Research Article Multiobjective Optimization Problem of...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 354206 8 pageshttpdxdoiorg1011552013354206

Research ArticleMultiobjective Optimization Problem of Multireservoir Systemin Semiarid Areas

Z J Chen12 Z J Cheng1 and X Q Yan1

1 Electrical Engineering College XinJiang University XinJiang Urumqi 830046 China2Dapartment of Automation Electrical Engineering College XinJiang University XinJiang Urumqi 830046 China

Correspondence should be addressed to Z J Chen chenzj1110163com

Received 27 June 2012 Accepted 6 September 2012

Academic Editor Zidong Wang

Copyright copy 2013 Z J Chen et alThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

With the increasing scarcity of water resources the growing importance of the optimization operation of the multireservoir systemin water resources development utilization and management is increasingly evident Some of the existing optimization methodsare inadequate in applicability and effectiveness Therefore we need further research in how to enhance the applicability andeffectiveness of the algorithm On the basis of the research of the multireservoir systemrsquos operating parameters in the Urumqi Riverbasin we establish a multiobjective optimization problem (MOP)model of water resources development which meets the require-ments of water resources development In the mathematical model the domestic water consumption is the biggest the productionof industry and agricultural is the largest the gross output value of industry and agricultural is the highest and the investment of thewater development is the minimum We use the weighted variable-step shuffled frog leaping algorithm (SFLA) to resolve it whichsatisfies the constraints Through establishing the test function and performance metrics we deduce the evolutionary algorithmswhich suit for solving MOP of the scheduling and realize the multiobjective optimization of the multireservoir system After thatusing the fuzzy theory we convert the competitive multiobjective function into single objective problem of maximum satisfactionwhich is the only solution A feasible solution is provided to resolve the multiobjective scheduling optimization of multireservoirsystem in the Urumqi River basin It is the significance of the layout of production the regional protection of ecological environ-ment and the sufficient and rational use of natural resources in Urumqi and the surrounding areas

1 Overview

Xinjiang is the most north-west province in China It islocated in the inland semiarid area which accounts for 3of water resources in China The average annual surfacewater runoff in Xinjiang is 882 billion cubic meters and5146 cubic meters per capita which is 225 times the averageof China [1] The average annual groundwater exploitationis about 251 billion cubic meters In addition the glaciersreserves in Xinjiang accounted for 50 percent of Chinarsquos totalYet Xinjiang is located in the hinterland of the Eurasiancontinent the climate is dry it is affected by seasonal factorseasily So the spatial and temporal distribution of the waterresource is extremely uneven [2]

As the capital and the center of the cultural economicof the Xinjiang Uygur autonomous region the population ofUrumqi city is nearly 5millionThe urbanwater resources are

mainly provided by the Urumqi River basin which belongsto the northern slope of Tianshan inland river system It con-sists of the Urumqi River Nanshan River Dongshan RiverToutunhe River and so forth Urumqi River originates in theGlacier number 1 of Tianger peak crosses the urban area ofUrumqi city and finally flows into Beishawo in the southernmargin of Junggar basin Its length is 214 km drainage areais 5803 square kilometers average annual runoff is 237million cubic meters Along the Urumqi River basin thereare more than 30 reservoirs as follows Daxigou reservoirUlabo reservoir Hongyanchi reservoir Mengjin reservoirand so on It provides domestic water farmland and greenspace irrigationwater industrial water and aquaculturewaterfor the residents in Urumqi city and the surrounding areaMeanwhile it also plays an important role in flood controldrought resistance and others [3]

2 Mathematical Problems in Engineering

In inland arid and semiarid regions such as Urumqi theshortage of water resources has become a major restrictingfactor of the cityrsquos development With the rapid developmentof the city as well as the gradual deterioration of the ecologicalenvironment severe water shortage has become the bottle-neck of the constraints and limitations of social and economicdevelopment in Urumqi In order to alleviate this problemin the beginning of this century ldquoYinEJiUrdquo project wasimplemented Using hydraulic engineering part of northernXinjiang Irtysh water was injected into the Urumqi RiverWith the implementation of this project a new water sourceis added into the Urumqi River basin It alleviated the statusof the serious water shortage in Urumqi to a certain extentHowever due to the limited increase of the water resourcesit cannot fully meet the demand Therefore from the man-agement point of view in addition to the plan of reservoirsconstruction in the upstream we can also apply the multiob-jective optimization operation In this way we can alleviatethe contradiction between the supply and the demand of thewater resource atmaximumand realize the optimal allocationof water resources in Urumqi river basin gradually

Since the early 1960s the research on the optimizationoperation of themultireservoir systemhas been carried out inChinaThe research developed rapidly after the 1980s and gota large amount of achievements Dong introduced the princi-ple and contents of the dynamic programming optimizationtechnique and its application in the hydro power stationreservoir (group) scheduling as well as the principle ofthe multiobjective decision Feng established the theoreticalfoundation for the multiobjective optimum scheduling of thereservoirs He also discussed the basic principle and methodof the large-scale system decomposition and coordinationsystematically Wang introduced the theory and applicationof the large-scale system optimization Using the fuzzyidentificationmode Li achieved the level characteristic of thesample and the contribution rate of the relevant factors inthe evaluation process of the evaluation which aims at thecomprehensive evaluation of the reservoir water resourcescarrying capacity At the same time he achieved assessmentof water resources carrying capacity in regional reservoir [4]

In the recent decades several researchers (HouckYakowitz and Needham et al) proposed methods appliedin optimization operation of multireservoir system such aslinear programming nonlinear programming and dynamicplanning Although those optimization methods provide away to resolve the problem of reservoir group schedulingthere are also some problems First some models need alot to simplify therefore their simulation performance ispoor Second some of the nonlinear methods are restrictedin practical application because of the lack of the boundaryconditions and more parameters Moreover those methodsare hard to simulate the experience of scheduling knowledgeConsequently the research on the optimization scheduling ismore and the practical application is less [5]

In the process of the multireservoir system operation theregulation and scheduling consist of the conflicting and theintereffecting multitargets In a specific field people oftenencounter the matching problem which requires multiplesolutions to achieve optimization at the same time It is

the multiobjective optimization problem MOP The valida-tion of MOP is whether the optimization target is more thanone and requires the simultaneous processing In engineeringapplications the MOP is very significant These engineeringapplications are very complicated and difficult They are alsoone of the main research fields Since the early nineteensixties the MOP has attracted more and more researchersrsquoattention from different fields For this reason it has veryimportant research value and practical significance to solveMOP [6]

There are a large number of monographs focusing on themethods to resolve the MOP In general these books can bedivided into two categories One uses the classical optimiza-tionmethods and the other uses themethods based on evolu-tionary algorithmwhich simulates the biological evolution oractivity behavior Because evolutionary algorithmrsquos constraintconditions are low and solution quality is high they havebeen widely applied in resolving the complex optimizationproblem At present the popular evolutionary algorithm toresolve the MOP is as follows the multiobjective geneticalgorithm [7] the artificial ant colony algorithm [8] the mul-tiobjective particle swarm optimization algorithm [9] andthe shuffled frog leaping algorithm [10]

