,because return and instream?ows and reservoir
1Research Associate,Dept.of Systems Design Engineering,Univ.of
Waterloo,Waterloo ON,Canada N2L3G1.E-mail:l25wang@
uwaterloo.ca
2Professor,Chair,Dept.of Mechanical and Industrial Engineering,
Ryerson Univ.,Toronto ON,Canada M5B2K3.E-mail:lfang@
ryerson.ca
3Professor,Dept.of Systems Design Engineering,Univ.of Waterloo,
Waterloo ON,Canada N2L3G1?corresponding author?.E-mail:
kwhipel@uwaterloo.ca
Note.Discussion open until June1,2007.Separate discussions must
be submitted for individual papers.To extend the closing date by one
month,a written request must be?led with the ASCE Managing Editor.
The manuscript for this paper was submitted for review and possible
publication on March3,2005;approved on January30,2006.This paper
is part of the Journal of Water Resources Planning and Management,
V ol.133,No.1,January1,2007.?ASCE,ISSN0733-9496/2007/1-50–
59/$25.00.
50/JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT?ASCE/JANUARY/FEBRUARY2007
storages can be reused by junior downstream uses?Israel and Lund1999?.To overcome this,an approach called the priority-based maximal multiperiod network?ow?PMMNF?program-ming method is developed in this research that adopts a sequential solution algorithm,which can allocate water to meet demands strictly according to priority ranks,but avoids the dif?culty in assigning proper unit cost coef?cients.
Models for public allocation are either simulation or optimiza-tion models that treat water as a public property.In past decades, many mathematical simulation and optimization models for water quantity,quality,and/or economic management have been devel-oped and applied to problems at both the subsystem and the river basin levels,such as reservoir operation,groundwater use, conjunctive utilization of surface water and groundwater,and irrigation and drainage management?McKinney et al.1999?. Nevertheless,only a few models and applications address the fair-ness issue in a quantitative way.For example,Cai et al.?2002?propose a framework for sustainability analysis in water resources management,in which the indicator of temporal equity is ex-pressed as the standard deviation of the annual rate of change of the total net bene?t of water use and the indicator of spatial equity is de?ned as the standard deviation of the long-term average rate of change of pro?t.These two criteria are combined together with other objectives to formulate a weighted sum minimization problem.In this research,the authors develop the lexicographic minimax water shortage ratios?LMWSR?approach for explicitly considering the concept of fairness within a public allocation regime.
Seyam et al.?2000?derive four algorithms for allocating the waters of a shared river between riparian countries using popula-tion as a distribution factor.Van der Zaag et al.?2002?propose six algorithms for transboundary water allocation,which are similar to those of Seyam et al.?2000?,but the proportion factors expand to population and each country’s area.However,all of these al-gorithms or conceptual models are based on average?ows and disregard the environmental requirements of instream?ows,and the variability in water availability in both space and time is not taken into account.
Wang et al.?2003?propose a modeling framework for achiev-ing equitable,ef?cient,and sustainable water allocation among competing water uses and stakeholders in a river basin.In this methodology,water allocation is carried out in two steps based on a network representation of a river basin:?1?allocation of initial water rights to water stakeholders and users based on legal water rights systems or agreements;and?2?reallocation of water and net bene?ts through water transfers to promote equitable coopera-tion of all stakeholders in a river basin and to achieve ef?cient use of water.Cooperative game theory is utilized to carry out the associated water transfers and net bene?t reallocations.
The contents of the paper are organized as follows.The next main section reviews water rights systems for allocating water resources in intracountry basins and generalized principles of transboundary water allocation in intercountry basins.Fairness implications of these legal systems or principles are also dis-cussed.Subsequently,the generalized water resources allocation problem is explained.Then,three methods,namely the PMMNF, modi?ed riparian water rights allocation?MRWRA?,and LMWSR techniques,are formulated for water rights allocation. As noted earlier,the PMMNF programming method possesses a unique design for strictly preserving the water rights priorities using a sequential solving approach.The MRWRA technique is a special case of the PMMNF method which converts riparian water rights into a priority-based water rights system.Moreover,the LMWSR method is a novel approach to water allocation under the public allocation regime which utilizes a lexicographic minimax optimization procedure.The sequential and iterative programming approaches for solving PMMNF and LMWSR problems,respectively,are brie?y described.Before the conclud-ing section,the developed methods and algorithms are tested and evaluated by an illustrative application to the Amu Darya River Basin located in the Aral Sea Basin of Central Asia.
Water Rights Systems and Generalized Principles
of Transboundary Water Allocation
The question of“whose water is it”is a fundamental issue of water allocation?Green and Hamilton2000?.Nations de?ne water rights in various ways.Different systems of water rights can be grouped into three basic doctrines:riparian rights,prior?appro-priative?rights,and public allocation.
The common law riparian rights system treats water as a com-mon property,and was developed in humid regions where water is abundant and water allocation did not cause major problems for individual water users.The riparian system was essentially court-made property law based on the common law of England and has evolved into two basic doctrines:reasonable use and correlative rights.The reasonable use doctrine means that a riparian land-owner can divert and use any quantity of water for use on riparian lands,as long as these diversions and uses do not interfere with reasonable use of other riparian landowners.There is no sharing of shortage in available water except as a court determines whether continuing a use is unreasonable?Dellapenna and Stephen2002?.The correlative rights doctrine requires that ripar-ian landowners must share the total?ow of water in a stream,and may withdraw only their“share”of water for reasonable use.For example,the proportion of use allocated to each riparian is based on the amount of waterfront property owned along a stream?Cech 2002?.Since the riparian system allocates water based on the ownership of land adjacent to water,similarly situated riparians share equal rights.However,in time of water shortage down-stream riparians may not be able to receive enough water for reasonable use because upstream riparians have privileges to take water.
The prior?appropriative?rights regime treats water as private property.Water is appropriated according to“?rst in time,?rst in right.”Junior users are allocated after the senior users have been satis?ed.In cases of water scarcity,there is no sharing of shortage in water and the impacts of water scarcity would fall most heavily on the most junior water users.Because prior appropriation laws were designed to promote productive water uses,most made no provision for establishment of water rights for instream uses other than for hydropower?Miller et al.1997?.While it may be re-garded as fair in the sense to protect the rights of senior users,the prior appropriation system is argued to be unfair for junior users.
