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Locating competitive facilities 63With this notation, for entry instants τ 1 , . . . , τ r fixed, the covering location problem to besolved is the following linear integer program∫ τk+1∫ τk+1r∑ { ∑Π r (τ 1 , . . . , τ r ) = max x kx,yv ρ(t)w v (t)dt − ∑ yfk c f (t)dt }k=1 v∈Vτ kf∈Fτ k∑yf r ≤ q (1)f∈Fx k v ≤ ∑f∈N vy k f, ∀v ∈ V, 1 ≤ k ≤ r (2)y k−1f≤ yf k , ∀f ∈ F, 2 ≤ k ≤ r (3)yf k , xk v ∈ {0, 1}, ∀f ∈ F, ∀v ∈ V, 1 ≤ k ≤ r. (4)We briefly discuss the correctness of the formulation. For the objective, within the time interval[τ 0 , τ 1 ], no benefit or cost is incurred, since no plants from A are operating. The interval[τ 1 , τ r+1 ] is split into the subintervals [τ k , τ k+1 ], k = 1, . . . , r. Within an interval [τ k , τ k+1 ], thetotal incomes obtained from consumer v are x k ∫ τk+1v τ kρ(t)w v (t)dt, whereas the total operatingcost incurred by facility at f is given by yfk ∫ τk+1τ kc f (t)dt.Constraint (1) imposes that the number of open facilities from A cannot exceed q. Constraints(2) impose that, if x k v = 1, i.e., if v is counted as captured by A in time interval [τ k, T ],then there must exist at least one facility f from A operating within such interval covering v.With constraints (3) we express that, if plant f is operating within [τ k−1 , T ], then it mustalso be so in [τ k , T ].Finally, constraints (4) express the binary character of the variables x k v and yf k for v ∈ V ,f ∈ F and 1 ≤ k ≤ r.Since the variables x k v and yf k are binary and the functions ρ(t)w v(t) and c f (t) are continuous,the optimization problem above is well defined, and its optimal value Π r (τ 1 , . . . , τ r ) isattained.The continuity of ρ(t)w v (t) and c f (t) enables us also to define their primitives,g v (t) =h f (t) =which are differentiable functions.Moreover, for each k, 1 ≤ k ≤ r, one has∫ τk+1∫ t0∫ t0ρ(s)w v (s)dsc f (s)ds,τ kρ(t)ω v (t)dt∫ τk+1= g v (τ k+1 ) − g v (τ k )τ kc f (t)dt = h f (τ k+1 ) − h f (τ k ).With this notation, we can express the optimal profit Π r (τ 1 , . . . , τ r ) for opening times τ 1 , . . . , τ rasr∑ { ∑Π r (τ 1 , . . . , τ r ) = max x k v[g v (τ k+1 ) − g v (τ k )] − ∑ yf k [h f (τ k+1 )) − h f (τ k )] } , (5)(x,y)∈S rv∈Vf∈Fk=1where S r denotes the set of pairs (x, y) satisfying constraints (1)-(4).Considering the instant times τ 1 , . . . , τ r as decision variables to be optimized yields theoptimal planning for locating at most q facilities in at most r different instant times. Indeed,