Genetic Algorithm Based Fuzzy Multi-objective Nonlinear Programming of Regional Water Allocation

LI Shurong, WANG Feng, MA Tao, YAN Wei College of Information and Control Engineering China University of Petroleum Dongying, China [email protected]

Abstract—A genetic algorithm (GA) based approach to solve Dongying city’s water allocation is built and the “fuzzy a fuzzy multi-objective nonlinear programming (FMNP) optimal solutions” are provided. problem is presented and applied to water resources allocation. A FMNP model of regional water allocation is established II. A FMNP MODEL OF REGIONAL WATER RESOURCE originally. The objective functions of the model include three ALLOCATION conflicting goals which are economic profits, environment goal A FMNP model of regional water resource allocation is and society goal. By turning the FMNP to an optimal level cut problem, the “fuzzy optimal solution” is solved by using a genetic established as follows. algorithm. An example of the Dongying’s water allocation A. Objective functions strategy is provided by using the proposed method. (i)Economic profit function Keywords—fuzzy multi-objective programming, fuzzy i optimal solution, water resource optimal allocation, genetic maxf 1 (x ) algorithm kkkk k kkk (1) =−+−∑∑[( ∑baxij ij ) ijα ij ∑ ( b pj ax pj ) pjα pj ] I. INTRODUCTION kj i p Water shortage is a serious problem in the north China where which affects the people’s life and economic development directly. Thus water management is one of the most important if ()x -a function which can be fuzzified, means that the problems for Chinese government. To satisfy the growing of value of objective function can vary in some domain. The the demand, a scientific programming is required to allocate domain is called the admissible error variable region. the limited water resource to meet the demand for some regions. f1()x - net profit of all regions water allocated. Many researchers have applied mathematical programming x -the decision variables. to water management. The traditional optimization methods attempted to obtain the optimal value to satisfy the objective k -the kth sub-area. function and constrains[2,5,6,11]. Grounder water quality i -the ith independent water resource of some subarea. management is investigated in [3,7]. Water supply capacity expansion problem is studied in [1]. Water distribution p-the pth common water resource. optimization problem is discussed in [4,5]. Generally, the k mentioned works are deterministic linear or nonlinear xij -the amount of water released from i source of kth sub- programming problems. Recently, some researchers are area to jth plant, the outflow of water. studying the water management by using fuzzy k programming[8, 12]. bij -the profit of per unit released water from i source of kth sub-area to jth plant. In this paper, considering some uncertainty and the different benefits of water allocation among industry, ak -the cost of per unit released water from i source of kth agriculture, residental consumption and environment in a ij region, a FMNP model is established. A GA based solving sub-area to jth plant. approach for FMNP is presented. By using the optimal level (ii)Environmental objective function cut idea, the FMNP is changed to be a single objective nonlinear programming problem which is easily solved with The environmental objective is determined by the water GA. To show the validity of the method, a FMNP model of quality and can be formulated by the amount of released

978-1-4244-1674-5/08 /$25.00 ©2008 IEEE CIS 2008 pollution water. Generally, the objective function is to k minimize the amount of some important matters such as σ1 = ()/∑ Ekj Biochemical Oxygen Demand (BOD) and Chemical Oxygen k ⎧ Dxxkkk−+() Demand (COD). jijpj∑∑ ∑ , ⎪ ki p 1,()−≥+Dkkkxx k ⎪ DLkk− j ∑∑ij ∑ pj i kk kk E j = ⎨ jj ki p Minf 2 (x )=+ (Pxij ij Px pj pj ) (2) ∑∑ ∑ ∑ ⎪ kkk kj i p ⎪1()Dj <+∑∑xxij ∑ pj ⎩ ki p where (10) f2 ()x -the amount of pollution matter (BOD and COD) of the region. P

k P0 P -the pollution matter of per unit released water from i σ 2 = , (11) ij GA source of kth sub-area to jth plant. GA0 (iii) Social objective function where

The social objective function is formulated by minimizing k water shortage of the residents and plants. Dj -the planning required water of jth plant of kth sub-area.