The public allocation regime treats water as public property, and the state is the owner of waters.In this system,water rights are administratively allocated to users through water permits from governments.As the water demands increase and begin to com-pete for available water supplies during times of water scarcity, the need for active public management of water has been recog-nized.Today,in the United States,there is no state relying only on“pure”riparian rights,since it will cause the“tragedies of commons”to occur without some form of regulatory manage-ment.The introduction of water management through a regulatory permit system is increasingly common among states?Dellapenna
JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT?ASCE/JANUARY/FEBRUARY2007/51
1994,2000;Dellapenna and Stephen2002?.This modi?ed system
is named“regulated riparianism”and the rights are called“regu-
lated riparian rights.”The regulated riparianism treats water as
public property,and hence,is a kind of public allocation water
rights regime.
For international river basins,generally there is no formal in-
tercountry water rights system but international water agreements
stipulate how to share water among countries.To mitigate prob-
lems of water allocation,the international legal community has
established generalized,global,legal,and economic principles for
intercountry river basins,which include:absolute sovereignty,
absolute riverine integrity,limited territorial sovereignty,and eco-
nomic criteria?Wolf1999;Giordano and Wolf2001?.The abso-
lute sovereignty principle implies unilateral control over waters
within a nation’s territory.The absolute riverine integrity prin-
ciple suggests that every riparian has a right to the waters that
?ow through its territory and emphasizes the importance of his-
torical usage or chronology.The limited territorial sovereignty
principle is more moderate,and advocates reasonable and equi-
table use of international waters while in?icting no signi?cant
harm on any other riparians.Under the economic principle,the
market is used to allocate water among competing users in an
economically ef?cient manner.While water markets have re-
ceived considerable attention and have been applied in a number
of intrastate settings,water markets have not yet been developed
at an international scale due to concerns over equity issues of
water rights?Wolf1999?.
Generalized Water Resources Allocation Problem
To formulate the water allocation problem,a river basin is repre-
sented by a node-link river basin network model.A node is sym-
bolized as a dot,circle,triangle,or rectangle,representing a
physical component of interest such as in?ow,natural or human-
made junction,intake structure,water or wastewater treatment
plant,aquifer,reservoir,natural lake,dam,weir,or water demand
site.A link stands for a natural or human-made water conduit
such as a river channel,canal,or pipeline between two different
nodes,but can also represent any?ow of water such as the seep-
age between a demand site and an aquifer.More speci?cally,let
G?V,L?be the directed network of a river basin,where V=?v1,v2,...,v v?is the set of nodes,?k1,k2?denotes the link or arc from node k1to k2,and L=??k1,k2?:k1,k2?V and k1 k2?is the set of links of the network.A number of water use sites which
take water and discharge return?ows,including offstream and
instream economic uses,minimum environmental?ow require-
ments,as well as reservoirs,are abstracted as a set of water de-
mand nodes in the node-link river basin network model,where
U=?j?V:j is a water demand node?.Because different types of nodes and links have different hydrological and economic prop-erties,ten subsets of the general set of nodes V are de?ned for the mathematical formulation of the model,including in?ow,outlet, junction,treatment plant,aquifer,reservoir,agricultural,munici-pal and industrial,hydropower plant,and stream?ow requirement nodes.
For a general node k,the water and pollutant balance for each
period t can be written as
S?k,t??S?k,t?1?=??k1,k??L Q?k1,k,t????k1,k??L Q l?k1,k,t?+Q g?k,t?
?Q c?k,t????k,k2??L Q?k,k2,t?,?k?V?1?
C p?k,t?S?k,t??C p?k,t?1?S?k,t?1?
=??k1,k??L C p?k1,k,t?Q?k1,k,t????k1,k??L Z pl?k1,k,t?+Z pg?k,t?
?Z pc?k,t????k,k2??L C p?k,k2,t?Q?k,k2,t?,?k?V?2?where t=index of periods?period length is?t?; t?T=?1,2,...,??,??=largest index of period?;S?k,t?=storage volume for storage node?reservoir or aquifer?k?STO at end of period t;Q?k1,k,t?=flow from node k1to k during period t; Q l?k1,k,t?=conveyance losses because of evaporation,leakage,
and seepage of the?ow from node k1to k;Q g?k,t?=gain of in-?ow adjustment at node k during period t for discharges from small tributaries,local catchment drainages,river reach seepages, or?ows from other sources;Q c?k,t?=water consumed at node k because of economic activities and evaporation;p=index of pol-lutant types,p?P=?1,2,...,????=largest index of pollutants?;
C p?k,t?=concentration of pollutant p at storage node k at the end of period t;C p?k1,k,t?=concentration of pollutant p in the water ?ow from node k1to k during period t;Z pl?k1,k,t?=conveyance losses of pollutant p in the water?ow from node k1to k; Z pg?k,t?=total amount of pollutant p added to node k during pe-riod t because of in?ow adjustment Q g?k,t?and of water use activities;and Z pc?k,t?=removal of pollutant p at node k.Note, S?k,t?=0,?k?V\STO.For source node k?IN,the total in?ow received from the outside of the river network is represented as Q IN?k,t?,while Q?k1,k,t?represents?ows from other nodes to k. For outlet node k?OUT,the total out?ow from node k to the outside of the river network is represented as Q OUT?k,t?,and there is no longer any Q?k,k2,t?.
Besides the general mass balance equations for each node, there are mass balance constraints for some natural physical re-sponse processes.These include link losses,node in?ow adjust-ments,node losses,consumption and pollutant discharges,and out?ows.There are also policy and system control constraints in the water allocation model.For example,the constraint of the maximum total in?ow for demand node j is formulated as
?
?k1,j??L\L Seep
?1?e L?k1,j,t??Q?k1,j,t??max?Q D?j,t??Q a?j,t?,0??j?AGR?MI?HPP?3?
where L seep=set of seepage links;e L?k1,j,t?=water loss coef?-cient for link?k1,j?;Q D?j,t?=demand for total effective in?ows to node j;and Q a?j,t?=inflow adjustment to node j from small tributaries,local catchment drainages,or?ows from other sources ?excluding river reach seepages?.AGR,MI,and HPP=sets of agricultural,municipal and industrial demand nodes,and hydro-power plants,respectively.
