, Volume 81, Issue 1, pp 393–417 | Cite as

Assortment Planning with Nested Preferences: Dynamic Programming with Distributions as States?

  • Danny SegevEmail author


The main contribution of this paper is to develop new techniques in approximate dynamic programming, along with the notions of rounded distributions and inventory filtering, to devise a quasi-PTAS for the capacitated assortment planning problem, recently studied by Goyal et al. (Oper Res 64(1):219–235, 2016). Motivated by real-life applications, their nested preference lists model stands as the only setting in dynamic assortment optimization where provably \(\epsilon \)-optimal inventory levels can be efficiently computed. However, these findings crucially depend on certain distributional assumptions, leaving the general problem wide open in terms of approximability prior to this work. In addition to proposing the first rigorous approach for handling the nested preference lists model in its utmost generality, from a technical perspective, we augment the existing literature on dynamic programming with a number of promising ideas. These are novel algorithmic tools for efficiently keeping approximate distributions as part of the state description, while losing very little information and while accumulating only small approximation errors throughout the overall computation. From a conceptual perspective, at the cost of losing an \(\epsilon \)-factor in optimality, we show how to dramatically improve on the truly exponential nature of standard dynamic programs, which seem essential for the purpose of computing optimal inventory levels.


Assortment planning Inventory management Dynamic programming Approximation scheme Rounded distributions 


  1. 1.
    Aouad, A., Farias, V.F., Levi, R., Segev, D.: The approximability of assortment optimization under ranking preferences. Oper. Res. (forthcoming). Available as SSRN report #2612947 (2015)Google Scholar
  2. 2.
    Bellman, R.: Dynamic Programming. Princeton University Press, Princeton (1957)zbMATHGoogle Scholar
  3. 3.
    Bertsekas, D.P.: Dynamic Programming and Optimal Control. Athena Scientific, Belmont (1995)zbMATHGoogle Scholar
  4. 4.
    Bront, J.J.M., Méndez-Díaz, I., Vulcano, G.J.: A column generation algorithm for choice-based network revenue management. Oper. Res. 57(3), 769–784 (2009)CrossRefzbMATHGoogle Scholar
  5. 5.
    Denardo, E.V.: Dynamic Programming: Models and Applications. Dover Publications, Mineola (1982)zbMATHGoogle Scholar
  6. 6.
    Desir, A., Goyal, V., Zhang, J.: Near-optimal algorithms for capacity constrained assortment optimization. Working paper, available as SSRN report #2543309 (2014)Google Scholar
  7. 7.
    Feldman, J.B., Topaloglu, H.: Capacity constraints across nests in assortment optimization under the nested logit model. Oper. Res. 63(4), 812–822 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fisher, M.L., Vaidyanathan, R.: An algorithm and demand estimation procedure for retail assortment optimization. Working paper (2007)Google Scholar
  9. 9.
    Gallego, G., Topaloglu, H.: Constrained assortment optimization for the nested logit model. Manag. Sci. 60(10), 2583–2601 (2014)CrossRefzbMATHGoogle Scholar
  10. 10.
    Gaur, V., Honhon, D.: Assortment planning and inventory decisions under a locational choice model. Manag. Sci. 52(10), 1528–1543 (2006)CrossRefzbMATHGoogle Scholar
  11. 11.
    Goyal, V., Levi, R., Segev, D.: Near-optimal algorithms for the assortment planning problem under dynamic substitution and stochastic demand. Oper. Res. 64(1), 219–235 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ho, T.H., Tang, C.S.: Product Variety Management: Research Advances. Kluwer, Alphen aan den Rijn (1998)CrossRefGoogle Scholar
  13. 13.
    Honhon, D., Gaur, V., Seshadri, S.: Assortment planning and inventory decisions under stockout-based substitution. Oper. Res. 58(5), 1364–1379 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Honhon, D., Gaur, V., Seshadri, S.: A multi-supplier sourcing problem with a preference ordering of suppliers. Prod. Oper. Manag. 21(6), 1028–1041 (2012)CrossRefGoogle Scholar
  15. 15.
    Hopp, W., Xu, X.: Product line selection and pricing with modularity in design. Manuf. Serv. Oper. Manag. 7(3), 172–187 (2005)CrossRefGoogle Scholar
  16. 16.
    Kök, A.G., Fisher, M.L., Vaidyanathan, R.: Assortment planning: Review of literature and industry practice. In: Retail Supply Chain Management: Quantitative Models and Empirical Studies, pp. 99–154. Springer (2009)Google Scholar
  17. 17.
    Lancaster, K.: A new approach to consumer theory. J. Polit. Econ. 74(2), 132–157 (1966)CrossRefGoogle Scholar
  18. 18.
    Lancaster, K.: Socially optimal product differentiation. Am. Econ. Rev. 65(4), 567–585 (1975)Google Scholar
  19. 19.
    Lancaster, K.: The economics of product variety: a survey. Market. Sci. 9(3), 189–206 (1990)CrossRefGoogle Scholar
  20. 20.
    Mahajan, S., van Ryzin, G.: Inventory competition under dynamic consumer choice. Oper. Res. 49(5), 646–657 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Mahajan, S., van Ryzin, G.: Stocking retail assortments under dynamic consumer substitution. Oper. Res. 49(3), 334–351 (2001)CrossRefzbMATHGoogle Scholar
  22. 22.
    Netessine, S., Rudi, N.: Centralized and competitive inventory models with demand substitution. Oper. Res. 51(2), 329–335 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ramdas, K.: Managing product variety: an integrative review and research directions. Prod. Oper. Manag. 12(1), 79–101 (2003)CrossRefGoogle Scholar
  24. 24.
    Rusmevichientong, P., Shen, Z.J.M., Shmoys, D.B.: A PTAS for capacitated sum-of-ratios optimization. Oper. Res. Lett. 37(4), 230–238 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Rusmevichientong, P., Shen, Z.J.M., Shmoys, D.B.: Dynamic assortment optimization with a multinomial logit choice model and capacity constraint. Oper. Res. 58(6), 1666–1680 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Rusmevichientong, P., Shmoys, D., Tong, C., Topaloglu, H.: Assortment optimization under the multinomial logit model with random choice parameters. Prod. Oper. Manag. 23(11), 2023–2039 (2014)CrossRefGoogle Scholar
  27. 27.
    Rusmevichientong, P., Topaloglu, H.: Robust assortment optimization in revenue management under the multinomial logit choice model. Oper. Res. 60(4), 865–882 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    van Ryzin, G., Mahajan, S.: On the relationship between inventory costs and variety benefits in retail assortments. Manag. Sci. 45(11), 1496–1509 (1999)CrossRefzbMATHGoogle Scholar
  29. 29.
    Saure, D., Zeevi, A.: Optimal dynamic assortment planning with demand learning. Manuf. Serv. Oper. Manag. 15(3), 387–404 (2013)CrossRefGoogle Scholar
  30. 30.
    Saure, D.R.: Essays in consumer choice driven assortment planning. Ph.D. thesis, Columbia University (2011)Google Scholar
  31. 31.
    Shaked, M., Shanthikumar, J.G.: Stochastic Orders. Springer, Berlin (2007)CrossRefzbMATHGoogle Scholar
  32. 32.
    Smith, J.C., Lim, C., Alptekinoğlu, A.: New product introduction against a predator: a bilevel mixed-integer programming approach. Naval Res. Logist. 56(8), 714–729 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Talluri, K., van Ryzin, G.: Revenue management under a general discrete choice model of consumer behavior. Manag. Sci. 50(1), 15–33 (2004)CrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of HaifaHaifaIsrael

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