The shuffled frog leaping algorithm (SFLA) is a biologicalevolutionary algorithm based on swarm intelligence whichwas proposed by Eusuff and Lansey in 2003 The SFLA hasthe characteristics of simple concept fewer parameters fastcalculation speed good global search ability easy to imple-ment and so on So at present it is wildly applied to thefields of the network of water resources allocation issuesthe function optimization themultiproduct pipeline networkoptimization the combinatorial optimization the image pro-cessing the multiuser detection and so forth Also themethod achieved good results As a kind of optimizationmethod the SHLAhas certain advantages in the optimizationoperation of the multireservoir system and has broad appli-cation prospects [11ndash13]

Thepurpose of this paper is to study the operation param-eters Based on these parameters we set up the operationmathematical model and use the SFLA to realize the opti-mization allocation of the multireservoir system in UrumqiRiver basin [14]

2 The Description of the MOP

Using words the MOP can be described as an optimizationproblem which consists of 119863 decision variable parameters119873 objective functions and 119898 + 119899 constraints The decisionvariables the objective function and the constraint conditionare function relation In noninferior solution set decisionmakers can only choose a satisfactory noninferior solutionas the final solution according to the specific requirements ofthe problemThemathematic form of MOP can be describedas follows

min (119910) = 119891 (119909) = [1198911(119909) 119891

2(119909) 119891

119899(119909)]

119899 = 1 2 119873

Mathematical Problems in Engineering 3

st 119892119894(119909) le 0 119894 = 1 2 119898

ℎ119895(119909) = 0 119895 = 1 2 119896

119909 = [1199091 1199092 119909

119889 119909

119863]

119909119889-min le 119909

119889le 119909119889-max 119889 = 1 2 119863

(1)

where 119909 is the decision-making relative to the amount of the119863-dimensional 119910 is the target vector 119892(119909) le 0 is the 119894thinequality constraint ℎ

119895(119909) = 0 is the 119895th equality constraints

119891119899(119909) is the 119899th objective function 119883 the decision space

formed by the decision vector 119892119894(119909) le 0 and ℎ

119895(119909) = 0 deter-

mine the feasible solution domain and 119909119889-min and 119909

119889-max areupper and lower searching limits of each dimension vector[15]

For optimal solution or noninferior optimal solution inthe MOP we can define the following

Definition 1 For any 119889 isin [1 119863] to meet 119909lowast119889

le 119909119889and exits

1198890

isin [1 119863] there is 119909lowast

119889le 1199091198890 then the vector 119909

lowast

= [119909lowast

1

119909lowast

2 119909

lowast

119863] dominates the vector 119909 = [119909

1 1199092 119909

119863]

If 119891(119909lowast) dominates 119891(119909) the following two conditionsmust be met

forall119899 119891119899(119909lowast

) le 119891119899(119909) 119899 = 1 2 119873

exist1198990 1198911198990(119909lowast

) lt 1198911198990(119909) 1 le 119899

0le 119873

(2)

The dominating relation of 119891(119909) is consistent with thedominating relation of 119909

Definition 2 The Pareto optimal solution cannot be domi-nated by any solution in the feasible solution set If 119909lowast is apoint in the searching space 119909lowast is the noninferior optimalsolution If and only if exit 119909 (in the feasible domain of thesearching space) it makes the establishment of (119909lowast) lt 119891(119909)119899 = 1 2 119873

Definition 3 Given a MOP of 119891(119909) if and only if for any 119909

(in the searching space) there is 119891119899(119909) lt 119891

119899(119909lowast) 119891(119909lowast) is the

global optimal solution

Definition 4 The solution set consisted of all the noninferioroptimal solutions is called Pareto optimal set of theMOP alsocalled acceptable solution set or efficient solution set

3 The Description of the Shuffled FrogLeaping Algorithm (SFLA)

The SFLA is a kind of cooperative evolutionary algorithmenlightened by natural biological mimics which simulatesthe information exchange classified by subgroup in the pro-cess of the frog group looking for food First we generatea group of initial solution (population) from solution spacerandomly Then we divide the whole population into morethan one subpopulation in which the frogs execute subpop-ulation internal search according to a certain strategy Afterthe end of the number of searches in defined subpopulation

mix all the frogs sort and divide the subpopulation again andexchange information between each subpopulation globallyThe sub-population internal search and the global informa-tion exchange continue alternating until they satisfy theconvergence condition or reach the maximum number ofevolutionThe following is mathematical model of the SHLA

31 Dividing Subpopulation Suppose that the number offrogs in the population (candidate solution) is119873 the numberof subpopulation is 119896 the number of candidate solutionsis 119899 119883

119895= (1199091198951 1199091198952 119909

119895119863) represents the 119895th candidate

solution 119895 (0 le 119895 le 119873) and 119863 represents the dimensionsof the candidate solution For the initial population 119878 =

(1198831 1198832 119883

119873) generated randomly after sorted according

to the fitness 119891(119909) in descending the first candidate solutionis assigned to the first subpopulation the second candidatesolution is assigned to the second subpopulation the 119896thcandidate solution is assigned to the 119896th subpopulation the(119896 + 1)th candidate solution is assigned to the first subpopu-lation and the (119896 + 2)th candidate solution is assigned to thesecond subpopulation repeat until119873 candidate solutions aredistributed

32 Internal Search in Subpopulation Suppose that 119883119887is

the candidate solution in which fitness is best in the sub-population 119883

119908is the candidate solution in which fitness is

worst in the subpopulation and119883119898is the candidate solution

in which fitness is best in the whole population For everysubpopulation we do internal search It is to update 119883

119908of

the subpopulation The search strategy is as follows

1198831015840= 119883119908+ 119877 times (119883

119887minus 119883119908) (3)

where 1198831015840 is the new solution produced by (3) and 119877 is a

random number between 0 and 1 If 1198831015840 is superior to 119883119908

then 119883119908

= 1198831015840 Otherwise 119883

119887is replaced by 119883

119898in (3) and

repeats the search strategy If1198831015840 is not always superior to119883119908

a random candidate solution is generated to replace 119883119908 We

repeat the above steps until the number of searches is greaterthan the largest number of the largest subpopulation internalsearch

33 Global Information Exchange When all the internalupdates of subpopulation are completed we do the subpop-ulation dividing and internal search again until it meet thetermination conditions (converges to the optimal solution orreaches the maximum number of evolving generation)

4 Reservoir Group Optimization Dispatchingof Urumqi River Basin

In the process of multireservoir system optimization opera-tion of Urumqi River basin we should consider the followingfive aspects of the requirements first to satisfy the demandfor residential water second to satisfy the water demand ofthe industrial and the agricultural production third to satisfythe water demand of the ecological environment fourth tocomplete the industrial and the agricultural planning output


target last the investment of water resources developmentis minimum under the premise to satisfy the demand ofwater We consider these five aspects which are mutual com-pensation to achieve the most satisfactory results under thecondition that satisfies the system performance [16]

41 The Objective Functions of Multireservoir System Opti-mization Operation With the above we take the followingas the goal function people living water consumption is thelargest the industrial and the agricultural production waterconsumption is the largest ecological environment waterconsumption is the largest the gross output value of industryand agriculture is the highest and the investment of waterresources development is minimum [17]