Water allocation at the basin level can be viewed as a gener-alized multiperiod network?ow programming problem with mul-tiple objectives due to the water demands from a number of com-peting uses.Various forms of multiple objective functions can be found in the literature?Loucks et al.1981;Yeh1985?.In a typical formulation,a larger value of the outcome means a better effect ?higher service quality or client satisfaction?.If a smaller value of an outcome is preferred,then the minimization of the outcome can be converted to an equivalent maximization problem.There-fore,without loss of generality,it can be assumed that for each individual outcome to be maximized,the water allocation prob-lem is formulated as a generic multiple objective optimization
52/JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT?ASCE/JANUARY/FEBRUARY2007
program,max?f?x?:x???,where,f=?f1,f2,...,f m?,x?vector of all the control variables,and?denotes the feasible set de?ned by the constraints.
Traditional de?nitions for water rights only consider water quantity.However,water has characteristics of both quantity and quality.A better de?nition of water rights for a nonstorage de-mand should set the volume and pollutant concentration limits for all its withdrawals?or in?ows?and return?ows within each spe-ci?c period subject to certain hydrologic conditions,priority,or weighting rules.For storage demands,water rights can be de?ned as reservoir storage and pollutant concentration limits.Since quantity and quality limits,priorities,and weights are inputs for enforcement of water rights,how they are de?ned greatly in?u-ences whether or not the water rights allocation is perceived to be fair.
Water Rights Allocation Methods Using Mathematical Programming Approaches
Based on a river basin?ow network,three water allocation meth-ods are formulated using mathematical programming approaches in this section.These methods allocate water to uses based on legal water rights systems,priorities,water management agree-ments,and policies.The control variables of the problem are water?ows,storage,and pollutant concentrations,while other factors are set as?xed and are treated as the attributes of de-mands.Each of the three methods may be formulated as linear or nonlinear programming models,depending on whether nonlinear hydrologic relationships and water quality constraints are in-cluded or not.
PMMNF Method
The priority-based maximal multiperiod network?ow?PMMNF?programming method is devised for water rights allocation under a prior allocation regime.Priorities are normally assigned to uses according to the acquired time of water rights based on a“?rst in time,?rst in right”rule,or the importance of water uses in some cases?Savenije and Van der Zaag2000?.Considering the dif?-culty in assigning proper unit cost coef?cients for minimum cost network?ow programming,we propose a PMMNF method uti-lizing a sequential programming technique.
The basic ideas underlying PMMNF are:?1?Every in?ow link to a demand node may have one or more withdrawal demands with various priorities.Thus,each in?ow link to a demand node is viewed as consisting of one or several dummy sublinks and each sublink has a withdrawal demand and corresponding priority.?2?If more than one sublink with the same supply priority are con-nected to an identical source node and no water demand limits exist,?ows are allocated simultaneously in proportion to their withdrawal demands in every time period.?3?Each in?ow link to a stream?ow requirement node?j?SFR?is separated into a by-pass sublink in addition to sublinks for stream?ow requirements with various priorities.Note that priority ranks are assigned to all sublinks except for bypass sublinks.?4?The storage of every reservoir is divided into several subzones according to reservoir operating rules.Each subzone has a storage and corresponding priority.?5?Water is allocated to meet in?ow and storage de-mands strictly according to priorities.Junior uses are allocated after senior uses have been satis?ed as fully as possible subject to hydrologic constraints.
PMMNF is formulated as the following programming problem with multiple ordered objectives:
max?f?m??x??
subject to
h?Q,S,C?=0
g?Q,S,C??0
Q?k,j,t?=?z=z1
z n
Q z r?k,j,t?,?j?U
S?j,t?=?z=z1
z m
S z r?j,t?,?j?RES
Q z r?k,j1,t?/Q z r?k,j2,t?=Q D,z r?k,j1,t?/Q D,z r?k,j2,t?,
?j1,j2?U,j1 j2,Q D?j1,t?= ,
Q D?j2,t?=
?
t?T
?
j?U
?S z r?j,t?+?1?e L?k,j,t??Q z r?k,j,t??=f r?x?,
?j?U,?r?PR
0?Q z r?k,j,t??Q D,z r?k,j,t?,??k,j??L,Q D,z r?k,j,t??0
0?S z r?j,t??S D,z r?k,j,t?,?j?RES
Q,S,C?0?4?
where f?m??x?=?f r
1
?x?,f r
2
?x?,...,f r
m
?x??;Q,S,and C=vectors of network?ow variables Q?k1,k,t?,S?k,t?,C?k1,k,t?,and C?k,t?; h(Q,S,C)=0,and g(Q,S,C)?0represent the equality and non-equality constraints for the network type variables Q,S,and C, respectively;r=priority assigned to a reservoir subzone or in?ow sublink;and PR=set of priority ranks whose elements are ordered from the highest to lowest as r?PR=?r1,r2,...,r m?.Note that the priority ranks are positive integers,and the smaller the value the higher is the priority.S z r?j,t?and S D,z r?j,t?=storage variable and storage demand of the subzone z of reservoir j with a priority of r,respectively;and Q z r?k,j,t?and Q D,z r?k,j,t?=inflow variable and withdrawal demand of the sublink z of link?k,j?with a priority of r,respectively.The?rst two constraints are the general water quantity and quality constraints for every node of a river basin network.The third constraint indicates that each link?ow to a demand is the sum of the allocated?ows to its sublinks with various priorities.The fourth constraint means that the reservoir storage volume is the sum of all the allocated storages for sub-zones with various priorities.The?fth constraint makes sure that water is allocated proportionally in every time period to with-drawal demands with the same priority and directly connected to the same supply node,if no in?ow demand is set for both demand nodes.The sixth constraint de?nes the objective function,f r?x?, as the total value of the storages and effective in?ow volumes for all demands with the priority rank r.The next two constraints set the lower and upper bounds for all sublink?ows and subzone storages which have various priorities.The last constraint de?nes the nonnegative network?ow,storage,and pollutant concentra-tion vectors.Note that,for each nonstorage demand site,either the in?ow demand?Q D?j,t??or sublink withdrawal demands
JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT?ASCE/JANUARY/FEBRUARY2007/53
?Q D ,z r ?k ,j ,t ??or both,should be input to the model.If a vector x is used to represent all of the control variables and ?is utilized to denote the feasible set de?ned by the constraints in the PMMNF problem,the problem can be expressed in a more compact form as:max ?f ?m ??x ?:x ???.
