Question 3,4,5

IEOR 4601 Assignment 5: Due March 11

1. Problem 1 from Chapter 3, page 70.

2. Problem 2 from Chapter 3, page 70.

3. Consider a two fare problem with dependent demands governed by a BAM with param- eters v0 = 1,v1 = 1.1,v2 = 1.2. Suppose that the fares are p1 = 1, 000 and p2 = 720 and that the total number of potential customers, say D, is Poisson with parameter Λ = 50.

a) Determine the sale rate πi and the revenue rate ri per arriving customer under action i = 1, 2, where E1 = {1} and E2 = {1, 2}.

b) For capacity values c ∈{10, 11, . . . , 30} solve the linear problem

ΛR(c/Λ) = max Λ[r1t1 + r2t2]

subject to Λ[π1t1 + π2t2] ≤ c t1 + t2 + t0 = 1

ti ≥ 0, i = 0, 1, 2,

and determine the number of units Λπiti sold under action i = 1, 2.

c) From your answer to part b), determine the optimal number of units sold for each fare ij = 1, 2 for each value of c ∈{10, . . . , 30}. What happens to optimal number of sales for each fare j = 1, 2 as c increases?

d) Find the largest integer, say yp, such that P(D1 ≥ y) > r where D1 is Poisson with parameter Λ1 = Λπ1, r = u2/q1, u2 = (r2 − r1)/(π2 −π1) and q1 = r1/π1 = p1.

e) For each c ∈{10, 11, . . . , 30}, check if c < yp + Λ(π2 −π1) and if so, let

yh(c) = max

{ y ∈N : y ≤

yp −β(c + 1) 1 −β

} ∧ c,

and set yh(c) = 0 otherwise.

f) For each c ∈{10, 11, . . . , 30}, use simulation to compute the expected revenue using protection level yh(c) for action 1 against action 2. Compare the expected revenues to the upper bound ΛR(c/Λ). For what value of c do you find the largest gap?

4. Suppose d(p) = λH(p) where H(p) = exp(−p/θ). Argue that p(z) = z + θ maximizes r(p,z) = (p−z)d(p) . Find r(z) = r(p(z),z) and verify that r(z) is decreasing convex in z. Suppose that capacity is c < λ. Find the market clearing price pc such that d(p) = c. What price would you select to maximizes profits with finite capacity c if pc < p(z)? If pc > p(z)? Why?

5. Finite Price Menu for Linear Demands. Suppose that the demand function is of the form d(p) = a− bp for some constants a > 0 and b > 0.

a) Find p(z) the maximizer of r(p,z) = (p−z)d(p) for z ≥ 0.

1

b) Find r(z) = r(p(z),z) for all z ≥ 0. c) Find a maximizer of r(p,z) = (p − z)d(p) if d(p) = d1(p) + d2(p) where a1 =

110,a2 = 140,b1 = 1,b2 = 2.

2