An Improved Augmented Neural-Network Approach for Scheduling Problems

n : Number of tasks m : Number of machines k : Current iteration T : Set of tasks = {1,..,n} M : Set of machines = {1,..,m} Tj : j task node, j ∈ T Mij : Node for machine i connected from Tj, i ∈ M, j ∈ T Lj : Level of Tj, j ∈ T (number of tasks in the remaining path till the final node) LETj : Level with estimated time of Tj, j ∈ T (processing rime of the remaining path) ωj : Weight on the link from Tj to machine nodes ωm : Weight on the links between machine Nodes α : Learning coefficient εk : Error in iteration k t : Elapsed time in the current iteration I : Initial dummy task node F : Final dummy task node τj : Threshold value of Tj = # of tasks immediately preceding Tj, j ∈ T ∪ F STj : Start time of Tj, j ∈ T PTj : Processing Time of Tj, j ∈ T LSTj : Latest Start Time of Tj, j ∈ T CPj : Whether task j is on the critical path, j ∈ T NISj : Number of immediate successor tasks of j, j ∈ T PRj : Set of tasks that immediately precede task j, j ∈ T ∪ F NPR : Set of tasks with no preceding tasks : { j | PRj is an empty set }, j ∈ T SUj : Set of tasks that immediately succeed task j, j ∈ T Winj : Winning status of Tj, j ∈ T