 to the distance, we can solve a reduced problem (which will be elaborated later) to obtain an initial searching point. Second, a local search procedure is applied around the neighborhood of the initial search- ing point to find the local optimal solution, where the neighborhood structure is defined in [29]. The local search procedure follows the idea of VNS, which changes the neighbor- hood of the initial searching point, both in descent toward local optimal and in the escape from the valleys that contain them. While the local optimal is found, problem (P) is then updated by adding a new objective cut to avoid revisiting the same local search area. Finite iterations of the aforementioned steps can lead to an optimal solution of problem (P). Otherwise, (P) is proved to be infeasible in a finite number of iterations [30]. Next, we define the distance between two points in the solution space of (P), saying x Superscript(1) and x2, as = -x21 where B is defined in Section 10.3.2. Similarly, the dis- tance between the jth element of x Superscript(1) and x2, which are x} and x2 respectively, is defined as we define the partial distance related to Bk between x Superscript(1) and x2 as where BB= B. Now, we introduce the reduced problem of P specifically. The reduced problem is derived from P except that those variables whose subscripts fall in Bk are fixed at x, where X = is a constant vector. The reduced problem denoted by P(x, Bk) is given as follows. (10.32) (10.33) hipx<y1>01,J (10.34) (10.35) (10.36) x E X K., (10.37) = = - fixing one or several variables, the resulting reduced problem, which has the same structure as (P) but a smaller solution space, is solved by Matlab ILP solver. The kth neighborhood of x Superscript(1) is defined as Nk(x1) = {x¹ |Dj(x1,x) k}, where k is the size of Bk. According to [30], we can linearize Nk(x) as (10.38) Performing local search in Nk(x1) means to add (10.38) as a new constraint to reduce the solution space of P. We use (P|(10.12)) to denote the problem of adding constraint (10.3