According to the MOP the objective function the deci-sion variables and the constraints of the reservoir groupoptimization dispatching can be described as follows

min max 119891119898(1199091 1199092 119909

119899) 119898 = 1 2 119872

st (1) 119892119896(1199091 1199092 119909

119899) ge 0 119896 = 1 2 119872

(2) ℎ119897(1199091 1199092 119909

119899) = 0 119897 = 1 2 119871

(3) 119909(119871)

119894le 119909119894le 119909(119880)

119894 119894 = 1 2 119899

(4)

where119883 sub 119877119899 is the decision space 119909 = (119909

1 1199092 119909

119899) isin 119883

and 119909119894

(119871) and 119909119894

(119867) are upper and lower bounds of variablesrespectively Further we can define collection Ω as the fea-sible domain of (4)

Ω =119909 | 119892119896(1199091 119909

119899) ge 0 ℎ

119896(1199091 119909

119899) = 0

119909(119871)

119894le 119909119894le 119909(119880)

119894 Ω sub 119883

(5)

Among the objective functions of multireservoir systemoptimization operation the investment of water resourcesdevelopment is the minimizing objective function and theothers are the maximum objective functions In order tomaintain unity by taking the negative of the water resourcesinvestment we convert all the problems into the maximumobjective functions MOP The objective functions can bedescribed as follows

max1198911=

119879

sum

119905=1

119868

sum

119894

11988611198941198752

119866119894(119905)+ 1205731119894119875119866119894(119905)

+ 1205741119894

max1198912=

119879

sum

119905=1

119869

sum

119895

11988621198951198752

119878119895(119905)+ 1205732119895119875119878119895(119905)

+ 1205742119895

max1198913=

119879

sum

119905=1

119870

sum

119896

11988631198961198752

119882119896(119905)+ 1205733119896119875119882119896(119905)

+ 1205743119896

max1198914=

119879

sum

119905=1

119871

sum

119897

11988641198971198752

119867119897(119905)+ 1205734119897119875119867119897(119905)

+ 1205744119897

max1198915=

119879

sum

119905=1

119872

sum

119898

11988651198981198752

119880119898(119905)+ 1205735119898119875119880119898(119905)

+ 1205745119898

(6)

For (6) 119868 119869119870 119871 and119872 denote the effects of five optimalgoal influence factors respectively 119875

119866119894(119905)is the water con-

sumption of the 119894th living water user in the 119905th time interval119875119878119894(119905)

is the water consumption of the 119895th industry and agri-culture water user in the 119905th time interval 119875

119882119896(119905)is the water

consumption of the 119896th ecological water user in the 119905th timeinterval 119875

119867119897(119905)is the output value of the 119897th production unit

in the 119905th time interval 119875119880119898(119905)

is the investment amount ofthe 119898th water resources development unit in the 119905th timeinterval 119886119899

lowast 120573119899lowast 120574119899lowast(119899 = 1 2 3 4 5

lowast

= 119894 119895 119896 119897 119898) is theinfluence coefficient of each factor 119879 is the scheduling cycle

42 The Constraint Condition of Multireservoir System Opti-mizationOperation In dealingwith the constraint conditionaccording to the objective function we make multireservoirsystem operation satisfaction as constraint condition In viewof reservoir group optimal model the constraint conditionsare as follows

Thewater consumption demandof people living ismainlyaffected by the climate change and daily change So theliving water consumption change and its rate of change arealways in a certain range As one of the important indexeswhich influence the reservoir group operation under thepremise of the whole objective optimization and reducingthe consumption of water as far as possible the constraintconditions of the living water consumption can be describedas follows

119875119866min lt sum119875

119866lt 119875119866max

Δsum119875119866lt Δ119875119866max

(7)

where sum119875119866is the living water consumption of the reservoir

group system in unit interval (usually a day) Δsum119875119866is the

variation of the living water consumption of the reservoirgroup system in unit interval 119875

119866min and 119875119866max are the

minimum and the maximum living water consumptionrespectively Δ119875

119866max is the largest living water quantityvariation in unit interval [18]

The industrial and agricultural water consumption isdirectly related to the production situation In addition tothis the agricultural water consumption is also affected by theweather the rainfall the plant growth period and other fac-tors The industrial water consumption is also affected by theenvironment temperature As another important indicator ofthe reservoir group operation satisfaction the constraint con-ditions of the industrial and agricultural water consumptioncan be described as follows

119875119878min lt sum119875

119878lt 119875119878max

Δsum119875119878lt Δ119875119878max

(8)

where sum119875119878is the industrial and the agricultural water

consumption of the reservoir group system in unit interval(usually a day) Δsum119875

119878is the variation of the industrial and

the agricultural water consumption of the reservoir groupsystem in unit interval119875

119878min and119875119878max are theminimumand

themaximum industrial and agricultural water consumption


respectively Δ119875119878max is the largest industrial and agricultural

water quantity variation in unit interval [19]The ecological environmental water consumption is the

other important satisfaction index of the reservoir groupoperation The constraint conditions of the ecological envi-ronmental water consumption can be described as follows

119875119882min lt sum119875

119882lt 119875119882max

Δsum119875119882

lt Δ119875119882max

(9)

wheresum119875119882is the ecological environmental water consump-

tion of the reservoir group system within the unit interval(usually a day) Δsum119875

119882is the variation of the ecological envi-

ronmental water consumption of the reservoir group systemwithin the unit interval 119875

119882min and 119875119882max are the mini-

mum and the maximum industrial and agricultural waterconsumption respectively Δ119875

119882max is the largest ecologi-cal environmental water quantity variation within the unitinterval The ecological environmental water consumption isinfluenced by the climate environment and external factorsand it relates to the living water consumption the industrialand agricultural water consumption

The gross output value of industry and agriculture is oneof the most important satisfaction indexes of the reservoirgroup operationWe can describe it with the social gross out-put So its constraint conditions can be described as follows

sum119875119867

ge 119875119867min (10)

where sum119875119867

is the gross output value of industry andagriculture in a unit of time (usually a month) and Δ119875

119867min isthe minimum gross output value in unit interval

It is the necessary conditions for the investment of thewater resources development to ensure the normal opera-tion of the reservoir group Under the general conditionthe minimum investment maximizes the benefits So theconstraint conditions of the investment of thewater resourcesdevelopment can be described as follows

sum119875119880le 119875119880max

sum 119875119880

Count (119875119880)ge 119875119880min

(11)

where sum119875119880is the investment of the water resources devel-

opment 119875119880max is the maximum investment of the water

resources development Count(119875119880) is the water resources

investment project number and119875119880min is theminimum social

product of the individual water resources of the investmentproject

43 Solving the MOP of the Reservoir Group OptimizationDispatching Using SFLA For evaluation of each frog (thesolution) the following fitness function can be used

119891 (119909) =1

119869 + 120572 (12)

where 119869 is themaximum value of the solutions which is max-imum satisfaction 120572 (0 lt 120572 lt 01) is a constant Obviously

Start

Set parameters and initial frog population

Calculate the fitness of each frog

in the population

Sort and divide the population according

to fitness in descending

Update each population in part

Mix the updated population and replace the

original group with new

Whether is satisfied the termination condition

N

Output the optimal solution

End

Y

Figure 1 The flow chart of SFLA

the smaller the value of the objective function the greater thefitness of frog

We generate 119871 frogs (the solution of the problem) as theinitial population the update strategies of each populationlocal depth search are as follows