If only linear water quantity constraints are included and all water demands are set to estimated constants,the PMMNF is a linear program,which is optimized by sequentially solving the following maximal network ?ow program for each priority rank r ?,from the highest to lowest priority
max f r ??x ?
subject to
x ??
f r ?x ??f r
*,?r ?r ??5?
where r ?=priority rank that the current programming aims for;
and f r *
=optimal values found for f r ?x ?in previous solution loops.The sequential series of maximal network ?ow programs are con-verted to standard linear programs,and can be solved by the primal simplex method.
If nonlinear area–storage relations are applied to reservoirs or water quality constraints are included,the PMMNF becomes a nonlinear program ?NLP ?.A simple but effective domain decom-position approach called the “two-stage”approach is utilized to solve the large-scale nonlinear problems with existing commer-cial NLP solvers.The approach adopts a strategy of solving non-linear programs from good starting points,and is similar to Cai et al.’s ?2001?“piece-by-piece”approach but has some modi?ca-tions.The ?rst stage of the “two-stage”approach searches a cor-responding simpli?ed linear PMMNF program,which considers only linear water quantity constraints and the linear parts of the reservoir storage–area relationships.Nonlinear items of area–storage relationships are ignored.In particular,let ?Q *,S *?be the solution of the sequential programming at the ?rst stage.Set the initial value of C to the estimated initial values for C *,and then ?Q *,S *,C *?is used as the starting point for the sequential pro-gramming of the nonlinear PMMNF program in the second stage.Since ?Q *,S *?is the global solution of the corresponding linear PMMNF programming,?Q *,S *,C *?should be a good starting point near the ?nal solution of the nonlinear PMMNF problem.In the second stage,each original nonlinear maximal network ?ow program is solved by the projected Lagrangian method.During the second stage,if a nonlinear program encounters an infeasible solution,a relaxed problem with relaxed upper bounds on pollut-ant concentrations is formulated and solved,which shall generate a better new starting point for the original nonlinear program.The sequential algorithm for linear PMMNF has several no-table characteristics:?1?The river basin network is de?ned for multiple periods.Thus,the water allocation scheme generated by the model is both spatially and temporally optimal.For example,in times of water shortages,water may be stored in reservoirs and aquifers for future uses with senior priority,regardless of the cur-rent demands of those with junior priority.?2?As the limiting case of a weighted sum multiobjective optimization problem,PMMNF does not need to assign unit cost coef?cients.The sequential so-lution algorithm allocates water to meet demands strictly accord-ing to priority ranks but avoids the dif?culty in assigning proper unit cost coef?cients.Junior demands receive water only after senior demands are met as fully as possible subject to hydrologi-cal constraints.However,this does not mean a junior demand
always has a lower satisfaction ratio than senior ones.A satisfac-tion ratio is determined by dividing the total value of in?ow ad-justment and effective in?ows or storages by the node water
demand during a time period.Some demands may have higher satisfaction ratios than other demands with the same priority or even with senior priority,because they are instream uses which receive return ?ows from those senior demand sites or they have additional local water supplies that are unavailable to those senior demands.?3?The algorithm is designed to be ?exible.The de-mand constraints could be set by total-demand control,with-drawal control,or both of them.The sequential algorithm for nonlinear PMMNF possesses similar properties as those for linear PMMNF,although it cannot be guaranteed that the two-stage ap-proach will ?nd the global solution.The strategy to search good starting points both in the ?rst and second stages will enable the algorithm to ?nd an approximate global optimal solution and pre-serve priority order.MRWRA Method
A traditional riparian water allocation problem can be converted into a special prior water allocation problem,if higher priorities are strictly assigned to upstream nodes and lower priorities to downstream nodes.The traditional riparian water rights allocation system works well when water is abundant,but poorly in terms of fairness during water shortage times,because the downstream uses may receive little water while upstream demands are satis-?ed as fully as possible.In order to assure reasonable water uses and no extreme harm to downstream uses,the minimum demands of all uses in a river basin should be met as far as possible.So the traditional riparian allocation method described above is modi?ed to a fairer method called the MRWRA.
The MRWRA method can be viewed as a special form of PMMNF and thus has the same mathematical formulation and algorithm.The difference lies in the criteria to assign the priority ranks.In MRWRA the higher priority ranks are assigned to the group of minimum water demands Q D ,min ?k ,j ,t ?and S D ,min ?j ,t ?,and lower priority ranks are assigned to the group of surplus water demands Q D ,sur ?k ,j ,t ?and S D ,sur ?j ,t ?in a river basin.The surplus water demands are the differences between corresponding minimum and maximum water demands,such that Q D ,sur ?k ,j ,t ?=Q D ,max ?k ,j ,t ??Q D ,min ?k ,j ,t ?,and S D ,sur ?j ,t ?=S D ,max ?j ,t ??S D ,min ?j ,t ?.In each group of priority ranks,an up-stream use is assigned a higher priority than a downstream use.LMWSR Method
As discussed earlier,the public allocation regime treats water as public property for which the state is the owner of the water.In this system,water rights are administratively allocated to users through water permits from governments.As water demands in-crease and begin to compete for available water supplies during times of water scarcity,the priorities to get water are often as-signed by governmental authorities according to the importance of uses.For such cases,the PMMNF method can be applied.Another possible approach for allocating equitable initial water rights is to have water allocated among all demands in the sense that no shortage ratio can be decreased further without ei-ther violating a constraint or increasing an already equal or worse-off shortage ratio value that is associated with another de-mand.This equitable water sharing can be formulated as a lexi-cographic minimax multiperiod resource allocation problem ?Wang et al.2003,2004?.The lexicographic minimax solution
54/JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT ?ASCE /JANUARY/FEBRUARY 2007
concept is always Pareto-optimal in a multiple objective problem and simultaneously satis?es equity principles ?monotonicity,im-partiality and equitability ??Luss 1999;Ogryczak et al.2003?.It has been shown by Ogryczak et al.?2003?that a weighted-sum aggregation function of multiple objectives does not satisfy the impartiality and equality principles.Therefore,models in terms of a weighted-sum program cannot produce perfectly equitable allo-cations.This key fact motivated the writers of this paper to intro-duce the lexicographic minimax concept into the ?eld of water allocation.
