Flow Shop Scheduling [chapter]

International Handbook on Information Systems  
Consider scheduling tasks on dedicated processors or machines. We assume that tasks belong to a set of n jobs, each of which is characterized by the same machine sequence. For convenience, let us assume that any two consecutive tasks of the same job are to be processed on different machines. The type of factory layout in the general case -handled in Chapter 10 -is the job shop; the particular case where each job is processed on a set of machines in the same order is the flow shop. The most
more » ... nly used performance measure will be makespan minimization. The latter is l<s<n J J J J equivalent to max{ Pij,Pij-P2j+Pij+i } ^ max{ py^^.p^j^^ -p^j^i +Py } which is equivalent to Plj^PiM ^^^ Pij-Pij-^Pij+i^PiM or Pij^Pij+i-Py+i-^Pij and Py-Py+Pij+i^Pij+i-Pij+i+Py-Thus, if py < mm{p^j^^, py} or P2j+i < min{pi^.+i, py}, or equivalently, if min{pij, P2j+i] ^ ^^MPij+i 5 P2j} then permutation TI; defines a schedule at least as good as %'. Among all permutations defining an optimal schedule, assume TC is a permutation satisfying J^ precedes Jj if minlp^^., P2j} < mm{p2i, Py}, for any two jobs J^ and Jj where one is an immediate successor of the other in the schedule. It remains to verify transitivity, i.e. if minipn, P2j} < min{p2i, Py} implies J^-precedes Jj and min{pij, P2k} < min{/?2;, Pul implies Jj precedes /^ then minipn, P2k} ^ min{/?2/? Pik) implies /^-precedes Jj^ in TI. There are 16 different cases to distinguish according to the relative values of the four processing time pairs p^.
doi:10.1007/978-3-540-32220-7_8 fatcat:lpbh66dg6rchlabnmjhdp5wnli