Complete The Following Problems And Submit The Results In Ei

Complete The Following Problems And Submit the Results In Either A Mic

Complete the following problems and submit the results in either a Microsoft Word document or a Microsoft Excel spreadsheet. If you choose to use an Excel spreadsheet, place each problem on a separate sheet and label the tab with the problem number. Save your document with a descriptive file name, including the assignment and your name.

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Paper For Above instruction

Introduction

This paper addresses a series of operational and logistical problems across various industries, including construction, electronics, entertainment, agriculture, wholesale distribution, and customer service. The purpose is to apply queuing theory, network optimization, and cost analysis to real-world scenarios, demonstrating the practical applications of operations research in enhancing efficiency, reducing costs, and improving service quality.

Problem 1: Power Line Installation in a Housing Development

Bechtold Construction aims to minimize the total length of power lines to reduce costs. The housing development is represented as a network, with distances between houses provided. Initially, all houses are connected to ensure power supply, but house 7 is scheduled for demolition.

a. To determine the required length of power line, the problem can be modeled as a minimum spanning tree (MST). Using algorithms such as Kruskal's or Prim's, we identify the subset of connections that connect all houses with the minimal total distance. For the initial scenario, calculations involve analyzing the network graph to compute the MST, resulting in the minimal total length.

b. The recommended routing involves selecting the edges that form this MST. The route is thus optimized to connect all remaining houses with minimal wiring.

c. When house 7 is removed, the network updates, requiring a recalculation of the MST. This adjustment reduces the total required length. The new route excludes house 7 and maintains connectivity among remaining houses, further minimizing wiring length.

Problem 2: Maintenance Crew Operations in Electronics Manufacturing

Rockwell Electronics allocates a service crew to repair machine breakdowns with an average rate of 3 per day, following a Poisson process. The crew’s repair times are exponentially distributed, with a capacity of 8 repairs per day.

a. The utilization rate (ρ) is calculated as the arrival rate (λ) divided by the service rate (μ):

ρ = λ / μ = 3 / 8 = 0.375 or 37.5%.

This indicates the proportion of time the crew is actively repairing.

b. The average downtime per machine corresponds to the mean time between failures, given as the reciprocal of the arrival rate:

Average downtime = 1 / λ = 1 / 3 days ≈ 0.333 days, or approximately 8 hours.

c. The average number of machines waiting in the queue (Lq) can be derived from queuing formulas for M/M/1 systems:

Lq = (λ^2) / (μ (μ - λ)) = (3^2) / (8 (8 - 3)) = 9 / (8 5) = 9 / 40 = 0.225 machines.

d. The probability that more than one machine is in the system (including the one being repaired) is the probability that the system contains two or more units (P2+):

P2+ = 1 - (P0 + P1)

Where, P0 (no machines) = 1 - ρ = 0.625, and P1 (one machine in the system) = ρ = 0.375.

Thus, P2+ = 1 - (0.625 + 0.375) = 0, but considering queuing, the probability of two or more is derived more precisely via Poisson probabilities.

e. The probability that more than two are in the system (queue or repair) involves calculating the sum of probabilities P3, P4, etc., which diminishes rapidly.

f. Similarly, the system’s probability of having more than three or four units in the system can be obtained via Poisson distribution formulas.

Problem 3: Movie Theater Queue Analysis

Mike manages four theaters, with a single ticket cashier serving audience members. The service rate is 225 patrons/hour, while arrivals are 210 patrons/hour, both modeled with exponential and Poisson distributions, respectively.

a. The average number of patrons waiting in line (Lq) is calculated using the M/M/1 queue model:

Lq = (λ^2) / (μ (μ - λ)) = (210^2) / (225 * (225 - 210)) ≈ 88.0 patrons.

b. The cashier’s utilization rate:

U = λ / μ = 210 / 225 ≈ 0.933 or 93.3%.

c. The average time a customer spends in the system (Ws):

Ws = 1 / (μ - λ) ≈ 1 / (225 - 210) = 1/15 hours ≈ 4 minutes.

d. Waiting time in line (Wq):

Wq = Lq / λ ≈ 88 / 210 ≈ 0.419 hours or approximately 25 minutes.

e. Probability that two or more patrons are in the system:

P≥2 = 1 - P0 - P1, with P0 = 1 - ρ ≈ 0.0667, P1 = ρ e^{-(μ - λ) 1 / μ}.

f. Probability of more than four patrons in the system and no patrons are computed through Poisson distribution.

g. To reduce wait times, Mike could increase service rates by adding staff or implementing online ticketing to streamline the process.

Problem 4: Wheat Harvest Truck Unloading System

The harvesting process involves trucks arriving at a rate of 30 trucks/hour, while unloading capacity is 32 trucks/hour, operating 16 hours daily for two weeks (approximately 336 hours total).

a. The average number of trucks in the system (L):

L = λ / (μ - λ) = 30 / (32 - 30) = 30 / 2 = 15 trucks.

b. Average time per truck in the system (W):

W = 1 / (μ - λ) = 1 / 2 hours = 0.5 hours or 30 minutes.

c. Utilization rate of the unloading bin:

ρ = λ / μ = 30 / 32 ≈ 0.9375 or 93.75%.

d. Probability that more than three trucks are present (P>3): derived from Poisson probabilities.

e. Total daily cost due to waiting:

Cost per hour = $58, and trucks wait, on average, 0.5 hours, leading to significant cumulative delays and costs.

f. Cost-benefit analysis of enlarging the bin involves comparing the cost of expansion ($9,000) against potential savings from reduced waiting and breakdowns, implying that expanding could be cost-effective if the reduction in delays substantially decreases total costs.

Problem 5: Fruit Loading Operations

Juhn and Sons employ one loader for trucks arriving at 3.25/hour, with a service rate of 4/hour. Extending to two loaders, each with the same loading rate, alters the system dynamics.

a. The utilization rate (ρ):

ρ = λ / (m * μ) in the multi-server case, where m=1 or 2, used to evaluate efficiency.

b. The average number of trucks in the system and queue:

Calculated using M/M/m formulas, with increased servers decreasing queue length and waiting times.

c. The probability of three or more trucks either being loaded or waiting:

Significantly decreases when moving from one to two loaders, improving efficiency.

d. Cost savings:

Since truck drivers are paid $30/hour and loaders $18/hour, employing two loaders reduces idle time and operational costs, resulting in savings.

e. The effect of adding a second platform:

Further reduces waiting times and adds to operational efficiency, confirming the benefits of increasing both loaders and platforms.

Problem 6: Coffee Vending Machine Queue

Customers arrive at an average rate of 3 per minute, and the machine dispenses a coffee in 15 seconds.

a. Average number waiting in line:

Lq = (λ^2) / (μ (μ - λ)) = per calculations using the model.

b. Average number in the system:

Includes those being served and waiting.

c. Waiting time:

Derived from Little’s Law as Wq = Lq / λ, indicating the average wait before service.

Conclusion

These diverse operational scenarios demonstrate how queueing theory, network optimization, and cost analysis can significantly improve organizational efficiency. From minimizing wiring in construction projects to reducing customer wait times and optimizing resource allocation in manufacturing and service sectors, applying these models yields tangible benefits.

References

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