i kkk P0 -the amount of BOD and COD yielding in the reference Minf 3 (x )=−+∑∑∑ [Dxxj (ij pj )] (3) kip year. where P- the amount of BOD and COD yielding in the planning year. f3 ()x -water shortage of the residental life and plants. GA0 -the gross regional product per capita in the reference k Dj -the planning required water of jth plant of kth sub-area. year. B. Constraints condition GA -the gross regional product per capita in the planning year. (i) Capacity constraints of water resources σ **,σ -the optimal ratios pre-assigned. Constraints of independent water resources 12

k ω1 ,ω2 -the weights. ∑ xij≤ W i . (4) k From(10), we hope that the amount of consumed water Constraints of common water resources of j th plant can meet the planed level. Equation (11) means that the relative increment of polluted matters is less than that k of the gross regional product per capita. ∑∑xpjp≤ W , (5) kj (iv) Non-negative constraints where Wi is the upper limit of the amount of i th water All decision variables should be non-negative. resource. k xij ≥ 0 (12) (ii) Constraints of water transportations III. FUZZY NONLINEAR PROGRAMMING (FNP) SOLVING kkkk VIA A GENETIC ALGORITHM LxxHj ≤+∑∑ij pj ≤ j (6) ip Generally, a FNP problem can be described in the (iii) Constraints of regional harmonious development. following form, ~

μμ12≥ 0.8 , (7) maxf (xfxxx )= (12 , ," n ) s.t.gx ( )< b , (13) where μ μ is the coefficient of harmonious development,  12 x ≥ 0 and μ1 , μ2 is defined as follows, T where x = (,xx12 ," xn )is n − dimensional decision variable μσ()exp((=− ωσ − σ∗ ))2 , (8) 11 11 1 vector, g()x is a vector of constraint conditions, i.e. ∗ 2 μσ22()exp((=− ωσ 22 − σ 2 )). (9) T g()xgxgxgx= (12 (),()," m ()),

IN (8) and (9), σ1 ,σ 2 are defined as follows,

T b is a constant vector, bbbb= (,12 ," ,m ). μμμμmin()x = min{(),(), 0xx 1 " m ()}. x The FNP (13) can be turned to be the following The fitness function is defined as follows. deterministic nonlinear programming model, ⎧ μμmin(),xx min ()≥ α 0 Fx()==μ  () x ⎨ (15) ⎧Max α S ⎩εμmin(),xx μ min ()< α 0 ⎪ ⎪s.t.μα0 (x )≥ , ⎨ (14) where α0 is a pre-assigned degree of membership, μα()x ≥= ,im 1,2,," ⎪ i and ε ∈[0,1] . ⎩⎪ x ≥≤≤0, 0α 1, From (15), if x j ∉ Sα , a much small degree of Where α is the optimal level cut (or degree) defined as follows, 0 membership will be set to it.

αμμ==min{0 (x ),i (xi ), 1, 2," , m }. The GA based method is outlined as follows. Before solving (13) and (14), we need explain the Step 1. Initialize the population in the search space. following notations. Substep 1.1. Input the pre-assigned degree of

Definition 1. A fuzzy optimum of problem (13) is defined membershipα0 , and the maximal iterative steps NG, and the as a following fuzzy set S , number of individuals NP.

 n + Substep 1.2. Input the index set CS = {0,1, 2," } which Sxxx=∈{( ,μ  ( )) | (\ ) , S represent the index of the objective function and constraints μμμ()xxxim== min{(),(), 1,2,," }}. S 0 i respectively. For each index rCS∈ , input the minimum and

