Ieor 4601 Assignment 5 Due March 11 Problem 1 From Chapter

Ieor 4601 Assignment 5 Due March 111 Problem 1 From Chapter 3 Page

Consider a two fare problem with dependent demands governed by a BAM with parameters 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, 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 Λ R(c/Λ) = max Λ [r1 t1 + r2 t2] subject to Λ [Ï€1 t1 + Ï€2 t2] ≤ c, with t1 + t2 + t0 = 1, t_i ≥ 0 for i = 0, 1, 2, and determine the number of units Λ Ï€ i t_i sold under action i = 1, 2.

c) From your answer to part b), determine the optimal number of units sold for each fare i = 1, 2 for each value of c ∈ {10, ..., 30}. What happens to the 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 = Λ Ï€1, r = u2 / q1, u2 = (r2 − r1) / (Ï€2 − Ï€1), and q1 = r1 / Ï€1 = p1.

e) For each c ∈ {10, 11, ..., 30}, check if c

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 Λ R(c/Λ). For what value of c do you find the largest gap?

Suppose d(p) = λ H(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 p(z)? Why?

Finite Price Menu for Linear Demands. Suppose that the demand function is of the form d(p) = a − b p 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.

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.

Paper For Above instruction

The problem set revolves around analyzing a two-fare dynamic pricing model where customer demand dependencies are modeled using the Bayesian Arrival Model (BAM). This model incorporates demand parameters v0, v1, and v2 to reflect the elasticity and substitution effects between fare levels p1 and p2, with the goal of maximizing revenue while managing capacity constraints systematically.

Demand Parameters and Sale Rates

Given the parameters v0=1, v1=1.1, v2=1.2, and fares p1 = 1,000, p2 = 720, with a Poisson customer potential D with mean Λ=50, the first step involves calculating the sale probabilities (πi) and revenue rates (ri) per customer under each fare action. Using demand functions rooted in the BAM, the sale probability πi can be derived based on the likelihood a customer accepts fare i. Typically, this involves examining the demand elasticity coefficients with respect to the fare levels. The revenue rate ri is obtained by multiplying the fare p_i with the sale probability πi, providing the expected revenue per arriving customer under each action.

Capacity-Constrained Revenue Optimization

For capacity c ∈ {10, 11, ..., 30}, the linear programming problem becomes central in determining optimal policies. The goal is to maximize the function Λ R(c/Λ), where R(·) is the revenue per unit demand scaled by capacity utilization. The constraints ensure that the total sale volume, derived from sale probabilities influenced by demand dependencies, does not exceed the capacity c. The decision variables t1, t2 indicate the portion of total potential demand allocated to each fare, under the condition that their sum and weighted sum satisfy capacity limits. The optimal number of units sold, Λ Ï€i t_i, can then be calculated for each fare class based on these solutions, revealing how the capacity affects sales volume distributions.

Demand Dynamics and Threshold Prices

Part d) involves calculating the largest integer yp such that the probability P(D1 ≥ y) exceeds a threshold r, where D1 is a Poisson distribution with mean Λ1 = Λ Ï€1. This threshold essentially indicates the demand level that can be reliably met with high probability, influencing capacity planning. The variables u2 and q1 relate to fare elasticity and search for an optimal price point that maximizes expected profit given capacity constraints, incorporating demand response functions.

Protection Policy and Simulation

Part e) proposes defining a protection level yh(c), adjusting the demand capacity dynamically based on the comparison between capacity c and the threshold yp plus expected demand. This ensures a conservative approach to guarantee a certain level of service while balancing revenue. Subsequently, Monte Carlo simulations provide estimates of expected revenue at different capacity levels, enabling comparison against the theoretical upper bounds. The largest discrepancy between simulation and upper bound signals the capacity levels where the model's assumptions are most sensitive.

Demand Function and Price Optimization

In later sections, the demand function is modeled as d(p) = λ H(p), with H(p) = exp(−p/θ), presenting an exponential demand elasticity framework. Recognizing that p(z) = z + θ maximizes the revenue function r(p,z), the analysis moves onto deriving the convexity properties of r(z) and identifying the optimal market-clearing price p_c. This price balances the capacity constraint c against demand, maximizing profit when p_c

Linear Demand Market and Price Setting

The demand function d(p) = a − bp introduces a linear demand model, where straightforward optimization yields the price p(z) that maximizes revenue for given z. Extending this to combined demand components (d1 + d2), with parameters a1, a2, b1, b2, the derivations produce optimal prices and expected revenue levels for each demand segment, informing revenue management strategies for segmented markets.

Conclusion

Overall, these problems exemplify the integration of demand modeling, capacity constraints, probabilistic demand thresholds, and simulation to devise optimal revenue management policies. They demonstrate how advanced mathematical tools like linear programming, Poisson demand estimates, and convex analysis play crucial roles in strategic decision-making for dynamic pricing and capacity planning in revenue management.

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