Unit 8 Problems Instructions Work Through The Problems In Ex

Unit 8 Problemsinstructionswork Through The Problems In Excel Enter

Work through the problems in Excel. Enter your answers in the following spaces below. Attach your Word file or Excel spreadsheet where indicated. YOU MUST SUBMIT YOUR EXCEL SPREADSHEET OR OTHER WORK IN ORDER TO RECEIVE CREDIT. FAILURE TO SUBMIT THE EXCEL SPREADSHEET OR OTHER WORK WILL RESULT IN A GRADE OF 0.

You will only have access to the problems once. Therefore, complete all of the problems in Excel, and then enter your answers and upload your Excel file.

Paper For Above instruction

Introduction

The following comprehensive analysis addresses a series of operational research and project management problems, emphasizing critical path analysis, probability estimation using the normal distribution, queuing theory, and system utilization. Each problem is tackled systematically, illustrating applications of project scheduling, probability calculations, and queuing models to real-world scenarios.

Problem 1: Critical Path and Project Duration

The determination of the critical path is vital for project scheduling as it identifies the sequence of activities that dictate the minimum project duration. Although specific diagrams are not provided here, generally, the process involves listing activities, their durations, and dependencies, then applying network analysis methods such as the Critical Path Method (CPM). Based on typical project data, the critical path comprises activities with the longest total duration, which you identify by summing activity durations along all pathways. The expected project duration is the total duration of this critical path.

For each project, the critical path would be estimated by analyzing the diagrams for dependencies, summing activity durations, and identifying the path with the maximum total. For example, in a hypothetical scenario, if project A's critical path was A-B-C-D with total 20 weeks, that becomes the project’s expected duration; for others, similar calculations are conducted.

Problem 2: Probability of Project Completion Before a Given Time

This problem involves calculating the probability that a project completes before a certain week, given expected durations and variances. Under the assumption of a normal distribution of total project duration, the z-score can be computed as:

z = (Target Week - Expected Duration) / Standard Deviation

where Standard Deviation is the square root of the sum of variances along the critical path branches.

Calculations involve summing variances for all paths, computing the combined standard deviation, then using standard normal tables or software to find the probabilities. For example, if the expected duration is 15 weeks with a variance of 0.04, then the standard deviation is 0.2, and the z-score for 16 weeks would be (16-15)/0.2 = 5, leading to a probability close to 1 that the project finishes before 16 weeks.

Problem 3: Project Activity Network and Probabilities

Constructing a network diagram involves mapping activities and their dependencies, then calculating project timelines via forward and backward passes. In this case, activities like A, D, E, F, G, B, I, J, K, C, M, N, O, H, G, K, O are sequenced based on precedence relationships, which can be modeled using an Activity on Arrow (AOA) or Activity on Node (AON) diagram.

Calculating project completion probabilities within a certain timeframe involves determining the distribution of project duration from activity estimates. The key is to sum variances along the critical path, then compute the probability of completing within 26 or 27 weeks, applying the normal distribution as above to derive probabilities of bonuses.

Problem 4: Queueing Theory Applications

The scenarios involve M/M/1 and M/M/c queueing models. For each case, the utilization (ρ), average number in system (L), average waiting time in system (W), and other metrics are calculated using standard formulas:

• Utilization: ρ = λ / (μ * c)
• Average number in system: L = ρ / (1 - ρ)
• Average time in system: W = 1 / (μ - λ)
• Probability of n customers in system: Pn = (1 - ρ) * ρ^n

Specific calculations depend on input parameters such as customer arrival rates, service rates, number of servers, and mean service times, illustrating how queue performance varies with load.

Problem 5: Customer Service at an ATM Using Poisson Process

The arriving customers follow a Poisson process with a rate of one every two minutes, and service times are exponentially distributed with an average of 90 seconds. The total time spent at the machine (including waiting) is derived from queueing models, particularly the M/M/1 model, using:

• Average waiting time in line: Wq = λ / (μ(μ - λ))
• Total time in system: W = 1 / (μ - λ)
• Probability of immediate service: P0 = 1 - (λ / μ)

These calculations provide the expected wait times and probabilities relevant to customer experience management.

Problem 6: Call Center Queue System

The system's performance is modeled with an M/M/1 queue with finite capacity (8 callers). Calculations involve using the traffic intensity (a = λ / μ), where λ is call arrival rate, and μ is service rate per agent. The probability of a busy signal (system full) is computed using the Erlang B formula, which accounts for finite queue capacity. The probability that a caller is put on hold is derived from the system's steady-state probabilities.

For example, with λ = 40 per hour, μ = 20 per hour per representative, and three representatives, these calculations assess the likelihood of system congestion and customer wait times.

Conclusion

In applying these various models—critical path method, probability distributions, and queueing theory—to real-world scenarios, managers and analysts are better equipped to optimize project timelines, resource allocation, and customer service operations. Accurate modeling and calculation of probabilities and system metrics facilitate informed decision-making, ultimately enhancing operational efficiency and customer satisfaction.

References

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• Kerzner, H. (2017). Project Management: A Systems Approach to Planning, Scheduling, and Controlling. Wiley.
• Gross, D., Shortle, J. F., Thompson, J. M., & Harris, C. M. (2008). Fundamentals of Queueing Theory. Wiley.
• Stevenson, W. J. (2020). Operations Management (13th ed.). McGraw-Hill Education.
• Taha, H. A. (2017). Operations Research: An Introduction (10th ed.). Pearson.
• Winston, W. L. (2004). Operations Research: Applications and Algorithms. Thomson/Brooks/Cole.
• Cohen, M. (2013). The Art of Project Management. Pearson Education.
• Simchi-Levi, D., Kaminsky, P., & Simchi-Levi, E. (2008). Designing & Managing the Supply Chain. McGraw-Hill.
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• Hillier, F. S., & Lieberman, G. J. (2021). Introduction to Operations Research. McGraw-Hill Education.