The d-Level Nested Logit Model: Assortment and Price Optimization Problems

We consider assortment and price optimization problems under the d -level nested logit model. In the assortment optimization problem, the goal is to find the revenue-maximizing assortment of products to offer, when the prices of the products are fixed. Using a novel formulation of the d -level nested logit model as a tree of depth d , we provide an efficient algorithm to find the optimal assortment. For a d -level nested logit model with n products, the algorithm runs in O ( d n log n ) time. In the price optimization problem, the goal is to find the revenue-maximizing prices for the products, when the assortment of offered products is fixed. Although the expected revenue is not concave in the product prices, we develop an iterative algorithm that generates a sequence of prices converging to a stationary point. Numerical experiments show that our method converges faster than gradient-based methods, by many orders of magnitude. In addition to providing solutions for the assortment and price optimization problems, we give support for the d -level nested logit model by demonstrating that it is consistent with the random utility maximization principle and equivalent to the elimination by aspects model.

[1]  Andrew Daly,et al.  On the Equivalence Between Elimination-By-Aspects and Generalised Extreme Value Models of Choice Behaviour: , 2006 .

[2]  Frank S. Koppelman,et al.  Modeling the competition among air-travel itinerary shares: GEV model development , 2005 .

[3]  A. Börsch-Supan.  On the compatibility of nested logit models with utility maximization , 1990 .

[4]  Florian Heiss,et al.  Discrete Choice Methods with Simulation , 2016 .

[5]  J. Kadane.  Structural Analysis of Discrete Data with Econometric Applications , 1984 .

[6]  G. Iyengar,et al.  Managing Flexible Products on a Network , 2004 .

[7]  Richard S. Varga,et al.  Proof of Theorem 5 , 1983 .

[8]  Kalyan T. Talluri,et al.  A Randomized Concave Programming Method for Choice Network Revenue Management , 2010 .

[9]  Ward Hanson,et al.  Optimizing Multinomial Logit Profit Functions , 1996 .

[10]  Guillermo Gallego,et al.  WORKING PAPER SERIES , 2011 .

[11]  N. S. Cardell,et al.  Variance Components Structures for the Extreme-Value and Logistic Distributions with Application to Models of Heterogeneity , 1997, Econometric Theory.

[12]  Michael Duncan,et al.  Residential Self Selection and Rail Commuting: A Nested Logit Analysis , 2002 .

[13]  Devavrat Shah,et al.  A Nonparametric Approach to Modeling Choice with Limited Data , 2009, Manag. Sci..

[14]  G. Ryzin,et al.  On the Relationship Between Inventory Costs and Variety Benefits in Retailassortments , 1999 .

[15]  Richard T. Carson,et al.  A Nested Logit Model of Recreational Fishing Demand in Alaska , 2009, Marine Resource Economics.

[16]  Richard Ratliff,et al.  A General Attraction Model and Sales-Based Linear Program for Network Revenue Management Under Customer Choice , 2015, Oper. Res..

[17]  Éva Tardos,et al.  Algorithm design , 2005 .

[18]  Peter Vovsha,et al.  Application of Cross-Nested Logit Model to Mode Choice in Tel Aviv, Israel, Metropolitan Area , 1997 .

[19]  Guang Li,et al.  A greedy algorithm for the two-level nested logit model , 2014, Oper. Res. Lett..

[20]  Hongmin Li,et al.  Pricing Multiple Products with the Multinomial Logit and Nested Logit Models: Concavity and Implications , 2011, Manuf. Serv. Oper. Manag..

[21]  H. Williams.  On the Formation of Travel Demand Models and Economic Evaluation Measures of User Benefit , 1977 .

[22]  David B. Shmoys,et al.  Assortment Optimization with Mixtures of Logits , 2010 .

[23]  Frank S. Koppelman,et al.  CLOSED-FORM DISCRETE-CHOICE MODELS* , 2007 .

[24]  Zhaosong Lu,et al.  Assessing the Value of Dynamic Pricing in Network Revenue Management , 2013, INFORMS J. Comput..

[25]  M. Fisher,et al.  Assortment Planning: Review of Literature and Industry Practice , 2008 .

[26]  Juan José Miranda Bront,et al.  A Column Generation Algorithm for Choice-Based Network Revenue Management , 2008, Oper. Res..

[27]  A. Tversky,et al.  "Preference trees": Correction to Tversky and Sattath , 1980 .

[28]  Dan Zhang,et al.  An Approximate Dynamic Programming Approach to Network Revenue Management with Customer Choice , 2009, Transp. Sci..

[29]  Garrett J. van Ryzin,et al.  A Multiproduct Dynamic Pricing Problem and Its Applications to Network Yield Management , 1997, Oper. Res..

[30]  Huseyin Topaloglu,et al.  Assortment Optimization Under Variants of the Nested Logit Model , 2014, Oper. Res..