119880 (119902) = 119875119882

+ 119878

119878 =

min int [rand (119875119861minus 119875119882)] 119878max

Positive directionmax int [rand (119875

119861minus 119875119882)] minus119878max

Negative direction

(13)

where 119880(119902) is the present moment solution 119875119882

is the lastmoment solution 119878 is the moving distance on the compo-nents rand is a random number between 0 and 1 and 119875

119861and

119875119882are the optimal and worst solution of the fitnessThe steps

of the SFLA are as follows

Step 1 Set parameters and initial frog population

Step 2 Calculate the fitness of each frog in the population

Step 3 Sort and divide the population according to fitness indescending

Step 4 Update each population in part

Step 5 Mix the updated population and replace the originalgroup with new

Step 6 If it meets the termination condition output the opti-mal solution and finish operation if not go Step 2

The program flow chart is shown in Figure 1

44The Improvement of the SFLATheWeighted Variable StepSFLA Through studying the basic principle of the SFLA and


The basic SFLA

1 2 5 3

4 5 1 4

3 4 3 4 2 2 4 3

1 2 5 3

4 5 1 4

3 3 4 4

Before Present

1 2 5 3

4 5 1 4

3 4 3 4

The improved SFLA

2 2 4 3

1 2 5 3

4 5 1 4

3 3 4 4

PW

PB

U(q)

+ + +

Figure 2 The effect of the improved SFLA

in order to solve the engineering problems using the SFLAit is necessary to improve algorithm efficiency and searchbreadth The method using the weighted variable step-sizesearch is adopted We call this method the weighted variablestep SFLA

In the process of the basic SFLA for each decision variableis the same value in rand So it will limit the random of thedecision variables to the optimal directionThis problem willbe avoided if each random decision variable generates differ-ent rand valuesThe step formula can be improved as follows

119878(119894)

=

min int [rand(119894) (119875(119894)119861

minus 119875(119894)

119882)] 119878max

Positive direction

max int [rand(119894) (119875(119894)119861

minus 119875(119894)

119882)] minus119878max

Negative direction

(14)

At the same time in order to keep solution with betterfitness and eliminate the solution with bad fitness to eachdecision variable rand is multiplied by a variable weightcoefficient which relates to the fitness In order to reflect theglobal search steps do not only relate to present value butalso relate to before value The step formula can be improvedas follows

119878(119894)

=

min int [120572119891(119909)

rand(119894) (119875(119894)119861

minus 119875(119894)

119882) + 120573119891(119909)

119878(119894minus1)

] 119878max

Positive direction

max int [120572119891(119909)

rand(119894) (119875(119894)119861

minus 119875(119894)

119882) + 120573119891(119909)

119878(119894minus1)

] minus119878max

Negative direction(15)

120572119891(119909)

+ 120573119891(119909)

= 1 (0 lt 120572119891(119909)

120573119891(119909)

lt 1)Figure 2 is a decision effect using the improved SFLA

compared with the basic SFLA [20]In the process of local search we can improve the search

method make improvements in the posterior half of thefrog in the population which is called 12 search methodUsing this kind of method the update is more efficientthe transformation can be ensured towards the high fitnessdirection the search range can be widened

The optimization process flow of improved SFLA isshown in Figure 3

Start

Code

Initial population

Calculation of fitness

Dividepopulation

Evolution

Evolutionary completed

Divide new population

Obtainsolution

Whether are satisfied the

convergence requirements

End

YN

Y

Acquisition of basic parameters

Access to the system parameters

Figure 3 The flow chart of improved SFLA

1

a b

μ(f (x))

f (x)

Figure 4 The objective membership function

45 The Greatest Satisfaction Transformation of the ReservoirGroup MOP There are five competing goals of the functionin the reservoir group optimization dispatchingMOPWe canuse the fuzzy set theory to translate them into single-objectiveproblem using the descending half-line as their membershipfunction [21 22] shown in Figure 4

The objective membership function can be described asfollows

120583 (119891 (119909)) =

1 119891 (119909) lt 119886

119887 minus 119891 (119909)

119887 minus 119886 119886 le 119891 (119909) le 119887

0 119891 (119909) gt 119886

(16)

where ldquo119886rdquo is the optimization target of the single-objectivefunction ldquo119886 + 119887rdquo is the maximum acceptable values and 120582 isthe minimum membership variables in all the membershipfunction called the degree of satisfaction

120582 = min 120583 (1198911(119909) 119891

2(119909) 119891

3(119909)) (17)

According to the principle of maximum degree of mem-bership we can translate the MOP into single-objectiveproblem with satisfaction maximization [23]

5 The Conclusion

For competitive multiobjective function of multireservoirsystem optimization operation in the inland river basin of


semiarid areas using fuzzy set theory we convert the MOPof the multireservoir system into single-objective problem ofmaximum satisfaction The only solution of fuzzy reasoningmakes each objective function achieve optimization [24]Using this method we simplify the water resources opti-mization scheduling problem It is advantageous for theMOPof the inland river basin of arid areas to accelerate theengineering process [25]

For the requirements of the hydrology resource and thereservoir scheduling optimization analysis there are moreand more new theories and new technologies applied to thefield of water resources In the example of Urumqi Riverbasin we resolved the MOP of multireservoir system opti-mization operation in the semiarid areas by the SFLA Theresearch can promote the optimal scheduling and scientificmanagement of water resources in the semiarid inland areas[26] With the development of the city the problem ofrational management is increasingly important It is of greatsignificance for the development of the cities to resolve theMOP of the water resource optimal scheduling

In the process of research on the optimization operationof the multireservoir system besides the appropriate opti-mization algorithm the processing of the input informationis essential The input information of the system is randomin essence So the system can be considered as a nonlinearstochastic system At present the latest research of theseissues focuses on the119867infinfiltering [27 28] and the distributedfiltering [29] The issue will be discussed in the late papers

Acknowledgments

This work was financially supported by Innovation Fund forSmall Technology-Based Firms 11C26216506453 Also thiswork was financially supported by Key Projects of the Depar-tment of Education of Xinjiang Uygur Autonomous RegionXJEDU2010I07

References

[1] B Ye D Yang K Jiao et al ldquoThe Urumqi River source GlacierNo 1 Tianshan China changes over the past 45 yearsrdquo Geo-physical Research Letters vol 32 no 21 pp 1ndash4 2005

[2] Z Li W Wang M Zhang F Wang and H Li ldquoObservedchanges in streamflow at the headwaters of the Urumqi Rivereastern Tianshan central Asiardquo Hydrological Processes vol 24no 2 pp 217ndash224 2010

[3] L H Li M J de la Paix X Chen et al ldquoStudy on productivityof epilithic algae in Urumqi River Basin in Northwest ChinardquoAfrican Journal ofMicrobiology Research vol 5 no 14 pp 1888ndash1895 2011

[4] X Zheng H Guo and B Zhang ldquoStudy on application ofthe SD-MOP mix model to urban water resource planning ofQinghuangdao cityrdquo Advances in Water Science vol 13 no 3pp 351ndash357 2002

[5] V E Efimov ldquoProblem of water resources in the Middle EastrdquoGigiena i Sanitariia no 1 pp 21ndash23 1994

