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GASUB: A genetic-like algorithm for discrete location problems 201the closest to a demand point i, then demand in i is divided so that a fixed proportion θ i ofcustomers buy at the new closest facilities, 0 ≤ θ i ≤ 1. The following decision variables aredefined:{1 if a new facility is opened in jy j =0 otherwise{1 if the new facilities capture demand point ix i =0 otherwise{1 if demand point i is dividedz i =0 otherwisethen the problem is formulated as follows:⎧max∑ w i x i + ∑i∈Ii∈Is.t. x i ≤ ∑y j ,⎪⎨(P 2 )⎪⎩j∈N < iz i ≤ ∑j∈N = ix i + z i ≤ 1,∑y j = sj∈Jy j ,x i ≥ 0, z i ≥ 0y j ∈ {0, 1}θ i w i z ii ∈ I ∗i ∈ I ∗i ∈ I ∗In MAXPROFIT, facilities take charge of transportation and deliver the product to customers.Each facility offers a specific price at each demand point. As result of price competition,the optimal price a new facility can offer at demand point i is the equilibrium price,which is given by p min + tD i , where p min is the minimum selling price at the firm’s door. Withequilibrium prices, only the closest facility to a demand point i can offer the lowest price in i.Then, the same rule as in MAXCAP is used for tie breaking when two or more facilities are theclosest to a demand point. Let p net = p min − p prod , where p prod is the production cost, which issupposed not to depend on site location. The following decision variables are defined:{1 if a new facility is opened in jy j =0 otherwisex ij = proportion of demand at i served from site j{1 if demand point i is dividedz i =0 otherwisethen the problem is formulated as follows:⎧max ∑ ∑[p net + t(D i − d ij )]w i x ij + ∑ p net θ i w i z ii∈I ∗ i∈I ∗s.a.∑j∈N < ix ij + z i ≤ 1,i ∈ I ∗⎪⎨(P 3 )⎪⎩j∈N < ix ij ≤ y j ,z i ≤ ∑∑j∈Jj∈N = iy j = sx ij ≥ 0, z i ≥ 0y j ∈ {0, 1}i ∈ I ∗ , j ∈ N i