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IEOR 4601: Dynamic Pricing and Revenue Management Midterm 2010

Professor Guillermo Gallego

INSTRUCTIONS:

1. Write all your answers on the question sheets. Answers written on blue books will not be evaluated. Show and explain your work. Points will be deducted for solutions that are not clearly explained.

2. Write your name, sign the honor pledge and indicate your degree program.

HONOR PLEDGE: I pledge that I have neither given nor received unauthorized aid during this examination.

Name: Signature: Degree Program :

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1. Static Model Independent Demands For the multi-fare problem with low-to-high arrivals we have shown that the optimal protection levels for fares {j, . . . , 1} is given by

yj = max{y ∈N : ∆Vj(y) > pj+1} j = 1, . . . ,n− 1.

a) What is the definition of ∆Vj(y)? Definition: ∆Vj(y) = Vj(y) −Vj(y − 1) for y ≥ 1. b) What is the interpretation of ∆Vj(y)? Interpretation: Expected marginal seat value.

c) Is ∆Vj(y) monotone increasing in y?

No: It is decreasing in y.

d) Is ∆Vj(y) monotone increasing in j?

Yes.

e) Write an explicit expression for ∆V1(y)?

∆V1(y) = p1P(D1 ≥ y). f) Is this formula true or false?: ∆V2(x) = max{∆V1(x),E min(p1P(D[1, 2] ≥ x),p2)}

Yes

g) For general j what is the approximation for ∆Vj(y) implied by the EMSR-b heuristic?

∆Vj(y) ' qjP(D[1,j] ≥ y) where qj = ∑j k=1 pkE[Dk]/ED[1,j].

2. Dynamic Allocation Model: Independent Demands Consider the dynamic capacity allocation model

V (t,x) = V (t− 1,x) + n∑ j=1

λjt[pj − ∆V (t− 1,x)]+.

a) What is λjt? It is the arrival rate of fare j at time t.

b) What is the interpretation of ∆V (t− 1,x)? It is the marginal value of the xth unit of capacity at time t− 1.

c) Which fares will you accept at state (t,x)?

Accept those fares in the set {j : pj ≥ ∆V (t− 1,x)}. d) What happens to a(t,x) = max{j : pj ≥ ∆V (t− 1,x)} as t increases?

As the time-to-go increases a(t,x) decreases.

e) What happens to a(t,x) as x increases?

As capacity increases a(t,x) increases.

f) List three disadvantages of this model?

Allows fares to open and reopen. Assumes Poisson arrival rates. Assumes independent demands.

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3. Dynamic Model: Independent Demands Monotonic Fares Consider the dynamic ca- pacity allocation model

Vj(t,x) = max k≤j

Wk(t,x)

where Wk(t,x) = ∑k i=1 λit[pi + Vk(t− 1,x− 1)] + (1 −

∑k i=1 λit)Vk(t− 1,x).

a) What is the interpretation of Vj(t,x)?

It is the maximum expected revenue from state (t,x) where only fares {j, . . . , 1} are allowed and when closing a fare forces it to remain closed until the end of the horizon.

b) What is the interpretation of Wk(t,x)?

It is the maximum expected revenue of using action Sk at state (t,x) and then following an optimal policy thereafter.

c) Show that Wk(t,x) = Vk(t− 1,x) + ∑k i=1 λit[pi − ∆Vk(t− 1,x)].

This is simple algebra reorganizing the terms in the definition of Wk(t,x).

d) What happens to fares {k + 1, . . . ,j} if Vj(t,x) = Wk(t,x) for some k < j? All such fares are closed and will remain closed.

e) Suppose Vj(t,x) = Wk(t,x) for k < j. What can you say about Vj−1(t,x)?

Then Vj−1(t,x) = Vk(t,x).

f) How would you modify V1(t,x) to impose a penalty for denying capacity to high fare customers?

I would write V1(t,x) = p1E min(D1(t),x) −ρE(D1(t) −x)+.

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4. Choice Models and Dynamic Model with Dependent Demands Consider a choice model where for each set S ⊂ Sn = {1, . . . ,n}, π(S) =

∑ i∈S πi(S) and r(S) =

∑ i∈S piπi(S).

a) What is the interpretation of π(S)?

It is the probability that the customer selects an item in S rather than the outside alternative.

b) What is the interpretation of r(S)?

It is the expected revenue when S is offered.

c) What is the interpretation of q(S) = r(S)/π(S)?

It is the average fare assuming something is sold.

d) Give formulas for πj,rj and qj and uj = (rj −rj−1)/(πj −πj−1) for S = Sj = {1, . . . ,j} for the MNL model.

πj = π(Sj),rj = r(Sj). For the MNL model uj = pj−rj−1 1−πj−1

.

e) Consider the dynamic program V (t,x) = V (t−1,x) + λt max0≤k≤n[rk −πk∆V (t−1,x)] for the MNL model. For what states (t,x) are you better off offering Sj rather than Sj−1?

For states (t,x) such that ∆V (t,x) ≤ uj.

5. Static Model with Dependent Demands Consider a choice model as in Problem 4 and suppose that there are D potential customers, where D is a random variable taking positive integer values. Suppose you want to allocate capacity between sets Sj+1 and Sj by protecting y units of capacity for sales under set Sj.

a) What is the total demand, say Dj+1, for fares {1, . . . ,j + 1} when set Sj+1 is offered? It is binomial with parameters D and πj+1.

b) What are the total sales under action Sj+1 if c units of capacity are available and y units are protected for sales under set Sj? min(c−y,Dj+1).

c) What is the expected revenue under action Sj+1?

It is qj+1E min(c−y,Dj+1). d) Write an expression for sales under Sj if Dj+1 < c−y, if Dj+1 > c−y.

If Dj+1 < c−y then we run out of customers and sales under Sj will be zero. Otherwise sales under Dj will be min(y,Uj(y)) where Uj(y) is binomial with parameters (Dj+1 − c + y)+, θj = πj/πj+1.

e) Write an expression for the expected revenue under action Sj when Dj+1 > c − y It is qjE min(y,Uj(y)).

f) Write an expression for the expected revenue under both actions Sj and Sj+1.

Just add: qj+1E min(c−y,Dj+1) + qjE min(y,Uj(y)).

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