n + maximum of corresponding objective function or constraint. Let Sxα =∈{()|()},[0,1].\ μααS x ≥ ∈ Then, Sα ,  Step 2. Produce a population and determine a membership which is called the α level cut of S , is an ordinary set. function randomly. * Definition 2. α is called the optimal level cut, if Step 3. Set k = 1. α * satisfies ∀≤≤0 α α * , S is nonempty, and ∀>α α * , S is α α Step 4. For individual jj(1,2,,)= " NP , calculate the empty. fitness function F()j and selection probability Pj(), For the level cut, with the augmentation of α , the domain * of Sα will be reduced till there exists an α such that F()j Fj()==μ  ( xj ), Pj () NP . ∀>δααδ0, =* + , S is an empty set. α * is the optimal level S α ∑ F()j cut. i=1 The idea of solving FNP via a genetic algorithm is to find a Step 5. Perform crossover according to the crossover * small neighborhood of α , such that for all x corresponding probability, new individuals x j (1,2,,)jNP= " will be to the α in neighborhood of α * is the required optimal obtained after mutation. decision variable, i.e. fuzzy optimal solution. Thus the GA Step 6. For each new individual, calculate the value of the based approach can be expressed as follows. One can first set memebership function μ (x ) . Update the optimal level an optimal level cut α , and randomly produce a population S j 0 cut μ , and adjust the minimum and maximum of objective including np individuals. By defining a fitness function, the max function or constraints. individual with degree of membership no less than α0 is selected. The selected individuals consist of a sub-population. Step 7. kk+1,→ if kNG≤ ,Then go to Step 4 or go to For reducing computing time, the individuals which have Step 8. degree of membership less than α will be set much small 0 Step 8. Output the optimal level cut μ and the minimum degree of membership, so that they have little chance to be max chosen in the next step. and maximum of objective function or constraints. After some iterative steps, the individuals with degree of If some constraints are inequality, penalty function can be applied to these inequalities. membership less than α0 will be eliminated. The rest is a kth sub-population in which each individual is with degree of IV. MODEL SOLVING OF FMNP OF WATER RESOURCE membership α no less than α . Thus SS⊆ , every k 0 αk α0 ALLOCATION element in S is the desired optimal solution. αk A. Fuzzification of the model For individual x , let For (1)-(3), the following membership functions are defined.

For economic profit, the membership function is defined as kkkk k kkk maxfbaxbax1 (x )=−+−∑∑ [ ∑ (ij ij ) ijα ij ∑ ( pj pj ) pjα pj ] ⎧1,fx ( ) ≥ z kj i p ⎪ 11g fx()− l kk kk ⎪ 11 or minf (x )= (Px+≤ Px ) h μ1111()x =≤≤⎨ ,lfxz () g 22∑∑ ∑ij ij ∑ pj pj ⎪ zl11g − kj i p ⎪0,fx ( ) ≤ z ⎩ 11g kkk or minf33 (x )= ∑∑∑ [Dxxhjijpj−+ ( )] ≤ For environment objective, the membership function is kip defined as s.t.

⎧1,fx22 ( ) ≤ zg ⎪ k ∑ xij≤ W i , ⎪hfx22− () k μ2222()x =≤≤⎨ ,zfxhg () ⎪ hz22− g ⎪ xk ≤ W , 0,fx22 ( ) ≥ zg ∑∑ pjp ⎩ kj For social objective, the membership function is defined as kkkk LxxHj ≤+∑∑ij pj ≤ j , ⎧1,fx ( ) ≤ z ip ⎪ 33g hfx− () ⎪ 33 μμ ≥ 0.8 , μ3333()x =≤≤⎨ ,zfxhg () 12 ⎪ hz33− g k ⎪ xij ≥ 0 . ⎩0,fx33 ( ) ≥ zg where C. Applications to the city of Dongying The Dongying city locates in the delta of the yellow river z1g -the optimal value of economic profit, which is a water-short city. According to the practical situation, the supplying water area can be divided into 6 sub-areas which l1 -the lower limit of economic profit, are the Dongying district, the Hekou district, the Guangrao county, the Lijin County, the Kenli county and the Shengli z2g -the optimum value of COD, oilfield. In the Dongying district, the Hekou district, the Guangrao county, the Lijin County and the Kenli county, the z3g -the optimum value of water shortage, water consumption includes industrial water, agricultural water, residental life water of the towns, residental life water of the h2 -the upper limit of COD, countryside and environmental water. Only industrial water is considered in the Shengli oilfield. In the Dongying city, there h -the upper limit of water shortage. 3 are three water resources which are the yellow river, the Then, according to above section, the fuzzy programming rainfall and the groundwater. The yellow river and the rainfall (1)-(12) can be turned to be a following deterministic are common water resources and the groundwater is an programming model, independent resource only in the Guangrao county. For the water allocation of the Dongying city, we set the ⎧max α lower limit of the economic profit to be h =×1.32 108 yuan, ⎪s.t. 1 ⎪ 4 the upper limit of the environmental goal to be l2 =×3.6 10 ⎪ μα1 ()x ≥ ⎪ μα()x ≥ ton, the lower limit of social goal to be the 10% of the ⎨ 2 residental consumption water. The other parameters in the ⎪μα()x ≥ ⎪ 3 model are displayed in the following table. ⎪Gx()≤ b ⎪ V. CONCLUSIONS x ≥≤≤0, 0α 1. ⎩ This paper establishes a multi-objective fuzzy nonlinear where α is the optimal level cut, and G(x) denotes the programming problem of regional water allocation. By some constraints of (4)-(12). manipulation, the multi-objective fuzzy nonlinear programming is turned to be an optimal level cut problem which is a single goal nonlinear programming problem and can B. Computation of ziig ,1,2,3= and hh23,. be solved by using a genetic algorithm. Applying this method The upper limit or lower limit of each goal can be obtained to the Dongying’s water allocation, a satisfaction solution by solving the following single goal programming. among multi-objectives is obtained. The optimization result

