# Study Notes: Pricing Analytics, Network Management, and Overbooking ## Overview This study guide covers key concepts in pricing analytics, network management, and overbooking strategies, with a focus on deterministic linear programming (DLP), bid pricing, and heuristic/cost-based approaches. Examples from industries such as hotels, airlines, and air cargo are used to illustrate these concepts. --- ## Learning Objectives - **Formulate Deterministic Linear Programming (DLP):** Understand how to model network revenue management problems using DLP. - **Evaluate Bid Pricing:** Learn the advantages and disadvantages of bid pricing as a decision-making tool. - **Understand Overbooking:** Explore why overbooking is critical in certain industries (e.g., airlines, hotels) and less relevant in others (e.g., event ticketing). - **Apply Overbooking Strategies:** Develop heuristic and cost-based approaches to manage overbooking effectively. --- ## Key Concepts ### Deterministic Linear Programming (DLP) DLP is a mathematical approach used to allocate constrained resources optimally across competing demands. It is widely applied in network revenue management. #### Components of DLP: - **Decision Variables ($x_j$):** Represent the allocation of capacity to each product. - **Revenue ($r_j$):** Revenue generated by product $j$. - **Expected Demand ($d_j$):** Forecasted demand for product $j$. - **Constraints:** Ensure capacity limits are not exceeded. #### Example: Westbrook Hotel - **Resources:** Each night is treated as a resource with a capacity of 200 rooms. - **Products:** Different combinations of arrival dates, length of stay, and room rates (e.g., Monday stay at $79, Monday-Tuesday stay at $79 per night). - **Objective:** Maximize total revenue while adhering to capacity constraints. **DLP Formulation:** $$ \text{Maximize } \sum_{j} r_j x_j \\ \text{Subject to: } \sum_{j} x_j \leq \text{Capacity}, \quad x_j \leq d_j, \quad x_j \geq 0 $$ --- ### Bid Pricing Bid pricing is a method to evaluate the opportunity cost of accepting a booking request. It is based on shadow prices derived from DLP. #### Key Idea: Accept a booking request if: $$ \text{Revenue from request} > \text{Opportunity cost (bid price)} $$ #### Strengths: - Intuitive and simple to implement. - Useful for multi-night stays, group evaluations, and origin-destination control in airlines. #### Weaknesses: - Not optimal under dependent demand. - Requires frequent updates. - Does not provide information on best pricing strategies. **Illustration: Airline Bid Pricing** A group wants to book 10 seats from A to C at $80 per person. Should the request be accepted? - **Forecasted Demand:** Discount fare ($60), full fare ($100). - **Remaining Capacity:** 70 seats. - **Shadow Prices:** $60 for discount fare, $0 for full fare. Decision: $$ 10 \times 80 > 10 \times 60 + 10 \times 0 \quad \text{(Accept the group)} $$ --- ### Overbooking Overbooking involves selling more reservations than available capacity to account for cancellations and no-shows. It is critical in industries with perishable inventory, such as airlines and hotels. #### Applicability: Overbooking is relevant when: 1. **Capacity is constrained and perishable:** E.g., airline seats, hotel rooms. 2. **Bookings are accepted for future use:** Allows cancellations and no-shows. 3. **Cost of denying service is relatively low:** Includes monetary and goodwill costs. #### Industry Examples: - **Airlines:** High importance; compensation and rebooking offered. - **Hotels:** High importance; re-accommodation at other hotels. - **Cruise Lines:** Low importance; cancellations are rare and costly. - **Event Ticketing:** Very low importance; tickets can often be resold. --- ## Overbooking Strategies ### Heuristic Approach Calculate booking limits based on capacity and expected no-show rates. #### Formula: 1. **Overbooking Pad:** $C \cdot p$ 2. **Virtual Capacity:** $C \cdot (1 + p)$ **Example:** - Capacity ($C$): 100 seats. - No-show rate ($p$): 20%. Overbooking Pad: $$ 100 \cdot 0.2 = 20 $$ Virtual Capacity: $$ 100 \cdot (1 + 0.2) = 120 $$ ### Cost-Based Approach Explicitly include costs of denied service and optimize overbooking decisions. #### Decision Rule: Accept an overbooking reservation if: $$ \text{Revenue} \geq (\text{Revenue} + \text{Cost}) \cdot P(\text{No-shows} < X) $$ **Example: Park Hyatt Hotel** - Capacity: 118 rooms. - Price per night: $159. - No-show distribution: Uniform (0–20). - Cost of denying service: $350. Critical Value: $$ P(\text{No-shows} < X) = \frac{159}{159 + 350} = 0.31 $$ Using uniform distribution: $$ 0.31 = \frac{X}{21} \quad \Rightarrow \quad X = 6 $$ Overbooking Pad: $$ 118 + 6 = 124 \text{ reservations} $$ Probability of walking a customer: $$ \frac{6}{21} = 0.29 $$ --- ## Common Misconceptions - **Ignoring Length of Stay:** When calculating revenue, ensure multi-night stays are accounted for (e.g., $79 \times 2$ for a two-night stay). - **Overbooking Risks:** Overbooking does not guarantee full capacity utilization; unexpected show-ups can lead to costly compensations. --- ## Key Takeaways - **DLP:** Provides a structured way to allocate resources optimally. - **Bid Pricing:** Useful for quick decisions but requires frequent updates. - **Overbooking:** Balances spoilage costs with compensation costs; critical in industries with perishable inventory. - **Heuristic vs. Cost-Based:** Heuristic is simpler but less precise; cost-based accounts for denied service costs. --- ## Practice Problems ### Problem 1: DLP Formulation Formulate a DLP model for a hotel with 3 room rates ($79, $99, $129) and 2 length-of-stay options (1-night, 2-night). Capacity is 150 rooms per night. ### Problem 2: Overbooking Pad An airline has 200 seats and a no-show rate of 10%. Calculate the virtual capacity using both heuristic methods. ### Problem 3: Cost-Based Overbooking A hotel has 50 rooms priced at $200 per night. The no-show rate is uniformly distributed (0–10), and the cost of denying service is $500. How many rooms should be overbooked? --- **Tip:** Use Python or Excel to solve DLP problems efficiently. Automate decision variable generation for scalability. **Note:** Overbooking strategies vary by industry. Always consider the specific operational and customer dynamics.