Question 1,2,3,4

IEOR 4601 Homework 3: Due Wednesday, February 20

1. Find optimal protection levels for the following data and compute the optimal expected revenues V1(c),V2(c),V3(c) and V4(c) for c ∈{50, 55, 60, 65, 70, 75, 80} assuming Poisson demands.

j pj E[Dj] 1 $75 8 2 $100 21 3 $75 31 4 $60 20

2. Modify Problem 1 so that p1 = $125 and compute optimal protection levels and the value function V4(c) for c ∈{50, 55, 60, 65, 70, 75, 80}.

3. (Upper and Lower Bounds) Compute the upper bound V H (c) and V (c,µ) and the lower bound V L(c) and the spread V H (c)−V L(c) for Problem 2 for c ∈{50, 55, 60, 65, 70, 75, 80}.

4. (Time Varying Models) Use the discrete time dynamic programs to compute V (T,c) and Vj(T,c),j = 1, 2, 3, 4 for the data of Problem 2 for c ∈{50, 55, 60, 65, 70, 75, 80} for the following arrival rate models:

a) Uniform arrival rates, e.g, λtj = Λj = E[Dj] for 0 ≤ t ≤ T = 1. Be sure to rescale time so that T = a is an integer large enough so that

∑3 j=1 E[Dj]/a ≤ 0.01. What

accounts for the difference between V (T,c) and V4(T,c)? What accounts for the difference between V4(T,c) and V4(c) from Problem 2?

b) Low-to-high arrival rates: Dividing the selling horizon [0,T] = [0, 1] into 4 sub- intervals [tj−1, tj],j = 1, . . . , 4 with tj = j/4, and set λjt = 4Λj over t ∈ [tj−1, tj] and λjt = 0 otherwise. Again, be sure to rescale the system so that T = a is an integer large enough so that maxj maxt λjt/a ≤ 0.01. What accounts for the difference between V (T,c) and V4(T,c)? What accounts for the difference between V4(T,c) and V4(c) from Problem 2?

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