Scheduling Tasks on Unrelated Machines: Large Neighborhood Improvement Procedures

Two approximation algorithms are presented for minimizing the makespan of independant tasks assigned on unrelated machines. The rst one is based upon a partial and heuristical exploration of a search tree, which is used not only to build a solution but also to improve it thanks to a post-optimization procedure. The second implements a new large neighborhood improvement procedure to an already existing algorithm. Both algorithms nd solutions for large instances within a few minutes. Computational experiments show that they give near optimal solutions. We consider the problem of scheduling n independant tasks on m machines. Let the integer p i;j be the processing time of task T j (j = 1; : : :; n) on machine M i (i = 1; : : :; m). p i;j is a function of both the task and the machine: there is no idea of a fast machine requiring less time than a slow machine for every task. The machines are said to be unrelated. Each task is to be scheduled on exactly one machine. If T j is assigned to machine M i , we deene (j) = i. The completion time of M i is then C i = P (j)=i p i;j and the completion time or makespan of a schedule is C max = max 1im C i. The objective is to nd a schedule of minimum makespan. Following the notation of Graham et al. (1979), this problem is refered as RjjC max. This problem is NP-complete. Previous research studies have produced-approximation algorithms, that is algorithms delivering for any instance a schedule with makespan C such that C C ? where C ? is the optimal makespan. Such algorithms are of course very interesting but the following theoritical result gives an important restriction about. Lenstra, Schmoys and Tardos (1990) showed that it is NP-complete to decide if an instance of RjjC max has a schedule of length 2. So there is no polynomial-time-approximation algorithm with < 3 2 unless P = NP { see Lenstra and Schmoys (1995). The rst published algorithms are based on list scheduling. Ibarra and Kim (1977) show that a variety of simple algorithms are only m-approximation algorithms. A more elaborated variant, due to Davis and Jaae (1981), is shown to be a-approximation with 2 O(p m). Two 2-approximation algorithms based on linear programming were published later. The algorithm proposed by Potts (1985) …