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

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