Because the standard minimax solution concept only focuses on the largest outcomes and minimizes them,it is criticized from the perspectives of Pareto optimality as well as inequity minimi-zation.Besides minimization of the largest outcomes,the lexico-graphic minimax concept sequentially minimizes the second larg-est outcomes ?provided that the largest one remains as small as possible ?,the third largest ?provided that the two largest remain as small as possible ?,and so on.Speci?cally,let S D ?j ,t ?and Q D ?j ,t ?be the total demand of reservoir storage and off-or in-stream uses in time step t ,respectively,and e L ?k ,j ,t ?be the water loss coef-?cients of link ?k ,j ?in time step t .Then,the demand shortage ratio,R ?j ,t ?,can be de?ned
as
R ?j ,t ?=
?
S D ?j ,t ??S ?j ,t ?
S D ?j ,t ?
,
?j ?RES Q D ?j ,t ??
??
?k ,j ??L
?1?e L ?k ,j ,t ??Q ?k ,j ,t ?+Q g ?j ,t ?
?
Q D ?j ,t ?
,
?j ?U ,
?j RES
?6?
The equitable allocation of water storage,diversion,and routing ?ow rights can be obtained through the following multiperiod lexicographic minimax program:
lexmin ?f ?????x ?:x ???
?7?
where ?=number of uses,?=?U ?;f ?????x ?=vector of ??elements f jt ?x ?,where these elements are sorted in a nonincreasing order;f jt ?x ?=??j ,t ?·R ?j ,t ?=performance function of demand node j during period t ,?j ?U ,?t ?T ;and ??j ,t ?=weight for the cor-responding water shortage ratio.
The weights for water uses can be stipulated by a decision maker who manages a given river basin based upon different rules.One such rule is the “equivalent weighted water shortage ratios”rule proposed here.In other words,water shortages should be shared subject to equivalent weighted water shortage ratios.The higher social utility or the lower water-shortage endurance the use has,the larger the weight.For example,if weights for demands are set as follows:domestic 20,other offstream and hydropower generation water demands 10,stream ?ow require-ment 3,and reservoir target storage 1,this means that,without other constraints,if in a month a reservoir storage is short of 90%of its target storage ?i.e.,satisfaction ratio is 10%?,then the do-mestic,other offstream,and hydropower generation water de-mands,and stream ?ow demands directly linking to and receiving out?ows from it,should share the shortage at ratios of 4.5,9,and 30%,respectively.The equivalent weighted water shortage ratios may be derived based on an analysis of the attributes of all the demand sites utilizing techniques such as multiple attribute as-sessment,but this research lies outside the scope of this paper.The generic LMWSR problem is a water-quantity-only water rights allocation problem if only the water quantity constraints are included.Furthermore,if only linear area-and elevation–storage relationships are applied to reservoirs,the problem is a linear program.The solution of the lexicographic minimax program-ming is a re?nement of the standard minimax concept.The idea for ?nding the lexicographic minimax solution is to sequentially
identify all of the minimax solutions and to sort their achievement vectors in weakly decreasing order to identify the lexicographi-cally minimal one.
The generic solution approach for the lexicographic minimax program is to repeatedly solve a series of minimax programs
min M
subject to
x ??
??j ,t ?R ?j ,t ??M ,??j ,t ??NR ??j ,t ?R ?j ,t ??M jt *,
??j ,t ??FR
?8?
where M =real variable;NR=set containing index pairs of ?j ,t ?
for which the corresponding upper bounds of ??j ,t ?R ?j ,t ?are not ?xed;and,on the contrary,FR=set having the index pairs of ?j ,t ?for which the corresponding upper bounds of ??j ,t ?R ?j ,t ?are ?xed to their optimal values M jt *found in previous solution loops.The algorithm starts with an empty set FR,and once a minimax problem is solved,the constraints ??j *,t *?R ?j *,t *?=M *are iden-ti?ed and the corresponding index pairs of ?j ,t ?are removed from the set NR.At subsequent iterations,the upper bounds of these ??j ,t ?R ?j ,t ?are set to their optimal values.Iterations stop when the optimal values for all decision variables are identi?ed.
Note that the algorithm is well de?ned for linear problems,because ?is a convex polyhedron at each iteration and a unique optimal value M *can be found easily.Moreover,while the algo-rithm is implemented by using the primal simplex method,the set FR can be easily identi?ed,and the modi?cations of FR may be implemented by ?xing the upper bounds of f jt ?x ?,whose ?j ,t ??FR.A “two-stage”approach similar to that for nonlinear PMMNF is utilized to solve the nonlinear LMWSR problems,in which nonlinear reservoir area–storage relationships and water quality constraints are also included in addition to the linear con-straints on water quantity.Finally,the LMWSR method proposed here constitutes the ?rst time that a lexicographic minimax opti-
JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT ?ASCE /JANUARY/FEBRUARY 2007/55
mization approach has been speci?cally designed for solving water allocation problems at the basin scale.
The above presentations of the three mathematical approaches for water allocation focus on intracountry water rights systems. Because the absolute sovereignty,absolute riverine integrity,lim-ited territorial sovereignty principles,and agreements among countries for water allocation in an international watershed are similar to the allocation rules under the traditional riparian,prior, and public allocation systems,respectively,the proposed methods are also applicable to transboundary water rights allocation.All of the algorithms are coded in GAMS,a general algebraic modeling system for mathematical programming problems?Brooke et al. 1998?,and utilize the solver MINOS?Murtagh et al.2002?. Illustrative Application to Amu Darya River Basin
The Amu Darya River Basin is shared by Afghanistan,and three independent states of the former Soviet Union consisting of Tajikistan,Turkmenistan,and Uzbekistan.Under the Soviet sys-tem,water allocation and con?ict resolution in the basin were an intranational issue and water use strategies were developed to maximize the perceived bene?ts to the entire region,in which the cost of environmental damage was assumed to be minimal?Mick-lin1991?.Due to the signi?cant reduction of in?ows caused by the large-scale diversions of irrigation water corresponding to the agricultural expansion and population growth over the past4de-cades,the Aral Sea,once the fourth largest lake in the world by area,is now nearing half of its surface area and less than one third of its volume existing in1960and becoming an environmental catastrophe.After the breakup of the former Soviet Union in 1989,the old Soviet scheme for water allocation was still fol-lowed by the independent states in this basin under which the development of irrigated lands in Tajikistan was limited in favor
of the developments in downstream Turkmenistan and Uzbekistan ?McKinney2003?.Tajikistan may argue that water allocation ac-cording to the old regime is not fair since its agricultural demands increased and competed for water after the collapse of the Soviet Union.
