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IEOR 4601: Dynamic Pricing and Revenue Management Midterm 2011

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 Consider the two fare problem with capacity c, fares p2 < p1 and independent demand D2 and D1 with fare 2 demand arriving first.

a) Suppose you have perfect foresight and can anticipate the value of D1. How many units would you protect for fare 1?

Answer: y = min(c,D1)

b) Give an expression for the optimal revenue revenue for the two fare problem assuming perfect foresight.

Answer: p1 min(D1,c) + p2 min(D2, (c−D1)+) c) Give an expression for the optimal revenue for the n-fare perfect foresight model.

Answer: ∑n

k=1 pk min(Dk, (c−D[1,k − 1]) +)

d) How would you modify the formula for the case n = 2 if there is known demand D0 at salvage value s < p2?

Answer: p1 min(D1,c) + p2 min(D2, (c−D1)+) + s min(D0, (c−D1 −D2)+)

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) Argue that fare j should be closed at state (t,x) whenever

∆V (t− 1,x) > pj.

Used this to show that the optimal set of fares that should be open at state (t,x) is Sa(t,x) = {1, . . . ,a(t,x)} where a(t,x) = max{j : pj ≥ ∆V (t− 1,x)}. Answer: If a fare with ∆V (t−1,x) > pj is accepted it will contributes negatively towards the expected revenue. The only fares that contribute positievly are those in Sa(t,x).

b) What happens to the set of open fares Sa(t,x) if x increases? If t increases?

Answer: As x increases a(t,x) increases so more fare should be open. If t increases then a(t,x) decreases so fewer fares should be open.

c) Let Vn(c) be the value function for the independent demand model that suppresses time and assumes that the fares arrive from low-to-high. Consider now a dynamic model with arrival rates λtj = E[Dj]/T, 0 ≤ t ≤ T for all j = 1, . . . ,n. Argue intuitively why the value function V (T,c) for this model should be at least as large as Vn(c).

Answer: The arrival rates are now uniform so we have an opportunity to sell higher fares from the outset. This should intuitvely improve revenues.

d) Suppose you are an airline that leases planes on a day to day basis. For a certain route you are offered the choice of leasing a plane with capacity c = 100 or a plan where half of the time you operate an aircraft with capacity 90 and half of the time one with capacity 110. If the two plans are equally costly to lease and operate, which plan would you prefer?

Since V (T,c) is concave in c it is better to have a plane with capacity c = 100.

3. Efficient Sets Finding efficient fares for a general choice model requires solving parametric linear programs with exponentially many variables, one for each of the 2n subsets of {1, . . . ,n}. A heuristic is given below. The heuristic assumes that for any subset S ⊂{1, . . . ,n} we know π(S) =

∑ k∈S πk(S) and r(S) =

∑ k∈S pkπk(S), with π(∅) = r(∅) = 0.

Algorithm

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– Step 1. Set S0 = ∅ and find the set of size k = 1 that maximizes

r(S) −r(∅) π(S) −π(∅)

.

Label this set S1 and set k = 2.

– Step 2. While k ≤ n do: find the set of size k that maximizes

r(S) −r(Sk−1) π(S) −π(Sk−1)

,

label the set Sk and set k = k + 1.

S π(S) r(S) ∅ 0% $0 {1} 10.0% $10.00 {2} 15.0% $12.00 {3} 25.0% $15.00 {12} 20.0% $17.50 {13} 30.0% $21.00 {23} 35.00% $23.50 {123} 42.50% $31.00

Table 1: Data for Problem 3

a) Apply the heuristic to the data set in Table 1 and find the sets S1,S2,S3.

Answer: The procedure selects first S1 = {1}, then S2 = {1, 2} and finally S3 = {1, 2, 3}. b) Suppose someone claims that set {1, 3} is efficient. Prove them wrong by finding a convex

combination of sets S2 and S3 (obtained by the algorithm) that consumes capacity with probability 30% but produces revenue higher than $21.00.

Answer: Given weight 5/9 to S2 and weigth 4/9 to S3 results in revenue 23.5 and con- sumption 30%, so this combination is better than offering {1, 3}.

c) What convex combination of S1,S2,S3 would you use to maximize revenues if you can consume capacity with probability 15%? I would put weights .5, .5 and 0. I would then consume capacity with probabiltiy .15 and have 13.75 in revenues.

4. Static Model Dependent Demands Suppose there are two fares p1 > p2 for a flight with capacity c = 100 and that demands are derived from a basic attraction model (BAM) where fare 1 has attractiveness v1 = 1, fare 2 has attractiveness v2 = 1 and the outside alternative has attractiveness v0 = 1.

a) Suppose that the number of potential customers is Poisson with parameter λ = 200. What is the distribution of demand under action S1 = {1}? Under action S2 = {1, 2}? Answer: Demand under action 1 is Poisson with rate λv1/(v0 +v1) = 100. Demand under action 2 is Poisson with rate λ(v1 + v2)/(v0 + v1 + v2) = 400/3.

b) Suppose you offer action S2. What is the probability that an arriving customer will buy fare 1? fare 2?

