Are you ready for the job?
Tasks:
Q1. First assume fleet size
Q
= 1500
. Consider a
static
pricing strategy:
p
1
≡
p
1
1
=
p
1
2
=
p
1
3
=
p
1
4
= $40
, i.e., set the same price
$40
for four weeks. Simulate the operation of Tampa branch for
n
= 10
3
(sample size) months (sample path with 4 observations). Compute the expected fill rate
E
[
f
]
, total expect monthly profit
V
Q
(
p
1
)
, and its 95% confidence interval.
Q2. Assume
Q
= 1500
. Repeat question 1 for
p
1
∈ {
45
,
50
, . . . ,
70
}
. Graph
V
Q
(
p
1
)
against
p
1
and
find the optimal price
p
1
∗
that maximizes the total profit
V
∗
Q
= max
p
1
V
Q
(
p
1
)
.
Q3. Now change
Q
= 2000
, repeat questions 1 and 2. Find the optimal price
p
1
∗
and total profit
V
∗
Q
for fleet size
Q
= 2000
.
Q4.
Repeat questions 1 and 2, for
Q
=
{
2500
,
3000
, . . . ,
4000
}
.
For each
Q
, find the optimal
associated price
p
1
∗
and the optimal total profit
V
∗
Q
. Graph
V
∗
Q
against
Q
. Find the optimal fleet
size
Q
∗
.
Bonus (optional):
Now consider a
dynamic
pricing strategy. Suppose you may set the price
p
1
t
for each week differently, where
p
1
t
∈ {
40
,
45
, . . . ,
70
}
. By the similar approach in questions 1
3, find the optimal
price path
(
p
1
∗
1
, p
1
∗
2
, p
1
∗
3
, p
1
∗
4
)
for each fleet size
Q
∈ {
1500
,
2000
, . . . ,
4000
}
and
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its associated optimal profit
V
∗
Q
. Graph
V
∗
Q
against
Q
. Find the optimal
Q
∗
. Compare the total
profit under the dynamic pricing strategy with that under the static pricing strategy. Which one
performs better? By how much? Why?
7.3
Revenue Management
An airline offers two fare classes for coach seats on a particular flight: fullfare class at $440/ticket
and economy class at $218/ticket. There are
x
= 230
coach seats on the aircraft. Demand for
fullfare seats has a mean of 43, a standard deviation of 8, and the following empirical distribution
(it is not normal distribution, so we need to use inverse transform method to generate random
demand).
Economyclass customers must buy their tickets three weeks in advance, and these
tickets are expected to sell out.
For given capacity
x
, let
y
∗
(
x
)
be the protection level,
b
(
x
) =
x

y
∗
(
x
)
the booking limit for lowfare seats,
V
(
x
)
the maximal expected profit, and
p
(
x
) =
V
(
x
)

V
(
x

1)
the marginal value of
x
th
seat. Assume sample size
N
= 10
3
.
a)
Rationing:
For capacity
x
= 230
, find
V
(230)
,
y
∗
(230)
, and
b
(230)
via the following two meth
ods: analytical method (marginal value argument), and brute force simulation defined below.
6
b) For capacity
x
= 100
, find the profit
V
(100)
and protection level
y
∗
(100)
. Do
V
(
x
)
and
y
∗
(
x
)
change? Why or why not?
c)
Dynamic pricing:
Now compute
V
(
x
)
and
p
(
x
)
for each
x
= 1
,
2
, . . . ,
230
. Plot
V
(
x
)
and
p
(
x
)
against capacity
x
. Base one these results, if you were the manager, what price should you charge
for each seat?
Can you explain why the ticket price typically increases when approaching the
departure date?
d) Suppose that unsold seats may sometimes be sold at the last minute at a very reduced rate (sim
ilar to USAirways’ “Esaver” for lastminute travel). What effect will this have on the protection
level calculated in (a)?