[6] C Drake ldquoWater resource conflicts in the Middle Eastrdquo Journalof Geography vol 96 no 1 pp 4ndash12 1997

[7] H Ishibashi H E Aguirre K Tanaka and T Sugimura ldquoMulti-objective optimization with improved genetic algorithmrdquo inProceedings of the IEEE International Conference on SystemsMan and Cybernetics vol 5 pp 3852ndash3857 October 2000

[8] S Ruoying W Xingfen and Z Gang ldquoAn ant colony opti-mization approach to multi-objective supply chain modelrdquo inProceedings of the 2nd IEEE International Conference on SecureSystem Integration and Reliability Improvement (SSIRI rsquo08) pp193ndash194 July 2008

[9] E Granger D Prieur and J F Connolly ldquoEvolving ARTMAPneural networks usingmulti-objective particle swarmoptimiza-tionrdquo in IEEE Congress on Evolutionary Computation (CEC rsquo10)pp 1ndash8 July 2010

[10] L Yinghai D Xiaohua L Cuimei et al ldquoA modified shuffledfrog leaping algorithm and its application to short-term hydro-thermal schedulingrdquo inProceedings of the 7th International Con-ference on Natural Computation (ICNC rsquo11) pp 1909ndash1913 July2011

[11] L Yan X-F Kong and R-Q Hao ldquoModified shuffled frogleaping algorithm applying on logistics distribution vehiclerounting problemrdquo in Proceedings of the 4th International Con-ference on the Biomedical Engineering and Informatics (BMEIrsquo11) pp 2277ndash2280 October 2011

[12] A Khorsandi A Alimardani B Vahidi and S H HosseinianldquoHybrid shuffled frog leaping algorithm and Nelder-Mead sim-plex search for optimal reactive power dispatchrdquo IET Genera-tion Transmission and Distribution vol 5 no 2 pp 249ndash2562011

[13] MWang andWDi ldquoAmodified shuffled frog leaping algorithmfor the traveling salesman problemrdquo in Proceedings of the 6thInternational Conference on Natural Computation (ICNC rsquo10)pp 3701ndash3705 August 2010

[14] L Ping L Qiang andW Chenxi ldquoModified shuffled frog leap-ing algorithm based on new searching strategyrdquo in Proceedingsof the 7th International Conference on Natural Computation(ICNC rsquo11) pp 1346ndash1350 July 2011

[15] L DayongW JiganD Zengchuan et al ldquoStudy onmulti-objec-tive optimization model of industrial structure based on waterresources and its applicationrdquo in Proceedings of the InternationalConference on Information Engineering and Computer Science(ICIECS rsquo09) pp 1ndash5 December 2009

[16] H J Peng M H Zhou and H H Zhang ldquoResearch on multi-objective collaborative optimization model and algorithm ofsupply chain system based on complex needsrdquo in Proceedings ofthe WRI World Congress on Software Engineering (WCSE rsquo09)pp 424ndash428 May 2009

[17] R Bos ldquoWater resources development policies environmentalmanagement and human healthrdquo Parasitology Today vol 6 no6 pp 173ndash174 1990

[18] Z Sanyou C Shizhong Z Jiang et al ldquoDynamic constrainedmulti-objective model for solving constrained optimizationproblemrdquo in Proceedings of the IEEE Congress on EvolutionaryComputation (CEC rsquo11) pp 2041ndash2046 June 2011

[19] G Yirsquonan C Jian andM Xiaoping ldquoCoking optimization con-trol model based on hierarchical multi-objective evolutionaryalgorithmrdquo in Proceedings of the 6th World Congress on Intelli-gent Control and Automation (WCICA rsquo06) pp 6544ndash6548June 2006

[20] M-R Chen X Li and N Wang ldquoAn improved shuffledfrog-leaping algorithm for job-shop scheduling problemrdquo inProceedings of the 2nd International Conference on Innovations


in Bio-InspiredComputing andApplications (IBICA rsquo11) pp 203ndash206 December 2011

[21] S Bandyopadhyay ldquoMultiobjective simulated annealing forfuzzy clustering with stability and validityrdquo IEEE Transactionson Systems Man and Cybernetics Part C vol 45 no 5 pp 682ndash691 2011

[22] O Guenounou A Belmehdi and B Dahhou ldquoMulti-objectiveoptimization of TSK fuzzy modelsrdquo in Proceedings of the 5thInternational Multi-Conference on Systems Signals and Devices(SSD rsquo08) pp 1ndash6 July 2008

[23] C L Chen and C P Weng ldquoA multi-objective problem basedon fuzzy inference with application to parametric design of anelectrophotographic systemrdquo Expert Systems with Applicationsvol 38 no 7 pp 8673ndash8683 2011

[24] L Zhangjun and L Yaolong ldquoA new multi-objective optimiza-tion model based on fuzzy probability theoryrdquo in Proceedings ofthe 7th International Conference on Fuzzy Systems and Knowl-edge Discovery (FSKD rsquo10) pp 54ndash56 August 2010

[25] S Jain and K Lachhwani ldquoFuzzy programming approach tosolve multi objective linear fractional program with homo-geneous constraintsrdquo Proceedings of the National Academy ofSciences of India A vol 81 pp 29ndash38 2011

[26] J Zhang H Lu and L Fan ldquoThe multi-objective optimizationalgorithm to a simple model of urban transit routing problemrdquoin Proceedings of the 6th International Conference on NaturalComputation (ICNC rsquo10) pp 2812ndash2815 August 2010

[27] Z D Wang B Shen and X H Liu ldquoH-infinity filtering withrandomly occurring sensor saturations and missing measure-mentsrdquo Automatica vol 48 no 3 pp 556ndash562 2012

[28] W Zidong S Bo and S Huisheng ldquoQuantized H-infinity con-trol for nonlinear stochastic time-delay systems with missingmeasurementsrdquo IEEE Transactions on Automatic Control vol57 no 6 pp 1431ndash1444 2012

[29] D Hongli W Zidong and G Huijun ldquoDistributed filteringfor a class of time-varying systems over sensor networks withquantization errors and successive packet dropoutsrdquo IEEETransactions on Signal Processing vol 60 no 6 pp 3164ndash3173

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


In inland arid and semiarid regions such as Urumqi theshortage of water resources has become a major restrictingfactor of the cityrsquos development With the rapid developmentof the city as well as the gradual deterioration of the ecologicalenvironment severe water shortage has become the bottle-neck of the constraints and limitations of social and economicdevelopment in Urumqi In order to alleviate this problemin the beginning of this century ldquoYinEJiUrdquo project wasimplemented Using hydraulic engineering part of northernXinjiang Irtysh water was injected into the Urumqi RiverWith the implementation of this project a new water sourceis added into the Urumqi River basin It alleviated the statusof the serious water shortage in Urumqi to a certain extentHowever due to the limited increase of the water resourcesit cannot fully meet the demand Therefore from the man-agement point of view in addition to the plan of reservoirsconstruction in the upstream we can also apply the multiob-jective optimization operation In this way we can alleviatethe contradiction between the supply and the demand of thewater resource atmaximumand realize the optimal allocationof water resources in Urumqi river basin gradually

