An Exact Method for Assortment Optimization under the Nested Logit Model

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. For the NP-hard cases, we provide a general exact method that embeds a tailored Branch-and-Bound algorithm into a fractional programming framework. Contrary 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, and hence our approach can solve the most general problem setting. We show that the parameterized subproblem of the fractional programming scheme, which is a binary highly non-linear optimization problem, is decomposable by nests, which is a main advantage of the approach. To solve the subproblem for each nest, we propose a two-stage approach. In the first stage, we identify those products that are undoubtedly beneficial to offer, or not, which 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 up to 5,000 products per nest. The most challenging instances for our approach are those in which the dissimilarity parameters of nests can be either less or greater than one

Constrained Assortment Optimization under the Cross-Nested Logit Model

We study the assortment optimization problem under general linear constraints, where the customer choice behavior is captured by the Cross-Nested Logit model. In this problem, there is a set of products organized into multiple subsets (or nests), where each product can belong to more than one nest. The aim is to find an assortment to offer to customers so that the expected revenue is maximized. We show that, under the Cross-Nested Logit model, the assortment problem is NP-hard, even without any constraints. To tackle the assortment optimization problem, we develop a new discretization mechanism to approximate the problem by a linear fractional program with a performance guarantee of $\frac{1 - \epsilon}{1+\epsilon}$, for any accuracy level $\epsilon>0$. We then show that optimal solutions to the approximate problem can be obtained by solving mixed-integer linear programs. We further show that our discretization approach can also be applied to solve a joint assortment optimization and pricing problem, as well as an assortment problem under a mixture of Cross-Nested Logit models to account for multiple classes of customers. Our empirical results on a large number of randomly generated test instances demonstrate that, under a performance guarantee of 90%, the percentage gaps between the objective values obtained from our approximation methods and the optimal expected revenues are no larger than 1.2%

Bayesian Mechanism Design for Blockchain Transaction Fee Allocation

In blockchain systems, the design of transaction fee mechanisms is essential for stability and satisfaction for both miners and users. A recent work has proven the impossibility of collusion-proof mechanisms that achieve both non-zero miner revenue and Dominating-Strategy-Incentive-Compatible (DSIC) for users. However, a positive miner revenue is important in practice to motivate miners. To address this challenge, we consider a Bayesian game setting and relax the DSIC requirement for users to Bayesian-Nash-Incentive-Compatibility (BNIC). In particular, we propose an auxiliary mechanism method that makes connections between BNIC and DSIC mechanisms. With the auxiliary mechanism method, we design a transaction fee mechanism (TFM) based on the multinomial logit (MNL) choice model, and prove that the TFM has both BNIC and collusion-proof properties with an asymptotic constant-factor approximation of optimal miner revenue for i.i.d. bounded valuations. Our result breaks the zero-revenue barrier while preserving truthfulness and collusion-proof properties.Comment: 58 pages, CESC 202

Three Essays on Optimization and Decision-Making Solutions in Grocery Retail Operations

In this dissertation titled “Three Essays on Optimization and Decision-Making Solutions in Retail Operations,” we explore various techniques aimed at optimizing the operational efficiency in a grocery retail store. Specifically, the first essay examines a store manager’s decision of which stock-keeping units (SKUs) from a given category to assign to a promotional display space. We develop a decision support tool that consists of an estimation model and an optimization model. Using a grocery store sales transaction dataset, we introduce a methodology to measure the incremental lift in sales of placing a particular SKU on promotional display space. Our optimization model includes the incremental lifts (from the estimation method) combined with the estimated base-sales rates and profit margins of each SKU so that the profitmaximizing SKU can be chosen for a promotional display space for each week of the year. The second essay offers a novel methodological solution on the appropriate identification and analysis of submarkets in product categories. Our research contributes to the literature in the following ways. While a vast amount of literature in both marketing and operations management investigate retail decision tree structures, limited information exists on developing algorithms that allow to generate, analyze, and test data-driven decision trees. Understanding how decision trees may drive consumer preferences is critical to a retailer’s choice of product category assortment. We provide a methodology on empirically constructing and evaluating the best fitting decision tree structures using easily accessible and readily available scanner data. The third essay studies the mechanisms retailers can use to facilitate sales of reduced packaged products, which have a number of advantages that are attractive to retailers, manufacturers, and consumers. Large product packaging creates logistical and operational challenges for retailers who carry such products since these products require more space to be stored and displayed, and more manpower to handle it. In contrast, products in smaller packaging have fewer such problems, and, thus, positively contribute to the retailer’s operational efficiency. We discuss and empirically test two levers that retailers may utilize to influence the sales of reduced packaged products. Using sales data for liquid detergents, we show that retailers with market power are able to announce their preferences for reduced packaged detergents, which results in an industry-wide shift toward reduced packaged detergents. We also show that retailers, with varying degrees of market power, may select higher ratios of reduced packaged detergents and achieve convex levels of sales of reduced packaged detergents