A heuristic algorithm for identical parallel machine scheduling: splitting jobs, sequence-dependent setup times, and limited setup operators

Abstract

We examine a parallel machine scheduling problem with a job splitting property, sequence-dependent setup times, and limited setup operators, for minimizing makespan. Jobs are split into arbitrary (job) sections that can be processed on different machines simultaneously. When a job starts to be processed on a machine, a setup that requires an operator is performed, and the setup time is sequence-dependent. The number of setup operators is limited, and hence not all of the machines can be set up at the same time. For this problem, we propose a mathematical programming model and analyze a lower bound. We then develop a simple but efficient heuristic algorithm so that it can be used in practice, and analytically derive a worst-case bound of the algorithm. We finally evaluate the performance of the proposed algorithm numerically with various scenarios.

Appendices

Appendix 1

Choose job a where

• Selection Rule 3: $a = {{\,\mathrm{argmin}\,}}_{i \in N} \left[ \min _{j, k \in N, j \ne i, k \ne i, j \ne k} \left( s_{ji} + s_{ki}\right) \right]$.

• Selection Rule 4: $a = {{\,\mathrm{argmin}\,}}_{i \in N} \left[ \min _{j, k \in N, j \ne i, k \ne i, j \ne k} \left( s_{ij} + s_{ik}\right) + \min _{j, k \in N, j \ne i, k \ne i, j \ne k} \left( s_{ji} + s_{ki}\right) \right]$.

• Selection Rule 5: $a = {{\,\mathrm{argmax}\,}}_{i \in N} p_i$.

• Selection Rule 6: $a = {{\,\mathrm{argmin}\,}}_{i \in N} \left[ \min _{j \in N, j \ne i} s_{ji} \right]$.

• Selection Rule 7: $a = {{\,\mathrm{argmin}\,}}_{i \in N} \left[ \max _{j \in N, j \ne i} s_{ji} \right]$.

• Selection Rule 8: $a = {{\,\mathrm{argmax}\,}}_{i \in N} \frac{p_i}{s_{\bullet i}}$.

Appendix 2

See the Tables 7, 8, 9 and 10.

Table 7 Makespan and idle time changes according to r

Table 8 Computational results of the algorithm with selection rule 2 and $m=5$

Table 9 Computational results of the algorithm with selection rule 2 and $m=10$

Table 10 Computational results of the algorithm with selection rule 2 and $m=20$