Tamtam Proceedings - lamsin

108 Ellaia et al.to present first of all the system of the drinkable water supply of Kenitra city and thecomplex of exploitation of resources in ground water: Fouarat. In our case, the numberof drillings is equal to 10: (n=10) then the objective function (2) becomesZ =3∑K j .∆t jj=110∑i=1ρg.(α i .Q i ).(H gi + 1.1.∆H iL )η gi. (3)After simplification, the final formulation of this model is the following:⎧{ 3∑ }{ 10∑⎪⎨ Minimize Z = k j .∆t j [a i α i + c i αi 3 + b ∑iα i ( 10j=1i=1k=i∑subject to 14000 ≤ V 0 + ∆t.( 10(α i .Q i ) − Q d ) ≤ 28000⎪⎩i=1α i ∈ {0, 1}withL k;k+1 (k∑m=1D 5 k;k+1a i = ρg.Q i.H i, b i = 8.8ρλη giπ 2 · LiQ 3 iη gi Di5 , c i = 8.8ρλ′π 2 · Qiη giα mQ m) 2and k 1 = 0.48, k 2 = 0.712, k 3 = 1.0516. The parameters a i , b i , c i , Q m , L k;k+1 , andD k;k+1 are giving in [4].})](4)3. RPRGA for Global Optimization3.1. Penalty function method1. Definition α ′ is an integer point if its components α i ′ (i = 1, ....., n) are allintegers. For an integer point α ′ , the set N(α ′ ) = {α : ‖α − α ′ ‖ ∞ ≤ 1 5} is called a1/5-cubic neighborhood of the integer point α ′ .Let S = {α|V 1 ≤ V 0 + ∆t.( ∑ ni=1 (α i.Q i ) − Q d ) ≤ V max , 0 ≤ α i ≤ 1.5, i = 1, 2, ..., n}R. Ge and C. Huang [6] choose the penalty function φ(α, k) = f(α) − k ∑ ni=1 cos 2πα i,and studied the minimzation problemmin φ(α, k). (5)α∈SIf the global minimizer α ∗ of problem 5 is in a 1/5-cubic neighborhood of an integer pointᾱ in the feasible region S then ᾱ is the solution of (2); we need to make k large enoughin the computation.TAMTAM –Tunis– 2005