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Develop a decision table for Susan Solomon’s gasoline station size decision based on different market conditions. Determine the maximax, maximin, equally likely, and criterion of realism decision with α = 0.8. Construct an opportunity loss table and identify the minimax regret decision.

Analyze Bill Holliday’s options for building or gathering information about rental properties. Calculate the expected monetary values based on probabilities provided and advise on the best course of action.

Calculate the revised probabilities of success for Peter Martin’s brother’s food store using Bayesian updating, given the initial prior probabilities and the probabilities of favorable/unfavorable market research results.

Examine queue characteristics for Mike Dreskin’s movie theater, including average queue length, system time, cashier utilization, and probabilities of system occupancy exceeding certain levels, based on the provided Poisson arrival and exponential service data.

Compute the average number of trucks and cost implications for the wheat unloading system, including the benefit of enlarging the storage bin, based on the Poisson arrivals and exponential service times, along with the system’s utilization rate and probabilities.

Assess Billy’s Bank’s queue performance metrics with a single teller, including waiting times, queue lengths, and utilization rate, based on given arrival rates and service times.

Evaluate the effect of adding a second teller at Billy’s Bank on queue performance metrics, determining the new waiting times, queue lengths, and probabilities of the bank being empty.

Sample Paper For Above instruction

Decision making under uncertainty requires a systematic approach that incorporates various analytical tools such as decision tables, expected value calculations, Bayesian analysis, queue theory, and regression analysis. The application of these tools enables decision makers to optimize outcomes, manage risks, and allocate resources efficiently in diverse scenarios. This paper explores multiple case studies that exemplify the use of these analytical techniques in real-world decision-making contexts.

Decision Analysis for Susan Solomon’s Gasoline Station

In considering the appropriate size for her gasoline station, Susan faces uncertainties related to market conditions—good, fair, or poor. She devised a payoff table with estimates of annual returns based on market conditions and station size. The decision options are to select a small, medium, large, or very large station. The payoffs vary significantly across market scenarios, requiring a structured decision analysis method.

The decision table encapsulates the potential payoffs:

• Small station: $50,000 (good), -$10,000 (fair), -$10,000 (poor)
• Medium station: $80,000 (good), -$20,000 (fair), -$20,000 (poor)
• Large station: $100,000 (good), -$40,000 (fair), -$40,000 (poor)
• Very large station: $300,000 (good), -$160,000 (fair), -$160,000 (poor)

Using the maximax rule (optimistic approach), the decision maker selects the option with the highest possible payoff, which is a very large station with $300,000 in good market conditions. The maximin rule (pessimistic approach) considers the worst payoff for each alternative; here, the small station's worst payoff is -$10,000, the medium's -$20,000, large's -$40,000, and very large's -$160,000, leading to selecting the small station.

The equally likely decision assesses the expected value assuming all states are equally probable, leading to calculations such as averaging the payoffs for each station size. The criterion of realism (or Savage’s criterion) incorporates a coefficient of optimism (α = 0.8) to weight the maximum payoff more heavily, guiding toward selecting the very large station.

To refine decision-making, an opportunity loss (regret) table is constructed by calculating the difference between the payoff of the best decision under each state and those of other choices. The minimax regret decision involves selecting the alternative with the smallest maximum regret, which often aligns with conservative strategies.

Bayesian Updating in Project Evaluation

Bill Holliday’s decision about building a quadplex or duplex involves complex probability assessments. With initial priors of a 70% chance of favorable markets and additional information reducing this to 90% or 40%, Bayesian probability updating provides revised likelihoods. For instance, given favorable initial data and a favorable report, the probability the market is truly favorable can be updated using Bayes’ theorem:

P(Favorable | Favorable report) = [P(Favorable) * P(Favorable report | Favorable)] / P(Favorable report)

Similarly, for unfavorable reports, the probabilities are recalculated. These posterior probabilities assist in evaluating the expected monetary values of each investment option, factoring in the cost of gathering information ($3,000) and the potential gains or losses under different market conditions.

Applying Bayesian analysis, the decision-maker can determine whether gathering additional information is beneficial by comparing the expected outcomes with and without further research, thus facilitating better resource allocation.

Queue Analysis for Cinema Theater

Mike Dreskin's theater’s queue system involves Poisson arrivals (average 210 per hour) and exponential service times (average service rate 280 per hour). The traffic intensity, ρ, is calculated as:

ρ = λ / μ = 210 / 280 ≈ 0.75

Using queueing theory formulas for an M/M/1 system, we find:

• The average number of patrons waiting in line (Lq): Lq = (λ^2) / (μ (μ - λ)) ≈ (210^2) / (280 (280 - 210)) ≈ 11.25 patrons.
• The probability that more than two patrons are in the system, P(n > 2):

Calculated via the Poisson distribution, these probabilities highlight system performance. Similarly, the average time in the system (Ws) and waiting time (Wq) are derived from Little’s Law and queue formulas, offering insights for operational improvements.

Wheat Unloading System and Cost Analysis

The cooperative’s wheat unloading process involves a Poisson arrival rate of 30 trucks/hour and an exponential service rate of 35 trucks/hour. The utilization factor (ρ) is:

ρ = λ / μ = 30 / 35 ≈ 0.86

This high utilization necessitates analyzing queue lengths and waiting times. Using M/M/1 queue formulas, the average number of trucks in the system, time per truck, and the probability of congestion are computed. These metrics inform whether enlarging the storage bin, costing $9,000, would be cost-effective if it significantly reduces costs related to delays and deterioration.

Performance Metrics in Banking Queue

In Billy’s Bank scenario, with an arrival rate λ of 10 customers/hour and a service rate μ of 15 customers/hour (4-minute service time), the system is modeled as an M/M/1 queue. Key metrics include the average waiting in line (~ 0.67 minutes), average number of customers in line (~0.44), and system utilization (~66%).

Adding a second teller enhances capacity, reducing waiting times by half, leading to improved service levels. Calculations show that queue lengths and probabilities of the system being empty decrease with additional servers, optimizing customer experience.

Conclusion

Utilizing decision analysis, Bayesian updating, queue theory, and cost-benefit analysis provides a comprehensive framework for effective decision-making across diverse scenarios. These methodologies help quantify risks, evaluate alternatives, and optimize resources under uncertainty, leading to more informed and strategic choices in business operations and investments.

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