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IEOR 4601: Dynamic Pricing and Revenue Management Midterm 2013

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. You are allowed one page of notes on both sides.

3. 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

Consider a three fare problem with fares p1 = $500,p2 = $450, and p3 = $400 and Poisson demands with intensities λ1 = 3, λ2 = 3, and λ3 = 10. Suppose you are given the following partial table of marginal values ∆Vj(x) = Vj(x) −Vj(x− 1),j = 1, 2, 3 for x = 1, . . . , 5.

x ∆V1(x) ∆V2(x) ∆V3(x) 1 $475.10 2 $400.45 $447.55 3 $434.55 4 $401.15 5 $344.00 $400.00

Table 1: Data for Problem 2

a) What are the protection levels y1 and y2?

b) Compute V1(2)

c) Compute V2(5)

d) Compute V3(5).

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2. Dynamic Allocation Model: Independent Demands

Consider the independent demand, dynamic, capacity allocation model

V (t,x) = V (t− 1,x) + n∑

j=1

λjt[pj − ∆V (t− 1,x)]+, (1)

and assume that p1 > p2 > ... > pn. Let λt = ∑n

j=1 λjt be the overall arrival rate. Let

πjt = ∑j

k=1 λkt/λt and rjt = ∑j

k=1 pkλkt/λt for all j = 1, . . . ,n and define π0t = r0t = 0.

a) Interpret πjt and rjt in terms of the offer set Sj = {1, . . . ,j} for j = 1, . . . ,n.

b) Let qjt = rjt/πjt if πjt 6= 0 and let qjt = 0 otherwise. Interpret qjt when positive.

c) Let M = {0, 1, . . . ,n}. Argue that (1) can be written as

V (t,x) = V (t− 1,x) + max j∈M

λtπjt[qjt − ∆V (t− 1,x)].

d) Give a pricing interpretation to this model.

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3. Discrete Choice Models and Efficient Sets

Given a choice model, let (π(S),r(S)), be respectively, the sales rate and the revenue rate associated with offering set S ⊂ N = {1, . . . ,n}. Recall that π(S) =

∑ k∈S πk(S) and r(S) =∑

k∈S pkπk(S).

For any ρ ≥ 0 consider the linear program

R(ρ) = max ∑ S⊂N

r(S)t(S) (2)

subject to ∑ S⊂N

π(S)t(S) ≤ ρ

∑ S⊂N

t(S) = 1

t(S) ≥ 0 ∀S ⊂ N.

a) Is R(ρ) strictly increasing in ρ?

b) Is R(ρ) convex in ρ?

c) What can you say about R(π(S)) vs r(S) if S is an efficient set?

d) Consider a time homogenous dependent demand model with arrival rate λ and let ΛT = λT. Let c be the initial capacity. What can you say about ΛtR(c/ΛT )?

e) Name a class of models for which the efficient sets are nested.

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4. Dynamic Allocation Model: Dependent Demands Consider the discrete time, dynamic programming, model for dependent demands,

V (t,x) = V (t− 1,x) + λt max S⊂N

[r(S) − ∆V (t− 1,x)π(S)]. (3)

Let (πj,rj) be, respectively, the sales rate and revenue rate per arriving customer associated with efficient set Ej,j ∈ M = {0, 1, . . . ,m} and assume that the efficient sets have been ordered so that πj is increasing in j ∈ M.

a) How does problem (3) simplify if you know (πj,rj), j ∈ M?

b) Let a(t,x) = arg maxj∈M [rj − ∆V (t − 1,x)πj]. Which set of fares should be offered at state (t,x)?

c) Is a(t,x) increasing or decreasing in t?

d) Suppose that the efficient sets are nested. Explain how the dynamic program can reopen a fare once it is closed.

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5. Static Model: Dependent Demands Consider a two fare problem with dependent de- mands, and assume that the efficient fares are E0 = ∅,E1 = {1} and E2 = {1, 2}. Let πi and ri denote, respectively, the sales rate and the revenue rate per arriving customer under action i = 0, 1, 2, where action i corresponds to set Ei, i = 0, 1, 2. Let Di be Poisson with parameter Λπi, i = 0, 1, 2. Consider two artificial products with demands di = Di − Di−1 for i = 1, 2, and respective fares p̂1 = (r1 − r0)/(π1 − π0) and p̂2 = (r2 − r1)/(π2 − π1). It is possible to show that d1 and d2 are independent random variables. In industry, people have proposed to use Littlewood’s rule with the data corresponding to these artificial products.

a) What is the interpretation of di, i = 1, 2.

b) Consider the protection level for fare p̂1 against fare p̂2 given by Littlewood’s rule

ŷ1 = max{y ∈N : P(d1 ≥ y) > p̂2/p̂1}.

What do you think would go wrong if you protect ŷ1 units for sale for artificial product 1 (set E1) when capacity is large?

c) How much capacity would you reserve for set E1 if capacity is large?

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