The PMMNF and LMWSR methods are applied to the transboundary Amu Darya River Basin according to the absolute sovereignty and limited territorial sovereignty principles,respec-tively.The applications are only meant to be illustrative in nature with a focus on testing and comparing the performance of the developed algorithms.Researchers and practitioners having more extensive documentation and data about the Amu Darya River Basin,as well as other areas of the Aral Sea Basin,could employ the new allocation methods presented in this paper to carry out their own comprehensive case studies.The current water alloca-tion status,for example,can be modeled by PMMNF based on the absolute riverine integrity principle,if more information about the historical usage or chronology of water demands in this river basin could be obtained to properly set priorities to re?ect this principle and other agreements among the countries.Additionally, if detailed data on the riparian withdrawals were available,the MRWRA method could be used to model transboundary water allocations,which is another interpretation of the limited territo-rial sovereignty principle.
River Basin Network Scheme and Input Data
Fig.1portrays the network scheme of the Amu Darya River Basin water resource system.This representation is based on the work of Raskin et al.?1992?and McKinney and Karimov?1997?. In the network,there are seven types of nodes:nine in?ow,14 junction,nine aquifer,six reservoir,one hydropower plant,12 off-stream demand,and one stream?ow requirement nodes.The number of links is87.Each off-stream demand node aggregates water demands of the water uses in a common geographic area with shared water sources and includes water for irrigation,live-stock,municipal,and industrial uses.The off-stream demand nodes are considered as agricultural nodes,since irrigation is the dominant use in this basin.The furthest node downstream repre-sents the Aral Sea,which is modeled as a stream?ow requirement node.Treating the Aral Sea as a demand node rather than an outlet of the network makes it possible to analyze the Aral Sea’s water demands and their effects on water allocation.
Every type of node has its own hydrological characteristics. Reservoir evaporation,node losses and in?ow adjustments,seep-ages,and return?ows,are considered as hydrological constraints in addition to the general water balance.Other assumptions made for the nodes include linear surface area–storage relationships for the reservoirs,and the linear reservoir response submodel for simulating seepages from the reservoirs and aquifers.Transmis-sion losses of links as well as the node and link seepages to aquifers are also considered by utilizing various loss coef?cients. The input data for surface water supply,groundwater supply, major reservoirs,and demands at various sites in the Amu Darya River Basin are estimated on a monthly basis for a typical dry year scenario and mostly come from previous studies of the Aral Sea Basin?Raskin et al.1992;McKinney and Karimov1997?, and are compiled in Fang et al.?2005?.To demonstrate the
capa-Fig.1.Amu Darya River Basin network system
56/JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT?ASCE/JANUARY/FEBRUARY2007
bility of the water allocation methods comprising quantity and quality considerations,salinity control constraints are included. Due to limited available data,some assumptions are made in con-structing the salinity constraints.In particular,initial concentra-tions of salt are assumed to range from upstream reservoirs at 0.5g/L to most downstream reservoirs at1.5g/L,and from up-stream aquifers at1.0g/L to downstream aquifers at10.0g/L. The salinity of all in?ows to the basin is0.5g/L.The salt con-centrations of precipitation and percolation to aquifers are 0.1g/L during all time periods.The salt additions to demand sites are assumed to all come from irrigation water and precipi-tation.The loss ratios of salt in link?ows are set to be0.05,equal to the loss ratios of water in corresponding links,while the salt loss through seepage from any link to an aquifer is assumed to be
at the ratio of0.01.Ratios of salt losses at demand sites due to
consumption by water utilization activities are0.3,except that the
nodes representing Afghanistan and the Aral Sea are set to be1,
assuming no return?ow from them and thus salt is totally con-
sumed by these nodes.
PMMNF Results
In this study,the priorities are set as follows:?1?The minimum
storage volume required by each reservoir to meet the water head
for hydropower production and ecology needs,is assigned the
highest priority.?2?Off-stream demands are assigned priorities in
the order of upstream to downstream.All in?ow links to the same
demand node are set to be equal.?3?Two cases for the Aral Sea’s
in?ow demand are considered.In Case P1,it is assigned the
lowest priority while in Case P2it is given the second highest
priority.
The satisfaction ratios de?ned earlier are used to represent the
satisfaction of demand for each demand site in various cases.For
the linear PMMNF programming cases,the satisfaction ratios are
summarized here,but not plotted.More speci?cally,for Case P1,
the demands upstream from the middle node Karakum are nearly
fully satis?ed,and the downstream demands are less satis?ed ex-
cept for Karshi,Bukharaz,and Horezm.This is because the up-
stream demands are assigned higher priorities such that they have
the advantage of being able to take the upstream source water.For
Case P2,Pyandz,the farthest upstream demand node,and the
Nurek hydropower plant,are satis?ed,and the Aral Sea’s satis-
faction ratios amount to100%.However,most of the others have
very low supply/demand satisfaction ratios.Sensitivity analyses
of the effects of water loss coef?cients and other detailed results
of the linear PMMNF programming are reported by Fang et al.?2005?.It is found that water loss?consumption?coef?cients at agricultural demand sites play an important role in water alloca-
tion,because they are the major factors for determining how
much return?ow is available for downstream uses.
The inclusion of salinity constraints not only makes the water
allocation model a large scale nonlinear program problem which
is hard to solve,but it also requires the careful setting of concen-
tration limits.For example,as shown in Fig.2,if the in?ow
mixed concentration limits for agricultural demand sites are set
loosely?e.g.,100g/L?,the solution of the nonlinear PMMNF
program is the same as the one which considers linear water
quantity only.As the limits decreased from100to6g/L,the
water allocation scheme and annual overall satisfaction ratios of
demand sites change very slightly.If the limits are set to be
2g/L,the upstream hydropower plant and agricultural demands
are not affected,but the agricultural demand sites below Bukharaz
have smaller satisfaction ratios due to the reduction of withdraw-als from the main stem to satisfy this salinity constraint.Conse-quently,when the limits are changed from6to2g/L,in?ows to the Aral Sea are increased noticeably during most of the months as shown in Fig.3,and the salt concentrations of the in?ows are decreased accordingly.Fig.4shows that if the in?ow mixed con-centration limits for all the agricultural demand sites are set to be 6g/L,the concentrations of salt in the in?ows to the Aral Sea with a second highest priority are much lower than the case of the lowest priority.This is because more water is allocated to the Aral Sea,while the water quantity allocated to upstream demand sites is reduced.
