Assignment Needed In 3 Hours Help Please Columbia Pizza

Assignment Need In 3 Hours Help Plzcolumbia Pizza Has One Oven That

Columbia Pizza has one oven capable of making a whole pie in about 10 minutes. The shop experiences high demand among college students, with peak hours from 9 pm to 11 pm, averaging 8 pie orders per hour, and off-peak hours from 5 pm to 9 pm, averaging 5 pie orders per hour. Customer arrivals follow a Poisson distribution. If customer wait times exceed 20 minutes from entry to receipt, they are likely to leave without purchasing, risking a loss of sales worth $10 per customer. The owner considers purchasing a second oven and register, costing $75 per night to operate, to reduce wait times and capture more sales. The assignment involves calculating various queuing metrics, revenue opportunities, potential sales losses, and evaluating the financial viability of adding the second oven, especially considering process improvements that reduce making time to 6 minutes per pizza.

Paper For Above instruction

Columbia Pizza's operational challenge revolves around managing customer wait times while maximizing revenue amidst fluctuating demand. The current single-oven setup leads to longer wait times during peak hours, risking customer dissatisfaction and lost sales. Introducing a second oven could alleviate congestion, but entails additional operational costs. Analyzing this scenario requires applying queuing theory principles to quantify system performance, forecast revenue potential, and inform strategic decisions.

Queuing Analysis During Off-Peak (5 pm – 9 pm)

The demand during this period averages 5 orders per hour, which is approximately 0.0833 orders per minute. With a single oven, the service rate is 1 pie per 10 minutes, or 0.1 pies per minute. The traffic intensity (ρ) is given by the ratio of the arrival rate (λ) to the service rate (μ):

\[ \rho = \frac{\lambda}{\mu} = \frac{0.0833}{0.1} = 0.833 \]

Since ρ

• Average number in system, L = \(\frac{\rho}{1 - \rho} = \frac{0.833}{1 - 0.833} \approx 5 \)
• Average time in system, W = \(\frac{1}{\mu - \lambda} \approx \frac{1}{0.1 - 0.0833} = 60 \text{ minutes}\)
• Average number in queue, Lq = \(\frac{\rho^2}{1 - \rho} \approx \frac{0.833^2}{0.167} \approx 4.17\)
• Average wait in queue, Wq = \(\frac{Lq}{\lambda} \approx \frac{4.17}{0.0833} \approx 50\) minutes

This indicates customers wait, on average, about 50 minutes in queue during off-peak hours, exceeding the target of 20 minutes, suggesting need for improvement or additional capacity.

During peak hours (9 pm – 11 pm), with λ = 8 orders/hour or 0.1333 orders per minute, service rate remains 0.1 pies per minute (assuming one oven). The traffic intensity then is:

\[ \rho = \frac{0.1333}{0.1} = 1.333 \]

Since ρ>1, the queue will grow indefinitely under current conditions, indicating the system is unstable, and wait times will become unacceptable. This demonstrates the necessity of adding a second oven or improving service efficiency during peak hours.

Impact of Second Oven on Queuing Metrics

Adding a second oven effectively doubles the service rate. For the peak period (9-11 pm), with two ovens, the combined service rate becomes 0.2 pies per minute. The new traffic intensity becomes:

\[ \rho_{new} = \frac{0.1333}{0.2} = 0.666 \]

which is below 1, indicating a stable system.

Using M/M/2 queue formulas (with identical servers), the average number in the system (L), time in the system (W), number in queue (Lq), and wait times (Wq) can be calculated by standard queueing theory formulas for multi-server queues. Notably, the average queue length significantly reduces, and wait times drop below 20 minutes, aligning with customer satisfaction goals.

Total Revenue Opportunity Calculation

Assuming all customers are served without waiting, the potential daily revenue is based on the daily demand. Off-peak (5-9 pm): 4 hours at 5 orders/hour = 20 orders; peak (9-11 pm): 2 hours at 8 orders/hour = 16 orders. Total orders per day: 36. If each sale is $10, total potential revenue is $360 per day.

However, if wait times during peak hours extend beyond 20 minutes, some customers will leave, causing revenue loss. Introducing a second oven can eliminate this loss during peak hours, assuming all customers are served promptly.

Maximum Revenue with Two Ovens

With two ovens, all customers during peak hours can be served within the desired 20-minute window, maximizing revenue. Therefore, the total maximum revenue per day remains approximately $360, assuming demand stays constant and customers do not shift elsewhere due to wait times.

Sales Loss Due to Wait Times

Without intervention, during peak hours, the system's instability results in some customers leaving, especially during the busiest hours. If approximately 10% of customers abandon due to excessive wait, the daily loss is around 1.6 orders in peak hours, equaling roughly $16 daily.

Financial Justification for Buying the Second Oven

The extra operating cost is $75 per night. The additional revenue captured by serving previously lost customers during peak hours (assuming 10% abandonment rate and total peak demand of 16 orders, with 1.6 missed sales, equating to about $16 in potential lost sales) would justify the investment if the reduced wait times attract more customers or prevent loss of sales entirely.

Implementing a second oven during peak hours could ultimately prevent customer attrition, leading to higher sales and customer satisfaction, justifying the additional operating costs.

Impact of Pizza Making Time Reduction to 6 Minutes

Reducing pizza prep time from 10 to 6 minutes increases the service rate to approximately 10 pizzas every 6 minutes (~1 per 0.6 minutes), or 1.67 pizzas per minute with one oven, and 3.33 with two ovens. This acceleration dramatically improves system capacity and reduces queues and wait times substantially.

Under these improvements, the system becomes highly efficient and capable of handling peak demand comfortably, possibly enabling the store to serve more customers, increase revenue, and justify investment in process improvements without necessarily purchasing a second oven for capacity reasons.

Conclusion

Based on the queuing analysis, purchasing a second oven during peak hours offers significant benefits, primarily reducing wait times, increasing customer satisfaction, and preventing revenue loss. With the reduction of pizza preparation time to 6 minutes, the system's efficiency improves further, potentially eliminating the need for a second oven solely for capacity, and enabling the shop to handle higher demand levels profitably.

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