could be applied to aid in the police making of the city’s water allocation.

TABLE I. THE PARAMETERS IN THE MODEL (2.1)-(2.12) Industrial Agricultural Life water of the Life water of the Environmental Sub-area Paramters water water towns countryside water P1 425 23.8 —— —— —— P2 3.35 0.05 2.25 2.25 3.35 Dongying P3 0．08 —— 0．06 —— —— distrint P4 2820 5425 —— —— —— P5 2207 4611 —— —— —— P6 —— —— 3400 340 600 P1 307 8.7 —— —— —— P2 3.35 0.05 2.25 2.25 3.35 Hekou P3 0．1 —— 0．11 —— —— district P4 1768 11626 —— —— —— P5 1383 9882 —— —— —— P6 —— —— 1000 170 200 P1 322 13.6 —— —— —— P2 3.35 0.05 2.25 2.25 3.35 Kenli P3 0．1 0．11 —— —— county P4 4775 9419 —— —— —— P5 3737 8006 —— —— —— P6 —— —— 600 300 300 P1 296 9.1 —— —— —— P2 3.35 0.05 2.25 2.25 3.35 Lijin P3 0．1 —— 0．11 —— —— county P4 3524 24074 —— —— —— P5 2758 20463 —— —— —— P6 —— —— 550 580 600 P1 552 20.2 —— —— —— P2 3.35 0.05 2.25 2.25 3.35 Guangrao P3 0．13 —— 0．1 —— —— county P4 5803 16212 —— —— —— P5 4541 13780 —— —— —— P6 —— —— 780 1000 300 P1 625 —— —— —— —— P2 3.35 0.05 2.25 2.25 3.35 Shengli P3 0.1 —— —— —— —— oilfield P4 15569 —— —— —— —— P5 12184 —— —— —— —— P1- the profit per ton water（yuan/m3, P2- The cost per ton water（yuan/m3）, P3- The resulted amount of COD per ton water (kg/m3), P4- 3 3 3 3 H j in (6) (ten thousand m ), P5- L j in (6)(ten thousand m ), P6- Environmental water (ten thousand m )(ten thousand m ).

TABLE II. CAPACITY CONSTRAINTS OF THE THREE INDEPENDENT WATER RESOURCES Capacity constraints 90% of the maximal storage （ten thousand m3） Groundwater 9759 Rainfall 12134 Yellow river 78000

TABLE III. THE OPTIMUM VALUE OF THE OBJECTIVES

The objective Economic profit (Ten thousand yuan) Environment goal (Ten thousand ton) Social goal (Ten thousand m3) The value of the objective 14934000 3.391 335

TABLE IV. THE OPTIMUM DECISION VARIABLES

residental life residental life Environmental Sub-area Water resources Industrial water Agricultural water water of water of town water countryside Dongying rainfall 823 862 184 121 79 district Yellow river 1686 4517 3150 219 521 Hekou district rainfall 455 613 201 55 126 Yellow river 969 9992 734 115 74 Kenli county rainfall 704 780 156 109 1 Yellow river 3066 8136 396 191 303 Lijin county rainfall 727 744 146 171 78 Yellow river 2051 20519 346 409 522 Guangrao rainfall 787 756 59 82 11 county Yellow river 1724 8058 239 460 35 groundwater 2205 6456 385 458 255 Shengli oilfield rainfall 3305 Yellow river 9567

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