Since the early 1960s the research on the optimizationoperation of themultireservoir systemhas been carried out inChinaThe research developed rapidly after the 1980s and gota large amount of achievements Dong introduced the princi-ple and contents of the dynamic programming optimizationtechnique and its application in the hydro power stationreservoir (group) scheduling as well as the principle ofthe multiobjective decision Feng established the theoreticalfoundation for the multiobjective optimum scheduling of thereservoirs He also discussed the basic principle and methodof the large-scale system decomposition and coordinationsystematically Wang introduced the theory and applicationof the large-scale system optimization Using the fuzzyidentificationmode Li achieved the level characteristic of thesample and the contribution rate of the relevant factors inthe evaluation process of the evaluation which aims at thecomprehensive evaluation of the reservoir water resourcescarrying capacity At the same time he achieved assessmentof water resources carrying capacity in regional reservoir [4]

In the recent decades several researchers (HouckYakowitz and Needham et al) proposed methods appliedin optimization operation of multireservoir system such aslinear programming nonlinear programming and dynamicplanning Although those optimization methods provide away to resolve the problem of reservoir group schedulingthere are also some problems First some models need alot to simplify therefore their simulation performance ispoor Second some of the nonlinear methods are restrictedin practical application because of the lack of the boundaryconditions and more parameters Moreover those methodsare hard to simulate the experience of scheduling knowledgeConsequently the research on the optimization scheduling ismore and the practical application is less [5]

In the process of the multireservoir system operation theregulation and scheduling consist of the conflicting and theintereffecting multitargets In a specific field people oftenencounter the matching problem which requires multiplesolutions to achieve optimization at the same time It is

the multiobjective optimization problem MOP The valida-tion of MOP is whether the optimization target is more thanone and requires the simultaneous processing In engineeringapplications the MOP is very significant These engineeringapplications are very complicated and difficult They are alsoone of the main research fields Since the early nineteensixties the MOP has attracted more and more researchersrsquoattention from different fields For this reason it has veryimportant research value and practical significance to solveMOP [6]

There are a large number of monographs focusing on themethods to resolve the MOP In general these books can bedivided into two categories One uses the classical optimiza-tionmethods and the other uses themethods based on evolu-tionary algorithmwhich simulates the biological evolution oractivity behavior Because evolutionary algorithmrsquos constraintconditions are low and solution quality is high they havebeen widely applied in resolving the complex optimizationproblem At present the popular evolutionary algorithm toresolve the MOP is as follows the multiobjective geneticalgorithm [7] the artificial ant colony algorithm [8] the mul-tiobjective particle swarm optimization algorithm [9] andthe shuffled frog leaping algorithm [10]

The shuffled frog leaping algorithm (SFLA) is a biologicalevolutionary algorithm based on swarm intelligence whichwas proposed by Eusuff and Lansey in 2003 The SFLA hasthe characteristics of simple concept fewer parameters fastcalculation speed good global search ability easy to imple-ment and so on So at present it is wildly applied to thefields of the network of water resources allocation issuesthe function optimization themultiproduct pipeline networkoptimization the combinatorial optimization the image pro-cessing the multiuser detection and so forth Also themethod achieved good results As a kind of optimizationmethod the SHLAhas certain advantages in the optimizationoperation of the multireservoir system and has broad appli-cation prospects [11ndash13]

Thepurpose of this paper is to study the operation param-eters Based on these parameters we set up the operationmathematical model and use the SFLA to realize the opti-mization allocation of the multireservoir system in UrumqiRiver basin [14]

2 The Description of the MOP

Using words the MOP can be described as an optimizationproblem which consists of 119863 decision variable parameters119873 objective functions and 119898 + 119899 constraints The decisionvariables the objective function and the constraint conditionare function relation In noninferior solution set decisionmakers can only choose a satisfactory noninferior solutionas the final solution according to the specific requirements ofthe problemThemathematic form of MOP can be describedas follows

min (119910) = 119891 (119909) = [1198911(119909) 119891

2(119909) 119891

119899(119909)]

119899 = 1 2 119873


st 119892119894(119909) le 0 119894 = 1 2 119898

ℎ119895(119909) = 0 119895 = 1 2 119896

119909 = [1199091 1199092 119909

119889 119909

119863]

119909119889-min le 119909

119889le 119909119889-max 119889 = 1 2 119863

(1)





119895(119909) = 0 deter-






1198890



lowast

= [119909lowast

1

119909lowast

2 119909

lowast


1 1199092 119909

119863]


forall119899 119891119899(119909lowast

) le 119891119899(119909) 119899 = 1 2 119873

exist1198990 1198911198990(119909lowast

) lt 1198911198990(119909) 1 le 119899

0le 119873

(2)












119895= (1199091198951 1199091198952 119909



(1198831 1198832 119883








119908of


1198831015840= 119883119908+ 119877 times (119883

119887minus 119883119908) (3)



then 119883119908

= 1198831015840 Otherwise 119883


119898in (3) and











min max 119891119898(1199091 1199092 119909

119899) 119898 = 1 2 119872

st (1) 119892119896(1199091 1199092 119909

119899) ge 0 119896 = 1 2 119872

(2) ℎ119897(1199091 1199092 119909

119899) = 0 119897 = 1 2 119871

(3) 119909(119871)

119894le 119909119894le 119909(119880)

119894 119894 = 1 2 119899

(4)


1 1199092 119909

119899) isin 119883

and 119909119894

(119871) and 119909119894


Ω =119909 | 119892119896(1199091 119909

119899) ge 0 ℎ

119896(1199091 119909

119899) = 0

119909(119871)

119894le 119909119894le 119909(119880)

119894 Ω sub 119883

(5)


max1198911=

119879

sum

119905=1

119868

sum

119894

11988611198941198752

119866119894(119905)+ 1205731119894119875119866119894(119905)

+ 1205741119894

max1198912=

119879

sum

119905=1

119869

sum

119895

11988621198951198752

119878119895(119905)+ 1205732119895119875119878119895(119905)

+ 1205742119895

max1198913=

119879

sum

119905=1

119870

sum

119896

11988631198961198752

119882119896(119905)+ 1205733119896119875119882119896(119905)

+ 1205743119896

max1198914=

119879

sum

119905=1

119871

sum

119897

11988641198971198752

119867119897(119905)+ 1205734119897119875119867119897(119905)

+ 1205744119897

max1198915=

119879

sum

119905=1

119872

sum

119898

11988651198981198752

119880119898(119905)+ 1205735119898119875119880119898(119905)

+ 1205745119898

(6)


119866119894(119905)is the water con-



119882119896(119905)is the water






lowast




119875119866min lt sum119875

119866lt 119875119866max

Δsum119875119866lt Δ119875119866max

(7)








119875119878min lt sum119875

119878lt 119875119878max

Δsum119875119878lt Δ119875119878max

(8)











119875119882min lt sum119875

119882lt 119875119882max

Δsum119875119882

lt Δ119875119882max

(9)









sum119875119867

ge 119875119867min (10)

where sum119875119867




sum119875119880le 119875119880max

sum 119875119880

Count (119875119880)ge 119875119880min

(11)







119891 (119909) =1

119869 + 120572 (12)


Start



in the population







N


End

Y




119880 (119902) = 119875119882

+ 119878

119878 =



119861minus 119875119882)] minus119878max

Negative direction

(13)



