1. Lecture 13 Advanced Booking and Capacity Constraints

2. Lecture Plan HW3, HW 4 Dynamic Pricing Advanced Booking Overbooking
In class exercise

3. Dynamic Pricing
Dynamic pricing is a blanket term for any shopping experience where the price of an item fluctuates frequently based on complicated algorithms. 
A retailer might frequently change the price of an item based on consumer demand, price fluctuations at a competing retailer, or even the time of day and weather conditions.
Dynamic pricing can be found in a wide variety of industries.
Bezel-less/product/B01BMB3ANE?context=browse

4. Dynamic Pricing
One segment on the rise with dynamic pricing is professional sports with Real Time Pricing. 
E.g., the St. Louis Cardinals set their ticket price algorithms based on factors like team performance, pitching match ups, weather, and ticket demand. 

5. Dynamic Pricing
In certain grocery stores, the price consumers pay for the exact same product can differ based on personal data collected through loyalty card programs. 
At a Safeway in Denver, a 24-pack of Refreshe bottled water costs $ for Customer A. For Customer B, the price is $3.69.
The difference? The vast shopping data Safeway maintains on both customers through its loyalty card program.
Customer A has a history of buying Refreshe brand products, but not its bottled water, while customer B, a Smartwater partisan, is unlikely to try Refreshe.
A Safeway Web site shows Customer A the lower price, which is applied when she swipes her loyalty card at checkout

6. American: DFW-LAX All Tickets Sold in 2004Q4
Can see bands; even lowest fares sold close to departure

7. The “Prime Booking Window”
Don’t buy your ticket too early!
Best time to buy your ticket is 54 days in advance

8. Advanced Selling
Requires an inverse relationship between consumer price sensitivity and customer arrival time.
Less price sensitive customers are unwilling to purchase in the advance period so that advance purchases are made to only low-valuation customers
Similar to traditional models of second-degree price discrimination.

9. Advanced Booking
Consumers making reservations differ in their probability of showing up to collect the good or the service at the pre- agreed time of delivery.
Firms can save on unused capacity costs, generated by consumers’ cancellations and no-shows, by varying the degree of partial refunds
Airline companies in selling discounted tickets where cheaper tickets allow for a very small refund (if any) on cancellations,
Whereas full-fare tickets are either fully-refundable or subject to low penalty rates.

10. Advanced Booking and Partial Refunds
Partial refunds are used to control for the selection of potential customers who make reservations but differ with respect to their cancellation probabilities.

11. Capacity Constraints Examples of fixed supply – capacity constraints:
Travel industries (fixed number of seats, rooms, cars, etc).
Advertising time (limited number of time slots).
Telecommunications bandwidth.
Size of the Dyson business program.
Doctor’s availability for appointments.

12. The Park Hyatt Philadelphia
118 King/Queen rooms.
Hyatt offers a rL= $159 (low fare) discount fare targeting leisure travelers.
Regular fare is rH= $225 (high fare) targeting business travelers.
Demand for low fare rooms is abundant.
Let D be uncertain demand for high fare rooms.
Assume most of the high fare (business) demand occurs only within a few days of the actual stay.
Objective: Maximize expected revenues by controlling the number of low fare rooms sold.

13. Yield management decisions
The booking limit is the number of rooms to sell in a fare class or lower.
The protection level is the number of rooms you reserve for a fare class or higher.
Let Q be the protection level for the high fare class. Q is in effect while selling low fare tickets.
Since there are only two fare classes, the booking limit on the low fare class is 118 – Q:
You will sell no more than 118-Q low fare tickets because you are protecting (or reserving) Q seats for high fare passengers.
118
Sell no more than the low
fare booking limit, Q
Q seats protected for
high fare passengers

14. The connection to the newsvendor
A single decision is made before uncertain demand is realized.
D: Demand for high fare class;
Q: Protection level for high fare class
There is an overage cost:
If D < Q then you protected too many rooms (you over protected) ...
… so some rooms are empty which could have been sold to a low fare traveler.
There is an underage cost:
If D > Q then you protected too few rooms (you under protected) …
… so some rooms could have been sold at the high fare instead of the low fare.
Choose Q to balance the overage and underage costs.

