SCALED NEURAL MULTIPLICATIVE MODEL FOR TRACTABLE OPTIMIZATION

Abstract

Challenging decision problems in retail and beyond are often solved using the predict-then-optimize paradigm. An initial effort to develop and parameterize a model of an uncertain environment is followed by a separate effort to identify the best possible solution of an optimization problem. Linear models are often used to ensure optimization problems are tractable. Remarkably accurate Deep Neural Network (DNN) models have recently been developed for various prediction tasks. Such models have been shown to scale to large datasets without loss of accuracy and with good computational performance. It can, however, be challenging to formulate tractable optimization problems based on DNN models. In this work we consider the problem of shelf space allocation for retail stores using DNN models. We highlight the trade-off between predictive performance and the tractability of optimization problems. We introduce a Scaled Neural Multiplicative Model (SNMM) with shape constraints for demand learning that leads to a tractable optimization formulation. Although, this work focuses on a specific application, the formulation of the models are general enough such that they can be extended to many real world applications.

1. INTRODUCTION

The predict-then-optimize framework is ubiquitous in applied research. A predictive model is first developed to approximate the true dynamics of a system under consideration to a given level of accuracy. A mathematical programming formulation based on the predictive model is then used to help researchers identify optimal policies for challenging real-world decision problems. Despite the fact that predictive models that are estimated based on the historical data need not be causal, there are recent works that shed some light on the principles behind this approach (Bertsimas & Kallus, 2020). An alternative to the predict-then-optimize framework is integrating the two stages, predicting and optimizing at the same time, by utilizing a specific loss function in prediction models (Elmachtoub & Grigas, 2022) . For example, model-free Reinforcement learning (RL) approaches explore their environment while also exploiting it to make optimal decisions. Big-box retailers have thousands of stores located across the country. They use data-driven decision to optimize operations; e.g, assortment planning, pricing, and supply chain optimization. They work on problems related to shelf space allocation, making the best use of limited space in stores. Space planning for apparel is particularly challenging, due to the different shapes and sizes of the merchandise and the temporal shifts in brand importance. The problem that motivated our work in this area involves deciding how much space in terms of fixtures to assign to each of several brands within a category of products for sale. These problems are typically solved at the department or store level. Products are arranged into departments and categories; for example, 'activewear tops' is a category within the women's apparel department. In this work, we introduce a novel approach based on the predict-then-optimize framework for solving a shelf space allocation problem. In particular our contributions are as follows 1