119861and












The basic SFLA

1 2 5 3

4 5 1 4

3 4 3 4 2 2 4 3

1 2 5 3

4 5 1 4

3 3 4 4

Before Present

1 2 5 3

4 5 1 4

3 4 3 4

The improved SFLA

2 2 4 3

1 2 5 3

4 5 1 4

3 3 4 4

PW

PB

U(q)

+ + +




119878(119894)

=

min int [rand(119894) (119875(119894)119861

minus 119875(119894)

119882)] 119878max

Positive direction

max int [rand(119894) (119875(119894)119861

minus 119875(119894)

119882)] minus119878max

Negative direction

(14)


119878(119894)

=

min int [120572119891(119909)

rand(119894) (119875(119894)119861

minus 119875(119894)

119882) + 120573119891(119909)

119878(119894minus1)

] 119878max

Positive direction

max int [120572119891(119909)

rand(119894) (119875(119894)119861

minus 119875(119894)

119882) + 120573119891(119909)

119878(119894minus1)

] minus119878max


120572119891(119909)

+ 120573119891(119909)

= 1 (0 lt 120572119891(119909)

120573119891(119909)





Start

Code

Initial population


Dividepopulation

Evolution



Obtainsolution



End

YN

Y




1

a b

μ(f (x))

f (x)




120583 (119891 (119909)) =

1 119891 (119909) lt 119886

119887 minus 119891 (119909)

119887 minus 119886 119886 le 119891 (119909) le 119887

0 119891 (119909) gt 119886

(16)


120582 = min 120583 (1198911(119909) 119891

2(119909) 119891

3(119909)) (17)


5 The Conclusion






Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










st 119892119894(119909) le 0 119894 = 1 2 119898

ℎ119895(119909) = 0 119895 = 1 2 119896

119909 = [1199091 1199092 119909

119889 119909

119863]

119909119889-min le 119909

119889le 119909119889-max 119889 = 1 2 119863

(1)





119895(119909) = 0 deter-






1198890



lowast

= [119909lowast

1

119909lowast

2 119909

lowast


1 1199092 119909

119863]


forall119899 119891119899(119909lowast

) le 119891119899(119909) 119899 = 1 2 119873

exist1198990 1198911198990(119909lowast

) lt 1198911198990(119909) 1 le 119899

0le 119873

(2)












119895= (1199091198951 1199091198952 119909



(1198831 1198832 119883








119908of


1198831015840= 119883119908+ 119877 times (119883

119887minus 119883119908) (3)



then 119883119908

= 1198831015840 Otherwise 119883


119898in (3) and











min max 119891119898(1199091 1199092 119909

119899) 119898 = 1 2 119872

st (1) 119892119896(1199091 1199092 119909

119899) ge 0 119896 = 1 2 119872

(2) ℎ119897(1199091 1199092 119909

119899) = 0 119897 = 1 2 119871

(3) 119909(119871)

119894le 119909119894le 119909(119880)

119894 119894 = 1 2 119899

(4)


1 1199092 119909

119899) isin 119883

and 119909119894

(119871) and 119909119894


Ω =119909 | 119892119896(1199091 119909

119899) ge 0 ℎ

119896(1199091 119909

119899) = 0

119909(119871)

119894le 119909119894le 119909(119880)

119894 Ω sub 119883

(5)


max1198911=

119879

sum

119905=1

119868

sum

119894

11988611198941198752

119866119894(119905)+ 1205731119894119875119866119894(119905)

+ 1205741119894

max1198912=

119879

sum

119905=1

119869

sum

119895

11988621198951198752

119878119895(119905)+ 1205732119895119875119878119895(119905)

+ 1205742119895

max1198913=

119879

sum

119905=1

119870

sum

119896

11988631198961198752

119882119896(119905)+ 1205733119896119875119882119896(119905)

+ 1205743119896

max1198914=

119879

sum

119905=1

119871

sum

119897

11988641198971198752

119867119897(119905)+ 1205734119897119875119867119897(119905)

+ 1205744119897

max1198915=

119879

sum

119905=1

119872

sum

119898

11988651198981198752

119880119898(119905)+ 1205735119898119875119880119898(119905)

+ 1205745119898

(6)


119866119894(119905)is the water con-



119882119896(119905)is the water






lowast




119875119866min lt sum119875

119866lt 119875119866max

Δsum119875119866lt Δ119875119866max

(7)








119875119878min lt sum119875

119878lt 119875119878max

Δsum119875119878lt Δ119875119878max

(8)











119875119882min lt sum119875

119882lt 119875119882max

Δsum119875119882

lt Δ119875119882max

(9)









sum119875119867

ge 119875119867min (10)

where sum119875119867




sum119875119880le 119875119880max

sum 119875119880

Count (119875119880)ge 119875119880min

(11)







119891 (119909) =1

119869 + 120572 (12)


Start



in the population







N


End

Y




119880 (119902) = 119875119882

+ 119878

119878 =



119861minus 119875119882)] minus119878max

Negative direction

(13)



119861and












The basic SFLA

1 2 5 3

4 5 1 4

3 4 3 4 2 2 4 3

1 2 5 3

4 5 1 4

3 3 4 4

Before Present

1 2 5 3

4 5 1 4

3 4 3 4

The improved SFLA

2 2 4 3

1 2 5 3

4 5 1 4

3 3 4 4

PW

PB

U(q)

+ + +




119878(119894)

=

min int [rand(119894) (119875(119894)119861

minus 119875(119894)

119882)] 119878max

Positive direction

max int [rand(119894) (119875(119894)119861

minus 119875(119894)

119882)] minus119878max

Negative direction

(14)


119878(119894)

=

min int [120572119891(119909)

rand(119894) (119875(119894)119861

minus 119875(119894)

119882) + 120573119891(119909)

119878(119894minus1)

] 119878max

Positive direction

max int [120572119891(119909)

rand(119894) (119875(119894)119861

minus 119875(119894)

119882) + 120573119891(119909)

119878(119894minus1)

] minus119878max


120572119891(119909)

+ 120573119891(119909)

= 1 (0 lt 120572119891(119909)

120573119891(119909)





Start

Code

Initial population


Dividepopulation

Evolution



Obtainsolution



End

YN

Y




1

a b

μ(f (x))

f (x)




120583 (119891 (119909)) =

1 119891 (119909) lt 119886

119887 minus 119891 (119909)

119887 minus 119886 119886 le 119891 (119909) le 119887

0 119891 (119909) gt 119886

(16)


120582 = min 120583 (1198911(119909) 119891

2(119909) 119891

3(119909)) (17)


5 The Conclusion






Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













min max 119891119898(1199091 1199092 119909

119899) 119898 = 1 2 119872

st (1) 119892119896(1199091 1199092 119909

119899) ge 0 119896 = 1 2 119872

(2) ℎ119897(1199091 1199092 119909

119899) = 0 119897 = 1 2 119871

(3) 119909(119871)

119894le 119909119894le 119909(119880)

119894 119894 = 1 2 119899

(4)


1 1199092 119909

119899) isin 119883

and 119909119894

(119871) and 119909119894


Ω =119909 | 119892119896(1199091 119909

119899) ge 0 ℎ

119896(1199091 119909

119899) = 0

119909(119871)

