Approximation Algorithms for Flexible Job Shop Problems

Abstract

The Flexible Job Shop problem is a generalization of the classical job shop scheduling problem in which for every operation there is a group of machines that can process it. The problem is to assign operations to machines and to order the operations on the machines so that the operations can be processed in the smallest amount of time. This models a wide variety of problems encountered in real manufacturing systems. We present a linear time approximation scheme for the non-preemptive version of the problem when the number m of machines and the maximum number \x of operations per job are fixed. We also study the preemptive version of the problem when m and \x are fixed, and present a linear time approximation scheme for the problem without migration and a (2 + ^-approximation algorithm for the problem with migration. Int. J. Found. Comput. Sci. 2005.16:361-379. Downloaded from www.worldscientific.com by MCMASTER UNIVERSITY on 02/06/15. For personal use only.

A New Meta-Heuristic Algorithm for Solving the Flexible Dynamic Job-Shop Problem with Parallel Machines

“…In the same year, a hybrid genetic-based algorithm was proposed to minimize the C max in FJS [20]. After that, in 2001, a polynomial algorithm with/without considering interruption was proposed for FJS [21]. In 2002, for the first time, FJS was investigated in a multi-purpose case [22], and the proposed algorithm was a hybrid of fuzzy logic and evolutionary algorithms.…”

Optimal Scheduling of Two Communication Flows on Multiple Disjoint Packet-Type Aware Paths

“…During this interval, the packet (i,j) is in transit on path(i,j), so we will call it transit interval. A schedule is valid if for any two packets (i,j (1)) and (i,j(2)), j(1)<j(2), we have tfinish(i,j(1))≤tstart(i,j (2)), and if any two packets scheduled on the same path (disregarding their type) are assigned disjoint transit intervals. The first condition makes sure that each flow's packets are sent sequentially (in the order given by the starting time of the packets' transit intervals) and the second one makes sure that the packets scheduled on the same path are sent one at a time.…”

“…For each flow i, we define an ordering of the paths: po(i,1), po(i,2), …, po(i,P), such that ts(po(i,1),i)≤ ts(po(i,2),i)≤…≤ts(po(i,P),i). In the proofs of the following theorems we will frequently reassign a packet from a path po(i,q(1)) to a path po(i,q(2)), with q(2)<q (1). Such a reassignment does not change the starting time of the packet, but may decrease its ending time.…”

“…We will classify the type i(2)(≠i(1)) packets in the same three categories as in Theorem 2's proof and perform the same reassignments. If there are less than two packets in category 2 or if the two packets in category 2 are assigned to the same path or if one of them is assigned to path po(i(2),1) or to path po(i(2),4) then packet (i(1),j)'s transit interval intersects the transit intervals of packets scheduled on at most two distinct paths from the set {po(i(1),1), po(i(1),2), po(i(1),3)}, which allows us to reassign packet (i (1),j) to one of the paths in the set. The more difficult case occurs when there are two packets belonging to category 2, one of them assigned to the path po(i (2),2) and the other one to the path po(i(2),3) (see Fig.…”

“…If po(1,1)≠po (2,1), then all the type 1 packets will be scheduled on path po (1,1) and all the type 2 packets on the path po (2,1). The makespan will be:…”