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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)?

b) What is the interpretation of ∆Vj(y)?

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

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

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

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

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

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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?

b) What is the interpretation of ∆V (t− 1,x)?

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

d) What happens to a(t,x) = max{j : pj ≥ ∆V (t− 1,x)} as t increases?

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

f) List three disadvantages of this model?

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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)?

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

c) Show that Wk(t,x) = Vk(t− 1,x) + ∑k

i=1 λit[pi − ∆Vk(t− 1,x)].

d) What happens to fares {k + 1, . . . ,j} if Vj(t,x) = Wk(t,x) for some k < j?

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

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

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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)?

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

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

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.

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?

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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?

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?

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

d) Write an expression for sales under Sj if Dj+1 < c−y, if Dj+1 > c−y.

e) Write an expression for the expected revenue under action Sj when Dj+1 > c−y

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

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