The traditional dimensionality reduction methods can be generally classified into Feature Extraction (FE) and Feature Selection (FS) approaches. The classical FE algorithms are generally classified into linear and nonlinear algorithms. Linear algorithms such as Principal Component Analysis (PCA) aim to project high dimensional data to a lower-dimensional space by linear transformations according to certain criteria. The central idea of PCA is to reduce the dimensionality of the data set consisting of a large number of variables. In this paper, PCA was used to reduce the dimension of flow shop scheduling problems. This mathematical procedure transforms a number of (possibly) correlated jobs into a smaller number of uncorrelated jobs called principal components, which are the linear combinations of the original jobs. These jobs are carefully determined so that from the solution of the reduced problem multiple solutions of the original high dimensional problem can readily be obtained, or completely characterized, without actually listing the optimal solution(s). The results show that by fixing only some critical jobs at the beginnings and ends of the sequence using Johnson's method, the remaining jobs could be arranged in an arbitrary order in the remaining gap without violating the optimality condition that also guarantees minimum completion time. In this regard, Johnson's method was relaxed by terminating the listing of jobs at the first/last available positions when the job with minimum processing time on either machine attains the highest processing time on the other machine for the first time. By terminating Johnson's algorithm at an early stage, the method minimizes the time needed for sequencing those jobs that could be left arbitrarily. By allowing these jobs to be arranged in arbitrary order it gives job sequencing freedom for job operators without affecting minimum completion time. The results of the study were originally obtained for deterministic scheduling problems but they are more relevant on test problems randomly generated from uniform distribution with lower bound and upper bound and normal distribution with mean and standard deviation .
Dimension Reduction and Relaxation of Johnson’s Method for Two Machines Flow Shop Scheduling Problem
