Math 540 Week 3 Assignment Chapter 14 Jets

Math540 Week 3 Assignment Chapter 14 Jet Copies Suggested Template

In this assignment, the objective is to perform a manual simulation of the Jet Copies case study over a one-year period to estimate potential revenue loss due to machine downtime and determine whether purchasing a backup copier is justified. The case involves simulating random time intervals between breakdowns, repair times, daily demand for copies, and calculating resulting revenue losses. The process requires understanding and implementing probability distributions for repair times and demand, using Monte Carlo simulation techniques with random number generation, and applying Excel functions such as VLOOKUP for probability mapping. The final goal is to analyze the total revenue lost during the simulation period and assess whether it exceeds the cost of a backup copier, which is priced at $8,000, compared to the main copier at $18,000.

The setup provided includes the probability function for the time between repairs, f(x) = x/18 for 0 ≤ x ≤ 6, with the cumulative distribution and random number ranges to generate simulated repair times. Similarly, demand per day follows a uniform distribution between 2,000 and 8,000 copies, with a derived formula z = 6r + 2 to determine potential copies lost per day based on a random number. The simulation spans 52 weeks, with multiple breakdowns and repair cycles, where each week accumulates the duration of breakdowns and repairs, as well as the daily revenue loss calculations.

Throughout the simulation, random numbers will be generated (e.g., using Excel's RAND() function), converted into repair times via VLOOKUP, and cumulative repair times tracked to identify when the total exceeds 52 weeks, indicating the end of the year. For each breakdown cycle, the number of copies lost per day is calculated, converted to dollar losses at $0.10 per copy, and accumulated for the entire year. This process provides an estimate of total revenue lost, which is then compared to the $12,000 threshold specified in the case to decide if purchasing a backup copier is financially justified.

It is imperative to analyze the confidence level in the simulation results, considering the assumptions made about the distributions and the number of simulation runs. Limitations include the simplifying assumptions of uniform and quadratic probability distributions, potential variability not captured in the model, and the finite number of simulated weeks which might not encompass all real-world variability. Furthermore, the model assumes repairs and demand are independent events, which may not mirror complex operational realities. Despite these limitations, the simulation aims to provide a reasoned approximation to inform decision-making about investment in backup equipment.

Paper For Above instruction

The Jet Copies case study involves evaluating the need for a backup copier through a detailed Monte Carlo simulation that models the operational downtime and associated revenue losses over a one-year period. The primary objective is to determine whether the potential savings from preventing loss of revenue justify the cost of acquiring an additional copier, priced at $8,000, versus the main copier costing $18,000. The analysis hinges on simulating random breakdowns, repair durations, and daily sales demand, which collectively influence revenue loss due to machine downtime.

The first step involves modeling the time between breakdowns using the specified probability density function, f(x) = x/18 for 0 ≤ x ≤ 6. Using random numbers generated in Excel (via the RAND function), the times between breakdowns are obtained by applying the transformation x = 6 * sqrt(r1), where r1 is a uniform random number between 0 and 1. This formula produces the distribution of intervals in weeks, which are accumulated until reaching or exceeding 52 weeks, marking the simulation's endpoint for an entire year.

For repair times, the probability distribution is given by the model, with the probability function f(y) related to repair duration in days. Random numbers r2 are used in conjunction with a VLOOKUP table that maps these numbers to repair times y in days, based on the cumulative repair time distribution specified in the case. By generating multiple repair times and summing their durations, the simulation accounts for variability in repair durations, which is critical for estimating total downtime and associated revenue losses.

The demand for copies each day is modeled as a uniform distribution between 2,000 and 8,000 copies, with a likelihood of each demand level calculated from the density function, f(z) = 1/6. The derived demand formula z = 6r + 2 aligns a random number r to the number of copies sold, facilitating the calculation of potential revenue loss if the demand surpasses the copier’s capacity. The number of copies lost during downtime translates into revenue loss at a rate of $0.10 per copy.

The simulation proceeds week by week. For each breakdown, the simulation calculates the repair duration, updates the cumulative downtime, and assesses the demand each day during the downtime period. The total copies lost in each day are summed, converted into dollar amounts, and accumulated over all breakdown cycles within the 52-week span. This cumulative loss provides an estimate of annual revenue loss attributable to machine failure.

After simulating the entire year, the total revenue loss is compared to the threshold of $12,000 specified in the case. If the calculated loss exceeds this threshold, purchasing a backup copier is considered justified from a cost-benefit perspective. Conversely, if the loss remains below $12,000, the current setup may be deemed sufficient, and additional investment may not be warranted.

In considering confidence in the results, the randomness of the simulation introduces variability; thus, multiple iterations or sensitivity analysis could improve robustness. Limitations of the model include reliance on assumed probability distributions, which may not perfectly reflect real-world breakdown and demand patterns, as well as the finite simulation period that might not capture rare but impactful events. Moreover, operational factors such as maintenance quality and user behavior are not modeled, which could influence actual outcomes.

Despite these constraints, the simulation provides a valuable framework for estimating financial risk associated with equipment failure and guides decision-making regarding capital investments in backup machinery. By basing the analysis on stochastic modeling techniques, the case study demonstrates an application of probability and statistics principles in operational management and strategic planning.

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