IEOR 4601 Homework 4 Due Wednesday, February 27

Ieor 4601 Homework 4 Due Wednesday February 271 Consider A Flight Wi

Consider a flight with 3 fares p1 = 1100, p2 = 900, p3 = 600, quality attributes q1 = 1000, q2 = 850, q3 = 750, price sensitivity βp = -1, and quality sensitivity βq = 1.25. The utility of fare i is Ui = μi + εi, where μi = βp p_i + βq q_i, with εi being independent Gumbel random variables with parameter φ = 0.01. The assignment involves calculating expected utilities, attraction values, choice probabilities, and solving related optimization models under varying parameters, including simulations with the Multinomial Logit (MNL) model. Tasks include deriving explicit formulas, exploring the impact of parameter variations, computing subsets and revenues, and solving linear programs to determine optimal strategies, followed by examining dynamic programming approaches—culminating in analysis of revenue maximization under stochastic demand settings.

Paper For Above instruction

The problem set provided offers a comprehensive exploration of discrete choice models, specifically focusing on the application of the Multinomial Logit (MNL) framework to airline fare selection. This scenario involves a strategic analysis of consumer choice behavior based on fare prices, quality attributes, and the implications of varying model parameters on decision-making and revenue optimization. This essay discusses both theoretical and computational facets of these models, emphasizing utility calculation, decision subset evaluation, and optimization strategies under different parameter conditions.

Expected Utilities and Attraction Values

The first step involves calculating the expected utility μi for each fare class, where μi = βp p_i + βq q_i. Given the parameters, βp = -1 and βq = 1.25, along with the constants p_i and q_i, the calculation proceeds as follows:

  • μ1 = (-1) × 1100 + 1.25 × 1000 = -1100 + 1250 = 150
  • μ2 = (-1) × 900 + 1.25 × 850 = -900 + 1062.5 = 162.5
  • μ3 = (-1) × 600 + 1.25 × 750 = -600 + 937.5 = 337.5

Next, the attraction values v_i are computed as v_i = exp(φ μ_i), which serve as the core components in determining choice probabilities. With φ = 0.01, these are:

  • v_1 = exp(0.01 × 150) ≈ exp(1.5) ≈ 4.4817
  • v_2 = exp(0.01 × 162.5) ≈ exp(1.625) ≈ 5.078
  • v_3 = exp(0.01 × 337.5) ≈ exp(3.375) ≈ 29.246

These attraction values set the stage for calculating inclusion probabilities and potential choice outcomes.

Choice Probabilities and External Alternatives

The external or outside alternative is assigned an attractiveness v_0 = 3. Using the formula π_k(S_j) = v_k / v_{0,j} where v_{0,j} = v_0 + Σ_{k∈S_j} v_k, we compute choice probabilities over subsets S_j = {1, ..., j}. Specifically:

  • For j=1, v_{0,1} = 3 + 4.4817 ≈ 7.4817, so π_1 = 4.4817 / 7.4817 ≈ 0.598
  • For j=2, v_{0,2} = 3 + 4.4817 + 5.078 ≈ 12.5597, so π_2 = 5.078 / 12.5597 ≈ 0.404
  • For j=3, v_{0,3} = 3 + 4.4817 + 5.078 + 29.246 ≈ 41.770

Corresponding probabilities for each subset can then be derived to analyze market share and expectations under different models.

Effects of Parameter Variations on Choice Behavior

The sensitivity parameter φ significantly influences the distribution of choice probabilities. When φ tends to very small values, approaching zero (e.g., φ=0.001), the utility differences become less pronounced, resulting in more uniform choice probabilities across options, approaching equal likelihoods. Conversely, with larger φ values (e.g., φ=1), the model becomes more sensitive to utility differences, amplifying the preference for options with higher μ_i, which leads to more deterministic choice behavior favoring the highest utility fare. This sensitivity directly affects revenue predictions and strategic fare setting, as a higher φ accentuates the importance of small utility differences, whereas a lower φ evokes more random choice behavior.

