We propose new heuristics along with an augmented-neural-network (AugNN) formulation for solving the make-span minimization task-scheduling problem for the non-identical machine environment. We explore four task and three machine-priority rules, resulting in 12 combinations of single-pass heuristics. The task-priority rules are Highest-Level-First (HLF), Highest-Total-Remaining-Processing-Time-First (HTRPTF), Smallest-Latest-Finish-Time-First (SLFTF) and Minimum-Slack-First (MSF). For machine priority, we propose a greedy rule called Fastest-Available-Machine-First (FAMF), and two non-greedy rules: (1) Fastest-Available-Machine-First-With-Conditional-Wait-1 (FAMF-CW-1), and (2) Fastest-Available-Machine-First-With-Conditional-Wait-2 (FMF-CW-2). The AugNN approach integrates neural-network learning with domain and problem specific knowledge through heuristics, to produce good results. A single-pass heuristic is embedded in a neural network designed specifically for each problem. We give the AugNN formulation for each of the 12 heuristics and show computational results on 100 randomly generated problems of sizes ranging from 20 to 70 tasks and 2 to 5 machines. Results demonstrate that AugNN provides significant improvement over single-pass heuristics. The reduction in the gap between the obtained solution and the lower-bound due to AugNN over single-pass heuristics ranged from 24.4% to 50%. As far as heuristics, the non-greedy machine-priority rules performed significantly better than the greedy rule. The average gaps for the non-greedy rules ranged from 16.1% to 23.5% compared to 33.7% to 40.4% for greedy. For AugNN, for non-greedy rules, the gap ranged from 11.5% to 15.5% compared to 18.4% to 25.0% for greedy. The HLF and HTRPTF task priority rules performed better than the other two. The HTRPTF/FAMF-CW-1 combination gave the best results, closely followed by HLF/FAMF-CW-2 and HTRPTF/FAMF-CW-2 combinations. (c) 2005 Elsevier B.V. All rights reserved.