View - Universidad de Almería

Methods for obtaining an outer approximation of the efficient set 99In [7], a branch-and-bound method for obtaining the region of δ-optimality (of a general continuouslocation problem) is presented. In what follows we will refer to it as the δ-opt algorithm.It consists of two phases. The first one entails the determination of the optimal objectivevalue of (4) up to a prespecified relative precision ε. The second phase consists of the determinationof R δ , up to a prespecified precision η. The output of the algorithm is two lists ofsubsets, L 3 and L 4 . The union of the subsets in the first list gives an inner approximation ofR δ , whereas the union of the subsets in both L 3 and L 4 gives an outer approximation OR δof R δ , guaranteed to lie entirely within R δ+η(1+δ) , i.e., OR δ = ⋃ Q∈L 3 ∪L 4Q ∩ S satisfies thatR δ ⊆ OR δ ⊆ R δ+η(1+δ) .The algorithm can be easily carried out with the help of Interval Analysis. In that case, thesubsets Q will be boxes (multidimensional intervals), and the required bounds on them canbe obtained automatically with the use of inclusion functions.2.2 The constraint problemsGoing back to the determination of the efficient set of (1), we will use constraint problems ofthe form 1 (P i )mins.t.f 1 (x)f 2 (x) ≤ f (i)2x ∈ S(5)where f (i)2 is a given constant defined below. Let ˆx (i) denote an optimal solution of (P i ), andlet R (i)δbe the region of δ-optimality of (P i ), that is,R (i)δ= {x ∈ S : f 2 (x) ≤ f (i)2 , f 1(x) − f 1 (ˆx (i) ) ≤ δ · |f 1 (ˆx (i) )|}.In the first problem that we will consider, (P 0 ), we set f (0)2 = +∞. Thus, problem (P 0 ) is infact the single objective problemmin f 1 (x)s.t. x ∈ SOnce we have solved problem (P i ) and have obtained an outer approximation of R (i)δwith thehelp of the δ-opt algorithm mentioned in the previous subsection, the constant f (i+1)2 for thenext problem (P i+1 ) is given by (see Figure 1)f (i+1)2 = min{U 2 (Q) : Q ∈ L 3 ∪ L 4 , Q ∩ R (i)δ≠ ∅} (6)where U 2 (Q) is an upper bound on all objective values of f 2 at Q. However, from a computationalpoint of view, it can be better to setf (i+1)2 = min{U 2 (Q) : Q ∈ L 3 , Q ∩ S ≠ ∅} (7)although this is a worst (higher) value than the one obtained with (6). Using (7) we only haveto check whether a box Q in L 3 contains at least one feasible point. If so, we take that box intoaccount for calculating the minimum in (7).Let us denote by Q (i)Nthe subset at which the previous minimum (either (6) or (7)) is attained,i.e, f (i+1)2 = U 2 (Q (i)N ).Lemma 7. f 1 (ˆx (i+1) ) ≤ f 1 (ˆx (i) ) + δ|f 1 (ˆx (i) )|.1 We always minimize f 1 subject to a constraint on f 2 , but a similar process can be developed interchanging the functions.