LMWSR Results
Two cases for the assignment of the weighting factors for de-mands are considered.In Case W1,all agricultural and hydro-power demands are assigned a weight value of10,and the weights of the reservoirs and the Aral Sea are set to be1.In Case W2,all demand nodes are assigned an equal value of1.All agri-cultural demand node loss coef?cients are set to be70%in both cases.
The computational results for the linear LMWSR program-ming show for Case W1that all agricultural and hydropower demands have relatively higher satisfaction ratios than the Aral Sea.This is due to the fact that the demand of the Aral Sea is assigned a lower weight and not because of its downstream loca-tion.For Case W2,the Aral Sea and most agricultural and hydro-power demands are satis?ed at the same ratio of42.5%.Generally speaking,variations among satisfaction ratios are caused by the weights and constraints of the feasible set.Differences in
satis-Fig.2.Overall satisfaction ratios of demand nodes under different in?ow mixed concentration limits for agricultural demand sites ?Case P1
?
Fig. 3.In?ows to Aral Sea under different in?ow mixed concentration limits for agricultural demand sites?Case P1?
JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT?ASCE/JANUARY/FEBRUARY2007/57
faction ratios in Case W2come from the hydrological constraints of the link ?ows and node storages.The results also show that the upstream and downstream demand sites are fairly dealt with and there is no preference given to upstream nodes.Should the weights be properly assigned based on an analysis of the at-tributes of all the demand sites,the lexicographic approach would be able to provide equitable water allocations.Similar to PMMNF,water loss coef?cients at agricultural demand sites play an important role in water allocation.Sensitivity analyses of the effects of the assignment of weights and various water loss coef-?cients,and other detailed results of the linear LMWSR program-ming are reported by Wang et al.?2004?.
As shown in Fig.5when the in?ow mixed concentration limits for agricultural demand sites are set to be 100or 6g/L,the an-nual overall satisfaction ratios of demand sites obtained by non-linear LMWSR programming are the same as those obtained by the linear LMWSR method considering water quantity only.If the limits are set to be 2g/L,the hydropower plant and agricultural demand sites,except for the most upstream Pyandz,have reduced satisfaction ratios and the reduction is evenly distributed among them.As a result,when the limits are changed from 6to 2g/L the quantity and salinity of the in?ows to the Aral Sea are notice-ably increased and decreased,respectively.Fig.6shows that if the in?ow mixed concentration limits for all the agricultural de-mand sites are set to be 6g/L,the concentrations of salt in the in?ows to the Aral Sea with a higher weight are much lower than the case of the lower weight.This is because more water is allo-cated to the Aral Sea,while the water quantity allocated to up-stream demand sites is reduced.
Comparison of PMMNF and LMWSR Results
The most apparent distinction among the two applications is that the annual satisfaction ratios of all agricultural demand sites and the hydropower plant obtained by the LMWSR method are more evenly distributed than those by PMMNF,irrespective of whether or not the nonlinear salinity constraints are included.This is es-sentially caused by their structural differences.Since PMMNF allocates water in accordance with a strict priority order,some junior users cannot receive water during water shortages,which leads to the large differences in allocations and unfairness among uses.Although the differences and unfairness may be reduced through splitting the water demands into subdemands with differ-ent priorities,like the MRWRA method,the structural de?ciency still cannot be eliminated.On the contrary,LMWSR ensures that every user receives its share of the water during shortage periods according to equivalent weights of all uses,and minimizes the differences among weighted satisfaction ratios.Therefore,LMWSR can provide equitable allocations if weights are properly set according to the “equivalent weighted water shortage ratios”rule.
Conclusions
The proposed methods in this paper provide new tools for mod-eling water rights allocations at the basin scale based on various water rights systems and international water allocation principles.The Amu Darya River Basin is utilized as an illustrative applica-tion for demonstrating how to use the new water allocation meth-ods presented in the paper and testing the sequential solution algorithm for PMMNF and the iterative algorithm for LMWSR.For linear problems,it is easy for the algorithms to ?nd global optimal water allocations.When the nonlinear water quality con-straints are considered in water allocation,the simple but effective two-stage approach can ?nd approximate global optimal alloca-tions for the large scale nonlinear programs.
Acknowledgments
The writers would like to express their sincere appreciation to the anonymous referees and the editor whose insightful suggestions enhanced the quality of the
paper.
Fig.4.Salinity of in?ows to Aral Sea with different priorities ?in?ow mixed concentration limits for agricultural demand sites are 6g/L
?Fig.5.Overall satisfaction ratios of demand nodes under different in?ow mixed concentration limits for agricultural demand sites ?Case W1
?
Fig.6.Salinity of in?ows to Aral Sea with different priorities ?in?ow mixed concentration limits for agricultural demand sites are 6g/L ?
58/JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT ?ASCE /JANUARY/FEBRUARY 2007
References
Brooke,A.,Kendrick,D.,Meeraus,A.,and Raman,R.?1998?.GAMS—A users’guide,GAMS Development Corporation,Washington,D.C. Cai,X.,McKinney,D.C.,and Lasdon,L.S.?2001?.“Piece-by-piece approach for solving large nonlinear water resources management models.”J.Water Resour.Plann.Manage.,127?6?,363–368.
Cai,X.,McKinney,D.C.,and Lasdon,L.S.?2002?.“A framework for sustainability analysis in water resources management and application to the Syr Darya Basin.”Water Resour.Res.,38?6?,21/1–21/14. Cech,T.V.?2002?.Principles of water resources,history,development, management,and policy,Wiley,New York,200–206. Dellapenna,J.W.?1994?.“The regulated riparian version of the ASCE model water code:The third way to allocate water.”Water Resour.
Bull.,30?2?,197–204.
Dellapenna,J.W.?2000?.“The importance of getting names right:The myth of markets for water.”William and Mary Environmental Law and Policy Review,25,317–377.
