Bi-criteria Scheduling on Parallel Machines Under Fuzzy Processing Time

Job scheduling is concerned with the optimal allocation of scare resources with objective of optimising one or several criteria. Job scheduling has been a fruitful area of research for many decades in which scheduling resolve both allocation of machines and order of processing. If the jobs are scheduled properly, not only the time is saved but also efficiency of system is increased. The parallel machine scheduling problem is widely studied optimization problem in which every machine has same work function and a job can be processed by any of available machines. Optimising dual performance measures on parallel machines in fuzzy environment is fairly an open area of research. In real life situations, the processing times of jobs are not always exact due to incomplete knowledge or an uncertain environment which implies the existence of various external sources and types of uncertainty. Fuzzy set theory can be used to handle uncertainty inherent in actual scheduling problems. This paper pertains to a bi-criteria scheduling on parallel machines in fuzzy environment which optimizes the weighted flow time and total tardiness simultaneously. The fuzziness, vagueness or uncertainty in processing time of jobs is represented by triangular fuzzy membership function. The objective of the paper is to find the optimal sequence of jobs processing on parallel machines so as to minimize the secondary criterion of weighted flow time without violating the primary criterion of total tardiness. The bi-objective problem with total tardiness and weighted flow time as primary and secondary criteria respectively, for any number of parallel machines is NP-hard. A numerical illustration is carried out to the test efficiency of the proposed algorithm.