Homework 31: Consider A Two-Station Production Line With Sin

Homework 31 Consider A Two Station Production Line With Single Machin

Consider a two-station production line with single machines. Jobs arrive to the first station at an average rate of 5 per hour with a squared coefficient of variation equal to 1.96. Effective processing times average 0.125 hours with a squared coefficient of variation equal to 0.81. a) What is the first station’s utilization? b) What is the squared coefficient of variation of the arrivals to the second station? c) If an identical second machine is added to the first station, do you expect the SCV of arrivals to the second station to increase or decrease? Why? (Start by thinking about how utilization will change.) 2. Suppose jobs arrive at a single-machine workstation at a rate of 20 per hour and the average process time is 2.5 minutes. a) What is the utilization of the machine? b) Suppose that the inter-arrival and process times are such that and . i) What is the average time a job spends at the station (i.e. waiting plus process time)? ii) What is the average number of jobs at the station? c) Now suppose instead that the SCV of process times is 4, with a mean process time of 2.5 minutes (unchanged). i) What is the average time a job spends at the station? ii) What is the average number of jobs at the station? 3. Consider a workstation with 11 machines, each requiring one hour of process time per job with an SCV of 5. Each machine costs $10,000. Jobs arrive at a rate of 10 per hour with SCV equal to 1 and they must be filled. Management has specified a maximum allowable average response time (i.e. time a job spends at the station) of 2 hours. Currently it is just over 3 hours (check this). Analyze the following options and determine which is best for reducing average response time. a) Perform more preventive maintenance at a cost of $8000 (over lifetime of the machines). The effect of this on tr and tf is that ultimately ce2 is reduced to 1. b) Add another identical machine to the workstation at a cost of $10,000. c) Modify the existing machines to make them faster without changing the SCV, at a cost of $8500. The modified machines would have te = 0.96 hours and ce2 = 5. You may use the VUT spreadsheet used for the example in the video to help you in comparing the options.

Paper For Above instruction

Introduction

Manufacturing systems are complex and demand meticulous analysis to optimize efficiency, reduce costs, and improve service levels. This paper examines three distinct production scenarios, employing principles from queueing theory, work measurement, and production planning to analyze and propose optimal solutions. The first scenario explores a two-station production line, focusing on calculating utilization and variability impacts. The second scenario addresses a single-machine workstation with considerations on utilization, variability, and their influence on job processing times and queue lengths. The third scenario evaluates various strategies to reduce response times in a multichannel workstation, considering maintenance, expansion, and process improvements.

Analysis of the Two-Station Production Line

The first scenario involves a two-station production line, where jobs arrive at the first station at a rate of 5 jobs/hour, with a squared coefficient of variation (SCV) equal to 1.96. The processing time for each job is 0.125 hours with an SCV of 0.81. The tasks include calculating the utilization of the first station, the SCV of arrivals to the second station, and the impact of adding an additional machine to the first station.

To determine the utilization of the first station, the formula used is:

Utilization (ρ) = Arrival rate (λ) / Processing capacity (μ)

The effective processing time per job is 0.125 hours, so the processing capacity μ is 1 / 0.125 = 8 jobs/hour. Therefore, the utilization is:

ρ = 5 / 8 = 0.625 or 62.5%

This indicates that the first station is moderately utilized, which impacts queue lengths and variability. The SCV of arrivals to the second station depends on the variability introduced by the first station's output. The broader the variance of output, the higher the SCV received at downstream stations. Since the SCV of arrivals to the first station is 1.96, and the processing time SCV is 0.81, the propagation of variability can be modeled through queueing network formulas, which suggest that the SCV of arrivals to station two increases, further exacerbating variability issues downstream.

Adding a second identical machine to the first station will alter utilization and variability. Intuitively, increasing capacity reduces pressure on the station, which could decrease the variability of output (SCV) due to less congestion, thus likely decreasing the SCV of arrivals to station two. Additionally, the utilization at the first station would decrease, leading to a more balanced flow, contributing to a more stable production process.

