4.2. : * Jobs arrive dynamically over time. * Goal: Schedule the jobs on the machines to minimize the maximum lateness.
2.2. : * Sort the jobs in increasing order of processing time. * Schedule each job on the first available machine.
4.3. : * Multiple objective functions (e.g., makespan, lateness, and flowtime). * Goal: Schedule the jobs on the machines to optimize multiple objectives.
Please let me know if you need any further assistance.
The maximum lateness is 6.
The due dates are: 10, 12, 15, 18, 20.
1.2. : * Define the decision variables: $x_ij = 1$ if job $j$ is scheduled on machine $i$, and $0$ otherwise. * Define the objective function: Minimize $\max_j (C_j - d_j)$, where $C_j$ is the completion time of job $j$ and $d_j$ is the due date of job $j$. * Define the constraints: + Each job can only be scheduled on one machine: $\sum_i x_ij = 1$ for all $j$. + Each machine can only process one job at a time: $\sum_j x_ij \leq 1$ for all $i$. + The completion time of job $j$ is the sum of the processing times of all jobs scheduled on the same machine: $C_j = \sum_i p_ij x_ij$.
| Job | Machine 1 | Machine 2 | Machine 3 | | --- | --- | --- | --- | | 1 | 3 | 2 | 1 | | 2 | 2 | 3 | 4 | | 3 | 1 | 4 | 2 | | 4 | 4 | 1 | 3 | | 5 | 3 | 2 | 1 |