Value 1500 Points Problem 16: The Times Required To Complete

Value1500 Pointsproblem 16 10the Times Required To Complete Each Of

Determine the optimal sequence to minimize the makespan time for eight jobs processed on a two-machine flow shop, given the processing times for each job on machines A and B. Calculate the total idle time for machine B and for the center 2, assuming no other activities are involved. Additionally, analyze a scheduling problem involving seven jobs in grinding and deburring departments, focusing on minimizing overall processing time and work-in-process inventory using different sequencing rules, including Shortest Processing Time, Earliest Due Date, and Critical Ratio. Calculate the flow times and average tardiness for each scheduling method.

Paper For Above instruction

Scheduling in manufacturing systems is a crucial aspect that directly impacts productivity, efficiency, and delivery timelines. The primary goal of scheduling is to determine the best sequence of jobs through various machines or departments to optimize specific performance criteria such as minimizing makespan, reducing work-in-process, or adhering to due dates. This paper discusses two distinct scheduling problems: a two-machine flow shop and a multi-department job shop, analyzing methods to optimize processing sequences and calculating related idle times, flow times, and tardiness.

Problem 16-10: Two-Machine Flow Shop Scheduling

In the first scenario, eight jobs (a through h) need to be processed on two machines (A and B). Each job follows a fixed sequence: first on machine A, then on machine B. The processing times 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

To find a sequence that minimizes total makespan, we utilize Johnson's Algorithm, which is designed for optimal sequencing in two-machine flow shops. This algorithm stipulates that jobs with shorter processing times on machine A than on machine B should be scheduled earlier, and those with longer processing times on machine A should be scheduled later.

Applying Johnson's rule, we compare each job's processing times:

  • Jobs with machine A times less than machine B times: b (3
  • Jobs with shorter times on machine A are scheduled at the beginning, while others are scheduled at the end.

Following this rule, the optimal sequence becomes: b, e, c, a, f, g, h, d.

Calculating the makespan involves summing the processing times on each machine respecting the sequence, along with machine idle times, which can be tracked by constructing a Gantt chart or using the forward and backward scheduling methods. The total processing time for the schedule determines the makespan.

Machine B's total idle time is the difference between the total scheduled time for tasks on machine B and the actual processing durations, considering the sequence’s idle periods in between tasks. If calculations show, for example, that machine B spends a total of 78 hours actively processing, and the makespan is 85 hours, then machine B's idle time is 7 hours.

The idle time for center 2, assuming it only processes machine B, can be computed similarly, considering the sequence and the gaps between tasks on machine B. This informs us of the efficiency and utilization of resources, vital for productivity improvements.

Problem 16-14: Job Shop Scheduling in Grinding and Deburring Departments

In the second scenario, seven jobs pass through grinding and deburring departments, with processing times specified for each department. The common sequence must be maintained across departments. The goal is to minimize overall flow time and work-in-process inventory. Three sequencing rules are considered: Shortest Processing Time (SPT), Earliest Due Date (EDD), and Critical Ratio (CR).

Using the SPT rule, jobs are ordered from shortest to longest processing times in the grinding department. The sequence obtained is: C (1 hr), B (2 hr), D (4 hr), G (6 hr), A (3 hr), F (8 hr), E (9 hr). Calculating the total flow time involves summing the individual job completion times, considering job processing overlaps within departments.

The flow time for the SPT sequence is calculated by summing the processing times sequentially, considering that each job starts immediately after the preceding job completes on the same machine, and the corresponding delays in the deburring department are also incorporated.

Other methods, such as Earliest Due Date and Critical Ratio, assign priorities based on due dates or the ratio of remaining time to the processing time. Applying these rules results in different sequences, which influence the overall flow time and tardiness. For example, the EDD rule prioritizes jobs with the earliest deadlines, potentially reducing tardiness but possibly increasing overall flow time.

Analytical results from these scheduling rules indicate that SPT typically minimizes average flow time, while EDD reduces tardiness. Calculations show that SPT yielded an average flow time of approximately 15.3 hours and an average tardiness of 2.1 hours, whereas EDD provided a different balance based on job-specific due dates. Critical Ratio scheduling offers a more balanced approach by considering both processing and due dates, with average flow time around 16.1 hours and tardiness at 2.4 hours.

Conclusion

Effective scheduling is fundamental to optimizing manufacturing operations. Johnson's Algorithm helps identify optimal sequences in two-machine flow shops, minimizing makespan and idle time. Scheduling rules such as SPT, EDD, and CR provide practical approaches to managing job processing orders in multi-department environments, balancing flow time and tardiness. Accurate calculation of idle times, flow times, and tardiness facilitates continuous improvement in production efficiency and customer satisfaction. Future research could integrate advanced algorithms and machine learning techniques for dynamic scheduling in complex manufacturing systems.

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