We study the problem of finding an optimal assortment of products maximizing the expected revenue, in which customer preferences are modeled using a nested logit choice model. This problem is known to be polynomially solvable in a specific case and NP-hard otherwise, with only approximation algorithms existing in the literature. We provide an exact general method that embeds a tailored Branch-and-Bound algorithm into a fractional programming framework. In contrast to the existing literature, in which assumptions are imposed on either the structure of nests or the combination and characteristics of products, no assumptions on the input data are imposed. Although our approach is not polynomial in input size, it can solve the most general problem setting for large-size instances. We show that the fractional programming scheme’s parameterized subproblem, a highly non-linear binary optimization problem, is decomposable by nests, which is the primary advantage of the approach. To solve the subproblem for each nest, we propose a two-stage approach. In the first stage, we fix a large set of variables based on the single-nest subproblem’s newly-derived structural properties. This can significantly reduce the problem size. In the second stage, we design a tailored Branch-and-Bound algorithm with problem-specific upper bounds. Numerical results show that the approach is able to solve assortment instances with five nests and with up to 5000 products per nest. The most challenging instances for our approach are those with a mix of nests’ dissimilarity parameters, where some of them are smaller than one and others are greater than one.

我们研究了寻找最佳产品类别以最大化预期收入的问题，其中使用嵌套的logit选择模型对客户偏好进行建模。已知该问题在特定情况下可以多项式求解，否则在NP-hard条件下只能用文献中存在的近似算法解决。我们提供了一种精确的通用方法，该方法将量身定制的Branch-and-Bound算法嵌入分数编程框架。与现有文献相反，在现有文献中，对嵌套的结构或产品的组合和特征进行了假设，而对输入数据没有进行任何假设。尽管我们的方法不是输入大小多项式，但是它可以解决大型实例的最一般问题设置。我们表明分数编程方案的参数化子问题是一个高度非线性的二进制优化问题，可通过嵌套分解，这是该方法的主要优势。为了解决每个嵌套的子问题，我们提出了一种两阶段方法。在第一阶段，我们根据单巢子问题的新派生结构特性修复大量变量。这样可以大大减少问题的大小。在第二阶段，我们设计了具有特定问题上限的量身定制的分支定界算法。数值结果表明，该方法能够求解具有五个嵌套的分类实例，每个嵌套最多可容纳5000种产品。对于我们的方法而言，最具挑战性的例子是那些混合了嵌套差异参数的实例，