15. “Too much” and “too little” costs
As Q increases => Overage costs increase
As Q increases => Underage costs decrease
Overage cost:
If D < Q we protected too many rooms and earn nothing on Q - D rooms.
We could have sold those empty rooms at the low fare, so Co = rL.
Underage cost:
If D > Q we protected too few rooms.
D – Q rooms could have been sold at the high fare but were sold instead at the low fare, so Cu = rH – rL

16. Balancing the risk and benefit of ordering a unit
As Q increases by one more unit, the chance of overage increases
Expected loss on the Qth unit = Co x F(Q), where F(Q) = Prob{Demand <= Q)
Essentially: overage costs multiplied by probability of overage costs happening
The benefit of ordering one more unit is the reduction in the chance of underage:
Expected benefit on the Qth unit = Cu x (1-F(Q))
Essentially: underage costs multiplied by probability of underage costs happening
As more units are ordered,
the expected benefit from ordering one unit decreases
while the expected loss of ordering one more unit increases.

17. Graphical Analysis Expected gain or loss . Units
Expected marginal benefit
of understocking
Expected marginal loss
of overstocking

18. Expected profit maximizing order quantity
To minimize the expected total cost of underage and overage, order Q units so that the expected marginal cost with the Qth unit equals the expected marginal benefit with the Qth unit:
Rearrange terms in the above equation ->
The ratio Cu / (Co + Cu) is called the critical ratio.
Hence, to minimize the expected total cost of underage and overage, choose Q such that we don’t have lost sales (i.e., demand is Q or lower) with a probability that equals the critical ratio

19. Optimal protection level
Optimal high fare protection level:
Optimal low fare booking limit = 118 – Q*
Choosing the optimal high fare protection level is a Newsvendor problem with properly chosen underage and overage costs.
Recall: Co = rL; Cu = rH – rL

20. Hyatt example Critical ratio:
Demand for high fare is uncertain, but has a normal distribution with a mean of 30 and Standard deviation of 10.
See the Excel File Posted on the course website for calculations.
You can use normdist(Q,mean,st.dev, 1)=0.29 Excel function to solve for Q (see column E).
Answer: 25 rooms should be protected for high fare travelers. Similarly, a booking limit of = 93 rooms should be applied to low fare reservations.

21. Revenue Management: Overbooking

22. Hold the reservation!
related

23. Ugly reality: cancellations and noshows
Approximately 50% of reservations get cancelled at some point in time.
In many cases (car rentals, hotels, full fare airline passengers) there is no penalty for cancellations.
Problem:
the company may fail to fill the seat (room, car) if the passenger cancels at the very last minute or does not show up.
Solution:
sell more seats (rooms, cars) than capacity.
Danger:
some customers may have to be denied a seat even though they have a confirmed reservation.
Passengers who get bumped off overbooked domestic flights to receive a compensation

24. Hyatt’s Problem
The forecast for the number of customers that do not show up
( X ) is Normal distribution with mean 9 and Standard Deviation 3.
The cost of denying a room to the customer with a confirmed reservation is $350 in ill-will (loss of goodwill) and penalties.
How many rooms (y) should be overbooked (sold in excess of capacity)?
setup:
Single decision when the number of no-shows in uncertain.
Insufficient overbooking: Overbooking demand=X>y=Overbooked capacity.
Excessive overbooking: Overbooking demand=X <y=Overbooked capacity.

25. Overbooking solution Underage cost when insufficient overbooking
if X >y then we could have sold X-y more rooms…
… to be conservative, we could have sold those rooms at the low fare, Cu = rL.
Overage cost when excessive overbooking
if X <y then we bumped y-X customers …
… and incur an overage cost Co = $350 on each bumped customer.
Optimal overbooking level:
Critical ratio:

26. Optimal overbooking level
Normal Distribution
Mean=9
Standard Dev. 3
Optimal number of overbooked rooms is y=7.
Hyatt should allow up to reservations.
There is about F(7)=25.24% chance that Hyatt will find itself turning down travelers with reservations.