Three Assignment Questions: Thermostats Are Subjected To Rig

three Assignment Questions1 Thermostats Are Subjected To Rigorous

Analyze the process control and capability for the last five samples of thermostats. Evaluate whether the process is under control and if it meets 3 sigma quality performance standards.

Assess the probability that a line at a hominy stand, with customers arriving at a rate of 8 per hour and a service time of 5 minutes on average, will be longer than three people. Calculate the average waiting time in line and the average number of customers in line.

Optimize the production schedule for Boralis backpack manufacturing during peak demand months (March to June). The demand varies across these months, and the company can produce different quantities at different costs, including surplus and back-order penalties. Set up the transportation table, model the problem, find initial feasible solutions using Northwest corner, Least cost, and Vogel methods, and determine the optimal solution.

Sample Paper For Above instruction

Introduction

Effective quality control and operational efficiency are critical components in manufacturing and service industries. This paper analyzes three diverse operational scenarios: the quality process control in thermostat production, queuing analysis for a hominy stand, and production scheduling for a backpack manufacturer. Each scenario involves complex decision-making processes that can be optimized through statistical and operations research methods. By examining these scenarios, this paper illustrates how quality control charts, queuing theory, and linear programming can improve decision-making, optimize operations, and enhance overall productivity.

Process Control and Capability Analysis for Thermostats

The first scenario involves evaluating the production process of thermostats based on the last five samples. The primary goal is to determine if the process is under statistical control and capable of meeting the specified quality standards with a 3 sigma performance level. Control charts are instrumental in such assessments. The sample data should be plotted on X-bar and R charts; if all points fall within control limits and show no systematic patterns, the process is under control.

To assess capability, process sigma (σ) and the process mean (μ) are estimated. The process capability index (Cp) and process capability index adjusted for centering (Cpk) provide quantifiable measures. A Cp and Cpk greater than 1.33 typically indicate a capable process under a 3 sigma standard. For example, if the variation in the process is small and the mean aligns well within specifications, the process can be considered capable. Conversely, if points fall outside control limits or patterns emerge, process improvements are necessary.

In the given data, the five samples should be statistically analyzed to determine their mean and standard deviation, then compared against specification limits. If the process is found to be out of control or not capable, implementing corrective actions such as process adjustments or operator retraining can stabilize the process, reducing variation and ensuring product quality.

Queuing Analysis for a Hominy Stand

The second scenario considers a hominy stand at a busy intersection, where customer arrivals follow a Poisson process at a rate of 8 per hour, and the service time averages 5 minutes per customer. This scenario fits an M/M/1 queuing model. Calculating the probability that the line exceeds three people involves utilizing the steady-state probabilities in queuing theory. First, the traffic intensity (ρ) is calculated as λ / μ, where λ is the arrival rate (8 per hour) and μ is the service rate (12 customers per hour).

Since μ is 12 customers per hour (because 60 minutes / 5 minutes per customer), ρ = 8/12 = 0.6667. The probability that there are more than three customers in line is the sum of the probabilities for having four or more customers, obtained from the geometric distribution: P(n > 3) = ρ^4 + ρ^5 + ... . Summing these gives P(n > 3) = ρ^4 / (1 - ρ) = (0.6667^4) / (1 - 0.6667) ≈ 0.197.

The average waiting time in the queue (Wq) and the average number of customers in line (Lq) are derived from standard M/M/1 formulas: Wq = ρ / (μ - λ), and Lq = λ * Wq. Substituting the values yields Wq ≈ 2 minutes and Lq ≈ 1.33 customers. These metrics inform staffing and process improvements, such as reducing service time or increasing service capacity, to minimize wait times and improve customer satisfaction.

Production Scheduling for Boralis Backpack Manufacturing

The third scenario involves optimizing production schedules for Boralis backpacks over four months with fluctuating demand and various costs. Demand data for March to June are 100, 200, 180, and 300 units, respectively, while production capacities are 50, 180, 280, and 270 units for these months.

The problem is formalized as a transportation problem, where the demands are the quantities to be satisfied at each month, and the supplies are the maximum production capacities. Surplus and back-order costs influence the decision variables, which include production quantities in each month, surplus carryovers, and back-orders. The objective is to minimize total costs considering production, surplus, and penalty costs.

The transportation table is established by setting months as origins and demands as destinations. Initial feasible solutions are found through the Northwest Corner, Least Cost, and Vogel methods, which provide starting points for further optimization via the stepping stone or MODI method. The optimal solution balances production across months, minimizes surplus and back-order costs, and satisfies demand efficiently.

For instance, the initial solution using the Vogel method may assign production to months with the lowest costs first, then adjust allocations to meet demands while respecting capacity constraints. The final optimal schedule indicates the number of backpacks to produce each month, surplus quantities to carry over or back-order, and the overall minimal cost. This approach ensures a robust production plan that accommodates demand fluctuations and minimizes costs.

Conclusion

Analyzing these diverse operational scenarios demonstrates the importance of statistical process control, queuing theory, and linear programming in operational decision-making. Control charts ensure product quality, queuing models optimize customer flow and satisfaction, and production scheduling methods provide cost-effective resource allocation. These tools, when properly implemented, contribute to increased efficiency, reduced costs, and improved quality in manufacturing and service operations.

References

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