119894le 119909119894le 119909(119880)

119894 Ω sub 119883

(5)


max1198911=

119879

sum

119905=1

119868

sum

119894

11988611198941198752

119866119894(119905)+ 1205731119894119875119866119894(119905)

+ 1205741119894

max1198912=

119879

sum

119905=1

119869

sum

119895

11988621198951198752

119878119895(119905)+ 1205732119895119875119878119895(119905)

+ 1205742119895

max1198913=

119879

sum

119905=1

119870

sum

119896

11988631198961198752

119882119896(119905)+ 1205733119896119875119882119896(119905)

+ 1205743119896

max1198914=

119879

sum

119905=1

119871

sum

119897

11988641198971198752

119867119897(119905)+ 1205734119897119875119867119897(119905)

+ 1205744119897

max1198915=

119879

sum

119905=1

119872

sum

119898

11988651198981198752

119880119898(119905)+ 1205735119898119875119880119898(119905)

+ 1205745119898

(6)


119866119894(119905)is the water con-



119882119896(119905)is the water






lowast




119875119866min lt sum119875

119866lt 119875119866max

Δsum119875119866lt Δ119875119866max

(7)








119875119878min lt sum119875

119878lt 119875119878max

Δsum119875119878lt Δ119875119878max

(8)











119875119882min lt sum119875

119882lt 119875119882max

Δsum119875119882

lt Δ119875119882max

(9)









sum119875119867

ge 119875119867min (10)

where sum119875119867




sum119875119880le 119875119880max

sum 119875119880

Count (119875119880)ge 119875119880min

(11)







119891 (119909) =1

119869 + 120572 (12)


Start



in the population







N


End

Y




119880 (119902) = 119875119882

+ 119878

119878 =



119861minus 119875119882)] minus119878max

Negative direction

(13)



119861and












The basic SFLA

1 2 5 3

4 5 1 4

3 4 3 4 2 2 4 3

1 2 5 3

4 5 1 4

3 3 4 4

Before Present

1 2 5 3

4 5 1 4

3 4 3 4

The improved SFLA

2 2 4 3

1 2 5 3

4 5 1 4

3 3 4 4

PW

PB

U(q)

+ + +




119878(119894)

=

min int [rand(119894) (119875(119894)119861

minus 119875(119894)

119882)] 119878max

Positive direction

max int [rand(119894) (119875(119894)119861

minus 119875(119894)

119882)] minus119878max

Negative direction

(14)


119878(119894)

=

min int [120572119891(119909)

rand(119894) (119875(119894)119861

minus 119875(119894)

119882) + 120573119891(119909)

119878(119894minus1)

] 119878max

Positive direction

max int [120572119891(119909)

rand(119894) (119875(119894)119861

minus 119875(119894)

119882) + 120573119891(119909)

119878(119894minus1)

] minus119878max


120572119891(119909)

+ 120573119891(119909)

= 1 (0 lt 120572119891(119909)

120573119891(119909)





Start

Code

Initial population


Dividepopulation

Evolution



Obtainsolution



End

YN

Y




1

a b

μ(f (x))

f (x)




120583 (119891 (119909)) =

1 119891 (119909) lt 119886

119887 minus 119891 (119909)

119887 minus 119886 119886 le 119891 (119909) le 119887

0 119891 (119909) gt 119886

(16)


120582 = min 120583 (1198911(119909) 119891

2(119909) 119891

3(119909)) (17)


5 The Conclusion






Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119875119882min lt sum119875

119882lt 119875119882max

Δsum119875119882

lt Δ119875119882max

(9)









sum119875119867

ge 119875119867min (10)

where sum119875119867




sum119875119880le 119875119880max

sum 119875119880

Count (119875119880)ge 119875119880min

(11)







119891 (119909) =1

119869 + 120572 (12)


Start



in the population







N


End

Y




119880 (119902) = 119875119882

+ 119878

119878 =



119861minus 119875119882)] minus119878max

Negative direction

(13)



119861and












The basic SFLA

1 2 5 3

4 5 1 4

3 4 3 4 2 2 4 3

1 2 5 3

4 5 1 4

3 3 4 4

Before Present

1 2 5 3

4 5 1 4

3 4 3 4

The improved SFLA

2 2 4 3

1 2 5 3

4 5 1 4

3 3 4 4

PW

PB

U(q)

+ + +




119878(119894)

=

min int [rand(119894) (119875(119894)119861

minus 119875(119894)

119882)] 119878max

Positive direction

max int [rand(119894) (119875(119894)119861

minus 119875(119894)

119882)] minus119878max

Negative direction

(14)


119878(119894)

=

min int [120572119891(119909)

rand(119894) (119875(119894)119861

minus 119875(119894)

119882) + 120573119891(119909)

119878(119894minus1)

] 119878max

Positive direction

max int [120572119891(119909)

rand(119894) (119875(119894)119861

minus 119875(119894)

119882) + 120573119891(119909)

119878(119894minus1)

] minus119878max


120572119891(119909)

+ 120573119891(119909)

= 1 (0 lt 120572119891(119909)

120573119891(119909)





Start

Code

Initial population


Dividepopulation

Evolution



Obtainsolution



End

YN

Y




1

a b

μ(f (x))

f (x)




120583 (119891 (119909)) =

1 119891 (119909) lt 119886

119887 minus 119891 (119909)

119887 minus 119886 119886 le 119891 (119909) le 119887

0 119891 (119909) gt 119886

(16)


120582 = min 120583 (1198911(119909) 119891

2(119909) 119891

3(119909)) (17)


5 The Conclusion






Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










The basic SFLA

1 2 5 3

4 5 1 4

3 4 3 4 2 2 4 3

1 2 5 3

4 5 1 4

3 3 4 4

Before Present

1 2 5 3

4 5 1 4

3 4 3 4

The improved SFLA

2 2 4 3

1 2 5 3

4 5 1 4

3 3 4 4

PW

PB

U(q)

+ + +




119878(119894)

=

min int [rand(119894) (119875(119894)119861

minus 119875(119894)

119882)] 119878max

Positive direction

max int [rand(119894) (119875(119894)119861

minus 119875(119894)

119882)] minus119878max

Negative direction

(14)


119878(119894)

=

min int [120572119891(119909)

rand(119894) (119875(119894)119861

minus 119875(119894)

119882) + 120573119891(119909)

119878(119894minus1)

] 119878max

Positive direction

max int [120572119891(119909)

rand(119894) (119875(119894)119861

minus 119875(119894)

119882) + 120573119891(119909)

119878(119894minus1)

] minus119878max


120572119891(119909)

+ 120573119891(119909)

= 1 (0 lt 120572119891(119909)

120573119891(119909)





Start

Code

Initial population


Dividepopulation

Evolution



Obtainsolution



End

YN

Y




1

a b

μ(f (x))

f (x)




120583 (119891 (119909)) =

1 119891 (119909) lt 119886

119887 minus 119891 (119909)

119887 minus 119886 119886 le 119891 (119909) le 119887

0 119891 (119909) gt 119886

(16)


120582 = min 120583 (1198911(119909) 119891

2(119909) 119891

3(119909)) (17)


5 The Conclusion






Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Multiobjective Optimization Problem of...

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