Question 1,2,3,4

Please take a look at the questions first, then decide whether you want to do them or not, the name of this course is pricing and revenue optimization, I need detailed explaination, please upload solutions before deadline.

IEOR 460

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Homework 4: Due Wednesday February

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1. Consider a flight with 3 fares p1 = 1, 100,p2 = 900,p3 = 600, quality attributes q1 =
1000,q2 = 850,q3 = 750, price sensitivity βp = −1 and quality sensitivity βq = 1.25

.

Suppose that the utility of fare i is Ui = µi + �i where µi = βppi + βqqi, i = 1, 2, 3 and
the �is are independent Gumbel random variables with parameter φ = .01.

(a) Compute the expected utilities µi, i = 1, 2, 3.

(b) Compute the attraction values vi = exp(φµi), i = 1, 2, 3.

(c) Assume there is an outside alternative with attractiveness v0 = 3 compute π0(Sj) =
v0/v[0,j] and πk(Sj) = vk/v[0,j],k ∈ Sj = {1, . . . ,j} for j = 1, 2, 3.

(d) What can you say about the choice model when φ is very small, say φ = .001?
What if φ = 1?

(e) List all 8 subsets S ⊂ N = {1, 2, 3}. For φ = .01 and v0 = 3, compute the sales
rate π(S) =

∑
k∈S πk(S) and the revenue rate r(S) =

∑
k∈S pkπk(S).

(d) Let N denote the collection of all eight subsets of N. Solve the LP

RN(ρ) = max
∑
S∈S

r(S)t(S)

subject to
∑
S∈S

π(S)t(S) ≤ ρ
∑
S∈S

t(S) ≤ 1

t(S) ≥ 0 ∀S ∈N .

for all values of ρ ∈ [0, 1]. Here the decision variables are the t(S),S ∈N . You can
interpret RN(ρ) as the maximum revenue rate that can be obtained by a mixed
strategy if the sales expected rate is at most ρ. You can use Excel solver to do
this. First, solve for a small value of ρ and then invoke the sensitivity report to
see the range of values of ρ for which the current basis is optimal to identify the
next threshold where the basis changes. Repeat the procedure until you exhaust
the entire range ρ ∈ [0, 1].

(e) Repeat part (d) by computing RC(ρ) for ρ ∈ [0, 1] where the collection of sets is
now C = {S0,S1,S2,S3} (so only the nested subsets are included).

(f) Is RN(ρ) = RC(ρ) for all ρ ∈ [0, 1]?

2. For the MNL model, let πj =
∑

k∈Sj πk(Sj) and rj =
∑

k∈Sj pkπk(Sj).

(a) Show algebraically that

πj = πj−1
v[0,j − 1]
v[0,j]

+ 1
vj

v[0,j]
, j = 1, . . . ,n

rj = rj−1
v[0,j − 1]
v[0,j]

+ pj
vj

v[0,j]
j = 1, . . . ,n.

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(b) Notice that πj > πj−1 always, but rj ≤ rj−1 whenever pj ≤ rj−1. Thus pj ≤
rj−1 renders set Sj inefficient since it consumes more capacity and produces lower
expected revenues. Show that if Sj is inefficient then so are sets Sj+1, . . . ,Sn.

(c) Consider Problem 1 with fares p1 = 1000,p2 = 500,p3 = 475 and determine which
of the sets S1,S2,S3 are efficient?

3. Consider the dynamic program

V (t,x) = V (t− 1,x) + λt max
j

[rj −πj∆V (t− 1,x)]

with boundary condition V (t, 0) = V (0,x) = 0 for t ≥ 0 and x ∈N . Here rj and πj are
defined as in Problem 2, with r0 = π0 = 0.

Let

uj =
rj − rj−1
πj −πj−1

.

a) Show that uj = (pj − rj−1)/(1 −πj−1) and in particular that u1 = p1.
b) Show that it is optimal to offer set Sa(t,x) at state (t,x) where

a(t,x) = max{j : uj ≥ ∆V (t,x)}.

Hint: You may want to use the following two facts: 1) ∆V (t,x) ≤ p1 and 2) uj is
decreasing in j for the MNL model.

4. Modify the code you used for Homework 3 to solve the dynamic program

V (t,x) = V (t− 1,x) + λt max
j

[rj −πj∆V (t− 1,x)] (1)

with boundary condition V (t, 0) = V (0,x) = 0 for t ≥ 0 and x ∈N for the MNL model
with the data of Problem 1. You can assume that the arrival rates, already rescaled, are
λt = λ = .01, T = 10, 000. Find V (T,c) for c ∈{35, 40, 55, 60, 65, 70, 75, 80, 85, 90}.

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