Analysis of a Single-Machine Workstation

The second scenario involves a single-machine workstation where jobs arrive at 20 per hour, with an average process time of 2.5 minutes (0.04167 hours). The utilization is computed as:

ρ = Arrival rate / Service rate = 20 / (1 / 0.04167) = 20 * 0.04167 = 0.833 or 83.3%

This high utilization indicates a tightly loaded system, where queue lengths and wait times tend to increase. Under typical queueing models with SCV considerations, the average time a job spends at the station, including waiting and processing, can be estimated using the Pollaczek-Khinchine (P-K) formula for M/G/1 queues:

W = (ρ (C_s^2 + 1) T_s) / (2 * (1 - ρ))

Where C_s^2 is the SCV of service times, and T_s is the mean service time. In the initial case, if C_s^2 is approximated as 1 (assuming exponential distribution), the average waiting and processing time can be obtained. With C_s^2 = 1, the total average time (W) includes waiting time and service time.

When the SCV of process times increases to 4 while keeping the mean process time unchanged, the variability component increases markedly. This leads to a higher average total time in the system, which is calculated similarly but substituting the higher C_s^2. Resultantly, average waiting time increases significantly, illustrating how variability impacts throughput and latency.

Furthermore, the average number of jobs at the station, according to Little's Law, is:

N = λ * W

Higher variability inflates W, hence increasing N, leading to greater congestion and delayed responses.

Strategies to Reduce Response Time in a Multi-Machine Workstation

The third case examines an 11-machine workstation, each requiring an hour per job with an SCV of 5. The system faces a current average response time just over 3 hours, exceeding the management's maximum desired value of 2 hours. The costs of possible interventions include preventive maintenance, adding machines, or modifying existing machines for faster processing.

Option a) involves performing preventive maintenance, costing $8,000, which reduces variability in processing times from an SCV of 5 to 1, effectively stabilizing machine performance. This reduction in variability should decrease queue lengths and average response times. Using queueing models, the effect on the total system response time is significant, as process variability critically affects system throughput and delay.

Option b) adds an identical machine at a cost of $10,000. This increases system capacity directly, reducing utilization and queue lengths, which should decrease the average response time below the 2-hour mark. The impact can be quantified by analyzing the system as an M/M/c queue (with c=12) and recalculating waiting times accordingly.

Option c) involves modifying machines to perform faster, costing $8,500, with a new mean process time of 0.96 hours and an SCV of 5. This decreases individual machine processing times, thereby reducing overall system utilization and queueing delays. The higher process speed directly enhances throughput and reduces response times, assuming variability remains unchanged.

Simulation tools like VUT spreadsheets aid in contrasting these strategies by modeling different parameters' effects on queue lengths and wait times. While all options contribute to reducing response times, increasing capacity via added machines (option b) generally provides more immediate and predictable improvements, but at a higher incremental cost. Modifying machines for speed (option c) offers a less costly but possibly less predictable outcome due to the persistent high variability. Preventive maintenance (option a) enhances stability, which indirectly reduces delays but may be less impactful than capacity expansion.

Hence, considering cost-effectiveness and system impact, adding an extra machine appears to be the most straightforward and effective solution to meet the 2-hour response time constraint. Nonetheless, a combined approach involving maintenance and process improvements could optimize overall system performance and cost.

Conclusion

Analysis of production systems emphasizes the importance of balancing capacity, variability, and process stability to optimize throughput and response times. In the two-station line, increasing capacity reduces variability and improves flow. In single-machine systems, high variability significantly inflates wait times, necessitating process control or capacity increases. For multi-machine workstations, strategic interventions such as adding machines or improving process speeds are crucial in achieving targeted response times. These insights underscore the significance of comprehensive system analysis and tailored operational strategies in manufacturing management.

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