[31]  Leeat Yariv.  Online Appendix , 2008 .

[32]  A. Tversky.  Choice by elimination , 1972 .

[33]  Garrett J. van Ryzin,et al.  Revenue Management Under a General Discrete Choice Model of Consumer Behavior , 2004, Manag. Sci..

[34]  D. McFadden.  Conditional logit analysis of qualitative choice behavior , 1972 .

[35]  Garrett J. van Ryzin,et al.  On the Choice-Based Linear Programming Model for Network Revenue Management , 2008, Manuf. Serv. Oper. Manag..

[36]  Daniel McFadden,et al.  Modelling the Choice of Residential Location , 1977 .

[37]  A. Tversky.  Elimination by aspects: A theory of choice. , 1972 .

[38]  Guillermo Gallego,et al.  Multi-Product Price Optimization and Competition Under the Nested Logit Model with Product-Differentiated Price Sensitivities , 2013 .

[39]  Huseyin Topaloglu,et al.  A refined deterministic linear program for the network revenue management problem with customer choice behavior , 2008 .

[40]  Moshe Ben-Akiva,et al.  Discrete Choice Analysis: Theory and Application to Travel Demand , 1985 .

[41]  Yi Xu,et al.  Optimal and Competitive Assortments with Endogenous Pricing Under Hierarchical Consumer Choice Models , 2011, Manag. Sci..

[42]  Sumit Kunnumkal,et al.  Randomization Approaches for Network Revenue Management with Customer Choice Behavior , 2014 .

[43]  Warren H. Hausman,et al.  Technical Note: Mathematical Properties of the Optimal Product Line Selection Problem Using Choice-Based Conjoint Analysis.: Mathematical Properties of the Optimal Product Line Selection Problem Using Choice-Based Conjoint Analysis. , 2000 .

[44]  Panagiotis Kouvelis,et al.  Dynamic Pricing and Inventory Control of Substitute Products , 2009, Manuf. Serv. Oper. Manag..

[45]  G. Gallego,et al.  A general attraction model and an efficient formulation for the network revenue management problem , 2011 .

[46]  Ya-Xiang Yuan,et al.  Optimization theory and methods , 2006 .

[47]  R. Duncan Luce,et al.  Individual Choice Behavior: A Theoretical Analysis , 1979 .

[48]  Huseyin Topaloglu,et al.  Constrained Assortment Optimization for the Nested Logit Model , 2014, Manag. Sci..

[49]  R. Luce,et al.  Individual Choice Behavior: A Theoretical Analysis. , 1960 .

[50]  Kalyan T. Talluri,et al.  An Enhanced Concave Program Relaxation for Choice Network Revenue Management , 2013 .

[51]  Devavrat Shah,et al.  Sparse choice models , 2010, 2012 46th Annual Conference on Information Sciences and Systems (CISS).

[52]  Jing-Sheng Song,et al.  Demand Management and Inventory Control for Substitutable Products , 2007 .

[53]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[54]  H. Akaike.  A new look at the statistical model identification , 1974 .

[55]  Ya-Xiang Yuan,et al.  Optimization Theory and Methods: Nonlinear Programming , 2010 .

[56]  Stephen P. Boyd,et al.  Graph Implementations for Nonsmooth Convex Programs , 2008, Recent Advances in Learning and Control.

[57]  David B. Shmoys,et al.  Dynamic Assortment Optimization with a Multinomial Logit Choice Model and Capacity Constraint , 2010, Oper. Res..

[58]  Paul Waddell,et al.  Exogenous Workplace Choice in Residential Location Models: Is the Assumption Valid? , 2010 .

[59]  Zhaolin Li,et al.  A Single‐Period Assortment Optimization Model , 2009 .

[60]  F. Koppelman,et al.  The generalized nested logit model , 2001 .

[61]  Arne Strauss,et al.  Network revenue management with inventory-sensitive bid prices and customer choice , 2012, Eur. J. Oper. Res..

[62]  Wagner A. Kamakura,et al.  Book Review: Structural Analysis of Discrete Data with Econometric Applications , 1982 .

[63]  Greg M. Allenby,et al.  A Choice Model with Conjunctive, Disjunctive, and Compensatory Screening Rules , 2004 .

[64]  R. Varga,et al.  Proof of Theorem 4 , 1983 .

[65]  P. Rusmevichientong,et al.  Assortment Optimization under the Multinomial Logit Model with Random Choice Parameters , 2014 .

[66]  Jacob Pleune Gramlich.  Gas prices and fuel efficiency in the U.S. automobile industry: Policy implications of endogenous product choice , 2009 .

[67]  Ajay K. Manrai,et al.  Feature-based elimination: Model and empirical comparison , 1998, Eur. J. Oper. Res..

[68]  Kenneth E. Train,et al.  Discrete Choice Methods with Simulation , 2016 .