Question1,2

IEOR 4601 Assignment 7

1. Consider the following dynamic pricing problem with three market segments with time invariant arrival rates dmt(p) = dm(p) = λme

−p/θm,m = 1, 2, 3 and random request Zm,m = 1, 2, 3. Assume that λ1 = 10, λ2 = 5 and λ3 = 15, that θ1 = 500,θ2 = 550,θ3 = 600, and that the distribution of Zm is given by P(Zm = 1) = 0.8, P(Zm = 2) = .1 and P(Zm = 3) = P(Zm = 4) = .05, for m = 1, 2, 3. Finally, assume that T = 10 and that c = 70.

a) Compute z(T,c) by using formula (21) in the chapter on Dynamic Pricing.

b) Use the result of part a) to compute V̄ (T,c), the upper bound on the optimal expected revenue.

c) Simulate the performance of the quasi-static pricing policy Phm(t,x) given in formula (19). In the simulation you should use the distribution of Zm to generate the request sizes, and accept all requests at price Phm(t,x) of size Zm ≤ x.

d) Repeat part c), updating z of the quasi-static pricing policy at t = 5.

2. Consider a single market model with Poisson demands with dt(p) = λe −p/θ with λ = 10,

θ = 500, T = 100 and c = 150. Let tj = 50j for j = 1, 2 be the reading dates.

a) Simulate the performance of the quasi-static pricing policy updating z at each reading date.

b) Solve the dynamic policy in Section 5.2, and compare the performance of the semi- dynamic pricing policy to the results of part a)

c) Repeat parts a) and b) assuming that the reading dates are tj = 10j,j = 1, . . . , 10. Which policy performs better?

1