Subset Analysis and Revenue Computations

Listing all 8 subsets of the fare set N = {1, 2, 3} involves combinatorial enumeration of singleton, pairs, and the triplet:

  • {}
  • {1}
  • {2}
  • {3}
  • {1, 2}
  • {1, 3}
  • {2, 3}
  • {1, 2, 3}

For each subset S, calculating the sales rate π(S) = Σ_{k ∈ S} π_k(S) and revenue rate r(S) = Σ_{k ∈ S} p_k π_k(S) involves summing the respective choice probabilities and revenues. For example, for the full set S = {1, 2, 3}:

Using the computed probabilities π_k(S) and prices p_k, the total expected sales and revenues are determined, informing strategic decisions about which set of fares maximizes revenue under the model assumptions.

Optimization via Linear Programming Under Capacity Constraints

The LP formulation maximizes overall revenue rate R_N(ρ), considering the weighted combination of subsets with weights t(S), subject to the constraints Σ_{S ∈ N} π(S) t(S) ≤ ρ and Σ_{S ∈ N} t(S) ≤ 1, with t(S) ≥ 0. Solving this LP for various values of ρ ∈ [0,1] involves iterative computation and sensitivity analysis, typically using tools like Excel Solver. The analysis reveals the optimal mix of fare subsets for different capacity restrictions, enabling airline revenue management strategies that adapt dynamically to demand variability.

Nested Subset Optimization and Comparative Analysis

By constraining the collection to nested subsets C = {S_0, S_1, S_2, S_3}, the LP problem simplifies, and the solution R_C(ρ) can be computed efficiently. Comparing R_N(ρ) and R_C(ρ) assesses whether limiting the mixture to nested subsets impacts the maximum achievable revenue. Evidence often shows that restricting options to nested sets may approximate the full set solution, but potentially at a lower total revenue, highlighting the trade-offs involved in simplification versus optimality.

Behavioral Insights When Parameters Vary

Analyzing the models with different ρ values informs strategic fare and subset selections. For instance, at high ρ values—indicating high capacity—the models tend to favor broader subset options, while at low ρ values, more restrictive choices dominate. Understanding these dynamics is crucial for airlines seeking to optimize revenue under capacity and demand uncertainty, highlighting the influence of model parameters and decision restrictions on overall profitability.

Algebraic Derivations and Efficiency Analysis in the MNL Model

Algebraic expressions for πj and rj reveal the cumulative effects of including additional fare options. The conditions under which a set Sj is considered efficient depend on the comparison between p_j and the expected revenue r_{j-1}. If a set consumes more capacity but yields lower revenue (inefficient), the models show that subsequent larger sets tend to inherit inefficiency, guiding strategic inclusion or exclusion of fare options.

Dynamic Programming and Optimal Offering Strategies

The dynamic program V(t, x) evaluates the maximum expected reward for offering fare sets over time, factoring in the expected revenues and choice probabilities u_j. The derived formula uj = (p_j - r_{j-1}) / (1 - π_{j-1}) indicates when a particular set is optimal to offer, depending on the current state (t, x). The policy suggests choosing the set that maximizes the incremental value (uj), offering insights into optimal fare management tailored to capacity and demand over time.

Extension of Methods to Large-Scale Revenue Optimization

Modifying computational codes to solve the dynamic programming problem with large T (e.g., T=10,000) demonstrates the scalability of these models. The process involves iterative updates of V(t, x) and assessing the maximum value across different c values in {35, 40, 55, 60, 65, 70, 75, 80, 85, 90}. Such analyses enable airlines to develop adaptive pricing and capacity strategies, maximizing expected revenues across extensive time horizons under stochastic demand.

Conclusion

Overall, the analysis underscores the importance of accurate utility modeling, sensitivity to parameter changes, subset and revenue optimization, and dynamic decision rules in airline revenue management. These models assist in making informed strategic choices that adapt to fluctuating market conditions, capacity constraints, and consumer preferences, thus enhancing profitability.

References

  • Baker, J., & Seiler, H. (2015). Discrete Choice Models in Transportation Planning. Transportation Research Part B, 78, 610–620.

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