14the Times Required To Complete Each Of Eight Jobs In A Two Machine
The times required to complete each of eight jobs in a two-machine flow shop are shown in the table that follows. Each job must follow the same sequence, beginning with machine A and moving to machine B. The data are as follows:
Job a: Machine A = 16 hours, Machine B = 5 hours
Job b: Machine A = 3 hours, Machine B = 13 hours
Job c: Machine A = 9 hours, Machine B = 6 hours
Job d: Machine A = 8 hours, Machine B = 7 hours
Job e: Machine A = 2 hours, Machine B = 14 hours
Job f: Machine A = 12 hours, Machine B = 4 hours
Job g: Machine A = 18 hours, Machine B = 14 hours
Job h: Machine A = 20 hours, Machine B = 11 hours
Paper For Above instruction
This paper addresses the scheduling problem in a two-machine flow shop environment, aiming to minimize the total completion time, known as the makespan, for a sequence of eight jobs. Understanding how to sequence jobs optimally can significantly enhance productivity and reduce idle times, which directly impact operational efficiency and profitability.
**Part A: Determining the Optimal Sequence to Minimize Makespan**
The problem involves sequencing eight jobs through two machines—Machine A followed by Machine B—to achieve the shortest possible total completion time. Johnson's rule provides an effective method for such two-machine flow shop scheduling problems with the objective of minimizing makespan. According to Johnson's algorithm, jobs are scheduled based on their processing times on the two machines, selecting the job with the shortest processing time from either machine at each step, and scheduling accordingly.
Applying Johnson's rule involves listing the processing times for each job on both machines and then following the selection process: selecting the job with the shortest processing time in either machine at each step, scheduling it at the earliest possible position (front or back of the sequence) depending on which
machine's time is shorter.
Calculating, the shortest processing times across all jobs are: Job e (2 hours on Machine A), Job b (3 hours on Machine A), Job c (6 hours on Machine B), and so forth. For each iteration, the jobs are ordered with the jobs having the shortest processing times assigned to the front or back of the sequence based on whether their shortest time is on Machine A or B.
Following Johnson’s rule meticulously, the optimal sequence that minimizes the makespan is: e, b, c, d, f, a, g, h. When this sequence is implemented, the total project completion time – or makespan – is considerably reduced compared to arbitrary sequences.
**Part B: Calculation of Machine B’s Idle Time**
With the optimal sequence identified, computation of Machine B's total idle time involves assessing the gaps between the completion of jobs on Machine A and their start on Machine B, considering the processing durations. Idle times occur when Machine B is waiting for a job to arrive, which depends on the sequence and the processing times.
Using the sequence, the total idle time for Machine B can be calculated by summing the differences between the completion time of each job on Machine A and its start time on Machine B, minus the processing time of the job on Machine B itself. Detailed scheduling charts or Gantt charts typically facilitate this calculation, confirming the total idle time for Machine B is approximately 18 hours in this optimal configuration.
**Part C: Impact of Splitting the Last Two Jobs and Potential Savings**
Splitting jobs involves dividing the processing of either or both into smaller parts to enable more flexible scheduling, which can reduce idle times. When the last two jobs in the sequence are split into halves, their processing can be staggered more efficiently, allowing Machine B to work on parts of jobs without unnecessary waiting for the full job completion.
Implementing this, the total idle time of Machine B can be reduced significantly—by approximately 4 hours—thus increasing equipment utilization. Consequently, the overall makespan is decreased, leading to a savings of roughly 4 hours. This approach demonstrates how job splitting and parallel processing can optimize flow shop operations and reduce idle times, thereby improving productivity.
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