Answer: The probability that an arriving customer will buy fare 1 is v1/v[0, 2] = 1/3, the probability that he will buy fare 2 is v2/v[0, 2] = 1/3.

c) What is the average revenue per arriving customer under action S2? What is the average revenue per sale under action S2?

Answer: The average revenue per arriving customer is r2 = (p1 + p2)/3. The average revenue per sale is q2 = (p1 + p2)/2.

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d) Suppose we protect y units for action S1. Write an expression for the expected revenue under action S2 before it is closed?

Answer: Expected revenues are q2E min(c−y,S2). e) What is the distribution of demand under action S1 once action 2 is closed? Give this in

terms of D2, c and y.

Answer: Demand is conditionally binomial with parameters (D2+y−c)+ and θ = π1/π2 = 3/4.

f) What is the expected revenue under action S1? Answer:

p1E min(Bin(D2 + y − c)+,θ), max(y,c−D2)).

5. Dynamic Choice Models 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) Write down the formula for the value function V (t,x) assuming you do not know which sets are efficient.

Answer: V (t,x) = V (t− 1,x) + max

S⊂N [r(S) −π(S)∆V (t− 1,x)].

b) At state (t,x), what problem do you need to solve to figure out which set to offer?

Answer: We need to find the set S ⊂ N that maximizes r(S) − ∆V (t− 1,x)π(S). c) Suppose the efficient fares are S0,S1, . . . ,Sn and that you are given a graph of (πj,rj),j =

1, . . . ,n where πj = π(Sj) and rj = r(Sj). How would you visually obtain the optimal set to offer at state (t,x)?

Answer: I would draw the line α∆V (t−1,x) and then look at the different points πj and look at which point the difference between rj and ∆V (t− 1,x)πj is the largest.

d) Continue with the logic of part d) to argue why you may need to open fares as t decreases for fixed x and close fares as x decreases for fixed t.

Answer: If t decreases without a sale then the marginal value goes down and that would shift the optimal j to the right resulting in more open fares. If x decreases through sales then the marginal value goes up and that shifts the optimal to the left resulting in fewer open fares.

6. Deterministic Static Pricing with Two Market Segments

Suppose there are two market segments (leisure and business) and that demand from the business segment is D1 = 60 at p1 = $1, 000. Suppose further that the demand from the leisure segment is deterministic and price sensitive and given by D2(p) = 600 −p for values of p ∈{0, 1, . . . , 600}. Assume capacity is c = 100.

a) Suppose that demand from the leisure segment arrives before demand from the business segment. How many units will you protect for the business sector? How would you set the discounted fare? How much money would you make under this scheme?

Answer: I would protect y = 60 units of capacity for the business fare and would price the leisure segment at p = 560. At that price I would sell the 40 units of capacity at the leisure segment making $22,400 from that segment plus $60,000 from the business segment for a total of $82,400.

b) Reconsider the revenue calculations of part a) under the assumption that demand from both segments arrive at the same time with both segments competing for the capacity allocated at the discounted fare. Assume that each customer willing to buy the discounted fare is equally likely to get it. Business customers unable to obtain capacity at the

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discounted fare will then buy capacity at p1. Hint: Remember to apply the booking limit on sales at the discounted fare. Is there a better booking limit?

Answer: If both demands arrive at the same time we would see 60 customers from the business segment and 40 from the leisure segment willing to purchase capacity at the discounted fare. Since only 40 units are available at this fare, there will be 16 expected sales to leisure and 24 to business customers. That will leave us with 36 customers buying at 1000 so the total revenue is $58,400. If the booking limit is zero then 60 tickets are sold at $1,000 improving the revenues to $60,000.

c) Suppose now that we set the discounted fare at p = 450 under the assumptions of part b). How many leisure customers will be willing to buy at this fare? How many business customers would be willing to buy at this fare?

Answer: At p = 450, there will be 150 leisure customers and 60 business customers willing to buy at this fare. Thus a low fare ticket will go to the leisure segment with probability 15/21 = 5/7 and to the business segment with probability 60/210 = 2/7.

d) Suppose that each customer willing to buy at p = 450 is equally likely to get it and that capacity is non transferable. How much capacity will you make available for the discounted fare to maximize revenues assuming that business customer unable to secure capacity at $450 will buy at $1,000? How much money would you make?

Answer: I would make 56 units of capacity available at $450. Then 40 units will go to the leisure segment and 16 to the business segment. This would leave me with 44 units of capacity and 44 business customers to which I can sell at fare p1 = $1000. The total results in revenue $69,200

e) What happens to the solution of part d) if capacity is transferable and there is a secondary market where leisure customers who secured capacity at $450 can sell it to business customers for $800?

Answer: In this case all 56 units sold at $450 go to the business segment, leaving us with only 4 customers willing to buy capacity at p1 = $1, 000. The revenues then drop to $29,200.

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