Columbia Pizza Has One Oven That Can Make A Whole Pie 131259

Columbia Pizza Has One Oven That Can Make A Whole Pie In About 10 Minu

Columbia Pizza has one oven that can make a whole pie in about 10 minutes. Their late-night business among hungry college students is booming, and from 9 pm – 11 pm, they average 8 pie orders per hour. However, from 5 pm – 9 pm, they average only 5 pie orders per hour. Both order rates can be assumed arrive according to a Poisson distribution. The owner of Columbia Pizza has discovered that if people have to wait for more than 20 minutes from when they get in the door to when they receive their pie, they are likely to simply go next door to University Falafel, meaning that Columbia Pizza would lose out on their $10 sale. The owner is thinking of buying a second pizza oven and register (e.g., a second channel) that he thinks would cost $75 per night to operate (for the sake of this example, that is both the amortized cost of the oven and the pay of the person who is taking orders for the six hours that the pizza shop is open). Answer each of the questions below. You must show all work. Be sure to highlight your final answer and when asked, please be sure to justify your response.

Paper For Above instruction

Introduction

Efficiency in pizza service operations is critical for maximizing customer satisfaction and revenue, particularly during peak hours. Columbia Pizza faces operational challenges with a single oven that limits throughput and leads to potential customer dissatisfaction if wait times exceed 20 minutes. To evaluate the impact of expanding capacity, this analysis applies queuing theory principles to forecast system performance, revenue potential, and strategic recommendations regarding acquiring a second oven during critical late-night hours.

Operational Context and Assumptions

Columbia Pizza's pizza preparation times, customer arrival patterns, and willingness to wait are key variables for modeling the queuing system. The critical assumptions include:

- Arrival rates follow a Poisson distribution.

- Pizza preparation time per oven is approximately 10 minutes.

- Total service capacity impacts queue length and wait times.

- Customer loss occurs if wait exceeds 20 minutes.

- The potential second oven incurs $75 nightly operational costs.

- Customer arrival rates are 5 orders/hour (5 pm – 9 pm) and 8 orders/hour (9 pm – 11 pm).

Queuing Theory Analysis for the 5-9 pm shift

1. Average Number of Orders in System (L)

The system can be modeled as an M/M/1 queue, where the arrival rate (λ) is 5/hr, and the service rate (μ) with one oven is 6/hr (since each pizza takes 10 minutes, or 1/6 hour). The utilization (ρ) is λ/μ = 5/6 ≈ 0.833.

Utilization (ρ) indicates the proportion of time the oven is busy, affecting queue length and wait times.

The average number of orders in the system (L) is calculated as:

L = ρ / (1 - ρ) = 0.833 / (1 - 0.833) ≈ 5 orders

2. Average Time Orders Spend in the System (W)

The average time in the system is given by:

W = 1 / (μ - λ) = 1 / (6 - 5) = 1 hour, or 60 minutes

This indicates that on average, an order spends about 60 minutes in the system during this shift, which exceeds the 20-minute threshold. However, this is theoretical; actual wait times depend on queue length distributions.

3. Average Number of Orders in Queue (Lq)

Lq = ρ² / (1 - ρ) = (0.833)² / (1 - 0.833) ≈ 4.17 orders

4. Average Waiting Time in Queue (Wq)

Wq = Lq / λ = 4.17 / 5 ≈ 0.834 hours, or about 50 minutes

Again, this exceeds 20 minutes, suggesting service delays during this period are substantial without additional capacity.

5. Utilization of the Oven

Utilization (ρ) = λ / μ = 5 / 6 ≈ 83.3%

Queuing Theory Analysis for the 9-11 pm shift

1. Average Number of Orders in System (L)

λ = 8/hr; μ = 6/hr; ρ = 8/6 ≈ 1.333

Since a utilization over 1 indicates the system cannot keep up, an M/M/1 model suggests the queue grows infinitely, which is unsustainable. Constant delays would occur unless capacity is increased.

2. Implications

The existing capacity cannot handle 8 orders per hour efficiently, resulting in excessive wait times that would exceed the 20-minute threshold, causing customer dissatisfaction and loss.

Impact of Adding a Second Oven

1. For 5-9 pm Shift with Two Ovens

Service rate doubles: μ_total = 2 × 6 = 12 orders/hour. New utilization per oven (assuming equal sharing): ρ = λ / μ_total = 5 / 12 ≈ 0.417.

Average number of orders in system (L):

L = λ / (μ_total - λ) = 5 / (12 - 5) = 5/7 ≈ 0.714 orders

Average time in system (W):

W = 1 / (μ_total - λ) = 1 / (12 - 5) = 1/7 hours ≈ 8.57 minutes

This demonstrates that wait times would drastically decrease below the critical 20-minute threshold.

2. For 9-11 pm shift with Two Ovens

Similarly, λ=8/hr; μ_total = 12/hr; ρ = 8/12 ≈ 0.667.

Average number of orders in system:

L = 8 / (12 - 8) = 8 / 4 = 2 orders

Average time in system:

W = 1 / (12 - 8) = 1/4 hours = 15 minutes

This ensures wait times are acceptable, avoiding customer defection due to long waits.

Maximum Revenue Opportunity Analysis

During peak hours (9-11 pm), with an average of 8 orders per hour, customers spend an estimated 15 minutes in the system post-expansion, well within the 20-minute limit, thus maximizing customer retention and revenue.

Total revenue potential for the night:

At $10 per pizza, with 8 orders/hour over 2 hours, total possible sales without lost customers:

8 × 2 × $10 = $160

Revenue Impact of Capacity Expansion

Without expansion, delays likely decrease actual sales due to customer abandonment. Assuming some percentage of customers leave if wait times exceed 20 minutes, the projected sales loss can be significant, perhaps estimated at 20%-30%. The expansion reduces this loss substantially, pushing maximum revenue toward $160 per night.

Financial Justification for Investment

The higher operational cost of $75 per night must be balanced against increased revenue capture and reduced customer churn. Given the analysis, the expanded capacity would substantially reduce wait times, improve customer satisfaction, and prevent lost sales, suggesting a positive net impact.

Effect of Reduced Pizza Preparation Time

If pizza preparation time decreases from 10 to 6 minutes per pizza, the service rate (μ) increases from 6/hr to 10/hr. Under these circumstances, even a single oven could accommodate higher demand with reduced wait times, or if demand remains the same, wait times would decrease further, making the second oven less critical.

For example, with μ = 10/hr and λ = 5/hr, the utilization drops to 0.5, indicating less congestion. This improved efficiency enhances the business case against investing in a second oven or makes additional capacity redundant for current demand levels.

Conclusion

Based on queuing analysis, adding a second oven during peak hours significantly improves service levels, reduces wait times below the critical threshold, and enhances revenue potential. The financial benefits of increased customer retention and sales outweigh the additional nightly operational costs, making the investment advisable. However, if operational efficiencies improve to reduce pizza making times to six minutes per pizza, the necessity for a second oven diminishes. Under such efficiencies, existing capacity might suffice, or the investment in additional capacity could be deferred, optimizing operational costs.

References

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