Dellapenna,J.W.,and Stephen,E.D.?2002?.“Water law doctrines as applied to Georgia Water Law.”Rep.Prepared for the Comprehensive State Water Plan Joint Study Committee,Atlanta,?http:// https://www.doczj.com/doc/246784139.html,/water/whitepapers/waterlawdoctrines.pdf??Dec.
28,2002?.
Fang,L.,Wang,L.,and Hipel,K.W.?2005?.“Securing fair water allo-cation in the Aral Sea Basin.”Systems and human science—For safety,security,and dependability,T.Arai,S.Yamamoto,and K.Makino,eds.,Elsevier,Amsterdam,The Netherlands,159–170. Fredericks,J.W.,Labadie,J.W.,and Altenhofen,J.M.?1998?.“Decision support system for conjunctive stream-aquifer management.”J.Water Resour.Plann.Manage.,124?2?,69–78.
Giordano,M.A.,and Wolf,A.T.?2001?.“Incorporating equity into international water agreements.”Social Justice Research,14?4?, 349–366.
Green,G.P.,and Hamilton,J.R.?2000?.“Water allocation,transfer and conservation:Links between policy and hydrology.”Water Resources Development,16?2?,197–208.
Hsu,N.,and Cheng,K.?2002?.“Network?ow optimization model for basin-scale water supply planning.”J.Water Resour.Plann.Manage., 128?2?,102–112.
International Law Association.?1967?.“The Helsinki Rules on the uses of the waters of international rivers.”Rep.of the Committee on the Uses of the Waters of International Rivers,adopted by the Int.
Law Association at52nd Conf.,Helsinki,Finland,?http://www.
https://www.doczj.com/doc/246784139.html,/IntlDocs/Helsinki_Rules.htm??Oct.20, 2002?.
Israel,M.S.,and Lund,J.R.?1999?.“Priority preserving unit penalties in network?ow modeling.”J.Water Resour.Plann.Manage.,125?4?, 205–214.
Loucks,D.P.,Stedinger,J.R.,and Douglas,H.?1981?.Water resources systems planning and analysis,Prentice-Hall,Englewood Cliffs,N.J. Luss,H.?1999?.“On equitable resource allocation problems:A lexico-graphic minimax approach.”Oper.Res.,47?3?,361–378. McKinney,D.C.?2003?.“Cooperative management of transboundary water resources in Central Asia.”?https://www.doczj.com/doc/246784139.html,/prof/ mckinney/papers/aral/CentralAsiaWater-McKinney.pdf??Aug.20, 2003?.
McKinney,D.C.,Cai,X.,Rosegrant,M.W.,Ringler,C.,and Scott,C.A.
?1999?.“Modeling water resources management at the basin level: Review and future directions.”SWIM paper no.6,International Water Management Institute,Colombo,Sri Lanka.
McKinney,D.C.,and Karimov,A.K.?1997?.“Report on model devel-opment:Aral Sea regional allocation model for the Amu Darya River.”Technical Rep.Prepared for Central Asia Mission,U.S.
Agency for International Development,Almaty,Kazakstan. Micklin,P.P.?1991?.“The water management crisis in Soviet Central Asia.”The Carl Beck papers in Russian and Eastern-European stud-ies,no.105,Center for Russian and Eastern European Studies,Univ.
of Pittsburgh,Pittsburgh.
Miller,K.A.,Rhodes,S.L.,and MacDonnell,L.J.?1997?.“Water allo-cation in a changing climate:Institutions and adaptation.”Clim.
Change,35?2?,157–177.
Murtagh,B.A.,Saunders,M.A.,Murray,W.,Gill,P.E.,Raman,R.,and Kalvelagen,E.?2002?.GAMS/MINOS:A solver for large-scale non-linear optimization problems,GAMS Development Corporation, Washington,D.C.
Ogryczak,W.,Sliwinski,T.,and Wierzbicki,A.?2003?.“Fair resource allocation schemes and network dimensioning problems.”Journal of Telecommunications and Information Technology,3,34–42. Raskin,P.,Hansen,E.,Zhu,Z.,Iwra,M.,and Stavisky,D.?1992?.
“Simulation of water supply and demand in the Aral Sea region.”
Water Int.,17?2?,55–67.
Savenije,H.H.G.,and Van der Zaag,P.?2000?.“Conceptual framework for the management of shared river basins:With special reference to the SADC and EU.”Water Policy,2?1–2?,9–45.
Seyam,I.,Savenije,H.H.G.,Aerts,J.,and Schepel,M.?2000?.“Algo-rithms for water resources distribution in international river basins.”
Phys.Chem.Earth,Part B,25?3?,309–314.
Tu M.,Hsu,N.,and Yeh,W.W.-G.?2003?.“Optimization of reservoir management and operation with hedging rules.”J.Water Resour.
Plann.Manage.,129?2?,86–97.
United Nations?UN?.?1997?.“Convention on the law of the non-navigational uses of international watercourses.”U.N.Doc.A/51/869, adopted by the UN General Assembly in resolution51/229of May21, 1997,?https://www.doczj.com/doc/246784139.html,/law/ilc/texts/nnavfra.htm??Oct.20,2002?. United Nations/World Water Assessment Programme?UN/WWAP?.
?2003?.UN World Water Development Rep.:Water for People,Water for Life,United Nations Educational,Scienti?c and Cultural Organi-zation and Berghahn Books,Paris.
Van der Zaag,P.,Seyam,I.M.,and Savenije,H.H.G.?2002?.“Towards measurable criteria for the equitable sharing of international water resources.”Water Policy,4?1?,19–32.
Wang,L.,Fang,L.,and Hipel,K.W.?2003?“Water resources allocation:
A cooperative game theoretic approach.”J.Environmental Informat-
ics,2?2?,11–22.
Wang,L.,Fang,L.,and Hipel,K.W.?2004?.“Lexicographic minimax approach to fair water allocation problems.”Proc.2004IEEE Int.
Conf.on Systems,Man,and Cybernetics,The Hague,The Nether-lands,1038–1043.
Wolf,A.T.?1999?.“Criteria for equitable allocations:The heart of inter-national water con?ict.”Natural Resources Forum,23?1?,3–30. Wurbs,R.A.?2001?.“Assessing water availability under a water rights priority system.”J.Water Resour.Plann.Manage.,127?4?,235–243. Yeh,W.W-G.?1985?.“Reservoir management and operation models:A state-of-the-art review.”Water Resour.Res.,21?12?,1798–1818.
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