Suppose A Car Rental Agency Offers Insurance For A We 553113

Suppose That A Car Rental Agency Offers Insurance For Week T

Suppose that a car rental agency offers insurance for a week that will cost $10 per day. A minor fender bender will cost $1,500, while a major accident might cost $15,000 in repairs. Without the insurance, you would be personally liable for any damages. What should you do? Clearly, there are two decision alternatives: take the insurance or do not take the insurance.

The uncertain consequences, or events that might occur, are that you would not be involved in an accident, that you will involved in a fender bender, or that you would be involved in a major accident. Assume that researched insurance industry statistics and found out that the probability of a major accident is 0.05% and that the probability of a fender bender is 0.16%. What is the expected value decision? Would you choose this? Why or why not?

What would be some alternate ways to evaluate risk? (2) Suppose that the service rate to a waiting line system is 10 customers per hour (exponentially distributed). Analyze how the average waiting time is expected to change as the arrival rate varies from two to ten customers per hour (exponentially distributed). Please be sure your work is organized, legible, and your responses are substantive. You need to submit all details of your work including excel sheets used to arrive at the solution. It is not enough to attach your excel sheet. You MUST provide interpretation of results and describe conclusions.

Paper For Above instruction

The decision to purchase insurance when renting a vehicle involves risk evaluation under uncertain outcomes. In this context, analyzing the expected costs with and without insurance, based on the probabilities of different accident types, provides a rational basis for decision-making. Additionally, understanding how the average waiting time in a queue varies with changing arrival rates is crucial for optimizing service processes.

Part 1: Cost-Benefit Analysis for Insurance Purchase

The core question revolves around whether the additional cost of insurance is justified by the potential reduction in financial liability due to accidents. The insurance costs dollar250 per week (assuming 7 days at $10 per day). The key is to compare the expected costs associated with potential accidents against the insurance premium.

Let's denote:

• Cost of insurance per week: $70
• Probability of no accident: 1 - (probability of fender bender + probability of major accident) = 1 - (0.0016 + 0.0005) = 0.9979
• Probability of a fender bender: 0.0016
• Probability of a major accident: 0.0005

The expected cost if not insured can be calculated considering the probabilities:

• No accident: $0
• Fender bender: $1,500
• Major accident: $15,000

Expected value of potential damages without insurance (EV damages):

EV damages = (Probability of fender bender × Cost) + (Probability of major accident × Cost) = (0.0016 × 1500) + (0.0005 × 15000) = 2.4 + 7.5 = $9.9

Since the expected damage costs are approximately $9.90, which is lower than the weekly insurance premium ($70), an expected monetary perspective suggests that purchasing insurance may not be economically justified based solely on expected value considerations.

However, this analysis considers only the costs and probabilities, ignoring risk aversion and the psychological benefits of risk reduction. Many consumers might prefer to pay a fixed fee to avoid potentially catastrophic costs, even if the expected monetary value favors not purchasing insurance.

Alternative Risk Evaluation Methods

Beyond expected value calculation, other methods for evaluating risk include:

• Risk-averse decision models such as utility theory, which account for an individual's attitude toward risk.
• Monte Carlo simulations to forecast a range of possible costs based on probabilistic models.
• Cost-effectiveness analysis comparing the incremental costs and benefits of insurance coverage.
• Scenario analysis exploring different combinations of accident probabilities and costs to evaluate best- and worst-case outcomes.

Part 2: Analyzing Waiting Time in a Service System

The service rate of 10 customers per hour, modeled as an M/M/1 queue, determines the average waiting time depending on the arrival rate (λ). When λ is less than the service rate (μ=10), the system is stable; when λ approaches μ, wait times increase dramatically.

Using the standard queueing theory formula for the average waiting time in the queue (Wq):

Wq = λ / [μ (μ - λ)]

where:

- λ = arrival rate (customers/hour)

- μ = service rate (customers/hour) = 10

Analyzing how Wq varies from λ = 2 to λ = 10 demonstrates the impact of increasing arrival rates:

• At λ=2, Wq ≈ 0.222 hours or approximately 13.33 minutes.
• At λ=4, Wq ≈ 0.444 hours or approximately 26.67 minutes.
• At λ=6, Wq ≈ 0.75 hours or 45 minutes.
• At λ=8, Wq ≈ 1.78 hours or approximately 106.67 minutes.
• At λ=10, Wq approaches infinity, indicating system breakdown under maximum capacity.

These results underscore the nonlinear increase in waiting time as the arrival rate nears the service capacity. Management should ensure that the arrival rate remains well below service capacity to prevent excessive delays, which significantly affect customer satisfaction and operational efficiency.

Conclusion

Decision-making under risk in car rental insurance exemplifies the importance of probabilistic analysis to inform choices. While expected damages suggest that insurance may not be economically justified strictly from a monetary viewpoint, individual preferences and risk aversion can tilt the decision towards purchasing coverage for peace of mind. For queue management, understanding how waiting times depend on their input parameters enables organizations to better match capacity with demand, improving service levels and customer experience.

References

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