IEOR 4601 Assignment 5 Due March 11, Problem 1 From C 632472

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 Ï€i 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[Ï€1t1 + Ï€2t2] ≤ c, with t1 + t2 + t0 = 1, and t_i ≥ 0 for i = 0, 1, 2, and determine the number of units Λπi t_i sold under actions i = 1, 2. c) From your answer to part b), determine the optimal number of units sold for each fare j = 1, 2 for each value of c ∈ {10, ..., 30}. What happens to the optimal number of sales for each fare j as c increases? d) Find the largest integer, say y_p, such that P(D1 ≥ y) > r where D1 is Poisson with parameter Λ1 = Λπ1, 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 p(z)? Why? 5. Finite Price Menu for Linear Demands. Suppose 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. 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

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

This paper aims to analyze a complex revenue management scenario involving dependent demand modeling, capacity decision-making, and probabilistic assessment, as well as to evaluate various pricing and risk metrics. The scenario is rooted in the principles of revenue management and stochastic modeling, employing techniques from dynamic programming, probability theory, and operations research to optimize fare sales, assess risks, and determine optimal pricing strategies under capacity constraints.

Introduction

The problem at hand involves a two-fare system with dependent customer demands modeled via a Batch Arrival Model (BAM). The demand parameters are set with v0 = 1, v1 = 1.1, and v2 = 1.2, with fare prices p1 = 1,000 and p2 = 720 respectively. The customer potential, D, follows a Poisson distribution with mean Λ = 50, representing the total number of potential customers. The assignment encompasses multiple parts, beginning with calculating sale and revenue rates, followed by capacity planning through linear programming, demand analysis through probabilistic methods, and finally risk assessment and pricing strategies with simulations.

Part A: Sale Rate and Revenue Rate Calculation

The first step involves calculating the sale probability rate (Ï€i) and the corresponding revenue rate (ri) per customer for each action i under idealized conditions. Specifically, for action 1, where the sale event is limited to fare 1, the sale rate depends on the demand's dependency structure [[(1)]]. Conversely, for action 2, where demand can include both fares, the sale rate considers the joint distribution [[(2)]]. Using the BAM parameters and fare prices, these rates are computed by analyzing the demand's structure and the probability of each fare being accepted under different conditions [[(3)]]. The resulting rates inform revenue expectations per customer and help in capacity decisions.

Part B: Capacity Optimization via Linear Programming

Next, the capacity constraint c within the set {10, 11, ..., 30} is evaluated through a linear programming framework to maximize expected revenue. The problem involves optimizing the allocation of the customer potential Λ across different fare categories, considering the sale probabilities and revenue rates. Constraints ensure that the total sold units do not exceed capacity, and the decision variables T1, T2, and T0 are allocated accordingly [[(4)]]. Solving these linear programs yields the optimal units to be sold for each fare at each capacity level and reveals how sales quantity responds to capacity changes.

Part C: Demand and Sales Behavior as Capacity Increases

Analysis of the solutions illustrates how the optimal number of units sold per fare evolves as the capacity c progresses from 10 to 30. Typically, as capacity increases, the sales of higher-priced fares tend to increase or stabilize, reflecting their higher contribution to revenue, whereas lower fares might see reduced demand proportions [[(5)]]. This behavior indicates strategic capacity management and influences pricing policies for maximizing revenue over different capacity scenarios.

Part D: Probabilistic Risk Measurement

The risk metric involves calculating the largest integer y_p such that the probability P(D1 ≥ y) exceeds a threshold r. Using the Poisson distribution with mean Λ1 = Λπ1, u2, and q1, the analysis adopts probabilistic bounds to determine this cutoff point. The calculations are based on the demand's distribution and provide insight into the magnitude of potential demand shortfalls or surpluses [[(6)]]. This risk measure helps in understanding demand variability and capacity adequacy under uncertain conditions.

Part E: Adjusted Demand Thresholds Based on Capacity

For each capacity c, the model evaluates if c is less than a computed threshold involving y_p and demand variance. If so, the model calculates a safety threshold y_h(c) that guarantees service levels consistent with the probabilistic assessment, otherwise defaulting to zero. This modified demand threshold adapts to capacity levels and probabilistic risk assessments, offering a mechanism for managing demand uncertainty [[(7)]].

Part F: Simulation and Revenue Comparison

Simulation methods are employed to estimate expected revenues under different protection levels y_h(c), comparing these to upper bounds from the deterministic linear model. By iterating over capacities, the maximum gap between simulated revenues and theoretical bounds reveals the capacity points where pricing and capacity strategies are most sensitive to demand variability [[(8)]]. This empirical approach informs more resilient revenue management policies.

Theoretical Modeling of Demand with Exponential Distribution

In the second problem, demand is modeled with a function d(p) = λH(p), where H(p) = exp(-p/θ). The maximization of the revenue function r(p,z) = (p - z)d(p) leads to the determination of p(z) = z + θ as the maximizing price, leveraging calculus and properties of exponential functions [[(9)]]. The resulting analysis includes verifying the nature of r(z) as decreasing and convex, and identifying the market clearing price p_c where demand equals capacity c, as well as discussing profit maximization strategies based on the relative positioning of p_c and p(z) [[(10)]].

Pricing Strategies for Linear Demands

For linear demand functions d(p) = a - bp, the demand maximizer p(z) is derived analytically by setting the derivative of revenue to zero, which results in p(z) = (a + bz)/2. The corresponding revenue at this point, r(z), can be computed directly. For combined demand models, the maximizers are obtained by solving similar optimization problems, considering the sum of two demand functions with known parameters, providing insights into price-setting under composite demand scenarios.

Conclusion

This comprehensive analysis illustrates the integration of probabilistic demand modeling, linear programming for capacity decisions, and pricing optimization to maximize revenue under uncertainty. Probabilistic risk assessment and simulation approaches contribute to resilient decision-making, emphasizing the importance of understanding demand variability and capacity constraints in revenue management. These methods collectively support strategic planning, price setting, and capacity allocation in dynamic and uncertain environments.

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