Granite State Airlines serves the route between New York and Portsmouth, NH, wit
ID: 3321207 • Letter: G
Question
Granite State Airlines serves the route between New York and Portsmouth, NH, with a single-flight-daily 100-seat aircraft. The one-way fare for discount tickets is $100, and the one-way fare for full-fare tickets is $150. Discount tickets can be booked up until one week in advance, and all discount passengers book before all full-fare passengers. Over a long history of observation, the airline estimates that full-fare demand is normally distributed, with a mean of 56 passengers and a standard deviation of 23, while discount-fare demand is normally distributed, with a mean of 88 passengers and a standard deviation of 44. a) A consultant tells the airline they can maximize expected revenue by optimizing the booking limit. What is the optimal booking limit? (Hint: Use the standard normal cumulative distribution table) b) The airline has been setting a booking limit of 44 on discount demand, to preserve 56 seats for full-fare demand. What is their expected revenue per flight under this policy? (Hint: First find the expected revenue when b= 0. Here you can assume Probability{df = k} = Ff(k+0.5) – Ff(k-0.5) and use a spreadsheet. Then using the recursive formula, find the expected revenue if b is increased by 1 until it reaches b=44 using a spreadsheet) c) What is the expected gain from the optimal booking limit over the original booking limit?7 d) A low-fare competitor enters the market and Granite State Airlines sees its discount demand drop to 44 passengers per flight, with a standard-deviation of 30. Full-fare demand is unchanged. What is the new optimal booking limit?
Explanation / Answer
Full fare, Pf = 150 Discount fare, Pf = 100 Optimal protection level, F(y) = 1 - Pd/Pf = 1-100/150 = 0.333 (as per Littlewood's rule) z value corresponding to P(z) = 0.3333 = -0.4308
The standard normal distribution is a special case of the normal distribution. It is the distribution that occurs when a normal random variable has a mean of zero and a standard deviation of one.
Standard Score (aka, z Score)
The normal random variable of a standard normal distribution is called a standard score or a z-score. Every normal random variable X can be transformed into a z score via the following equation:
z = (X - ) /
where X is a normal random variable, is the mean of X, and is the standard deviation of X.
for discounted rate
z = (100 - 88) / 44 = 12/44 = 0.272
for normal rate
z = (150 - 56) / 23 = 94/23 = 4.09
0.607>p >0.999
Maximising the standard deviation or variance for a given mean is equivalent to maximising the sum of squares of the values for a given sum of the values.
Meanwhile if ba and >0,
(a)2+(b+)2
=a22a+2+b2+2b+2
=a2+b2+2(ba)+22
>a2+b2
so the sum of squares is maximised if and only if as many terms as possible are as large as possible, even if this makes others smaller.
1. Airlines establish booking limits B1 and B2.
2. Low-fare passengers arrive to their first-choice airlines and are accommodated up to the
booking limits.
3. Low-fare passengers not accommodated on their first-choice airlines 'spill' to the alternate
airlines and are accommodated up to the booking limits.
4. High-fare passengers arrive to their first-choice airlines and are accommodated with any
remaining seats, up to capacity C in each aircraft.
5. High-fare passengers not accommodated on their first-choice airlines 'spill' to the alternate
airlines and are accommodated in any remaining seats, up to capacity C in each aircraft.
To describe the problem in terms of customer demand and booking limits, define:
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