arXiv subjects

Ngan Ha Duong

Publications and source records attributed to Ngan Ha Duong.

4 recordsLinked to original sources

arXiv2026

Joint Binary-Continuous Fractional Programming: Solution Methods and Applications

In this paper, we investigate a class of non-convex sum-of-ratios programs relevant to decision-making in key areas such as product assortment and pricing, and facility location and cost planning. These optimization problems, characterized by both continuous and binary decision variables, are highly non-convex and challenging to solve. To the best of our knowledge, no existing methods can efficiently solve these problems to near-optimality with arbitrary precision. To address this challenge, we propose an innovative approach based on logarithmic transformations and piecewise linear approximation (PWLA) to approximate the nonlinear fractional program as a mixed-integer convex program with arbitrary precision, which can be efficiently solved using cutting plane (CP) or Branch-and-Cut (B&C) procedures. Our method offers several advantages: it allows for a shared set of binary variables to approximate nonlinear terms and employs an optimal set of breakpoints to approximate other non-convex terms in the reformulation, resulting in an approximate model that is minimal in size. Furthermore, we provide a theoretical analysis of the approximation errors associated with the solutions derived from the approximated problem. We demonstrate the applicability of our approach to constrained competitive joint facility location and cost optimization, as well as constrained product assortment and pricing problems. Extensive experiments on instances of varying sizes, comparing our method with several alternatives, including general-purpose solvers and more direct PWLA-based approximations, show that our approach consistently achieves superior performance across all baselines, particularly in large-scale instances.

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arXiv2024

Competitive Facility Location with Market Expansion and Customer-centric Objective

We study a competitive facility location problem, where customer behavior is modeled and predicted using a discrete choice random utility model. The goal is to strategically place new facilities to maximize the overall captured customer demand in a competitive marketplace. In this work, we introduce two novel considerations. First, the total customer demand in the market is not fixed but is modeled as an increasing function of the customers' total utilities. Second, we incorporate a new term into the objective function, aiming to balance the firm's benefits and customer satisfaction. Our new formulation exhibits a highly nonlinear structure and is not directly solved by existing approaches. To address this, we first demonstrate that, under a concave market expansion function, the objective function is concave and submodular, allowing for a $(1-1/e)$ approximation solution by a simple polynomial-time greedy algorithm. We then develop a new method, called Inner-approximation, which enables us to approximate the mixed-integer nonlinear problem (MINLP), with arbitrary precision, by an MILP without introducing additional integer variables. We further demonstrate that our inner-approximation method consistently yields lower approximations than the outer-approximation methods typically used in the literature. Moreover, we extend our settings by considering a\textit{ general (non-concave)} market-expansion function and show that the Inner-approximation mechanism enables us to approximate the resulting MINLP, with arbitrary precision, by an MILP. To further enhance this MILP, we show how to significantly reduce the number of additional binary variables by leveraging concave areas of the objective function. Extensive experiments demonstrate the efficiency of our approaches.

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arXiv2024

Competitive Facility Location under Random Utilities and Routing Constraints

In this paper, we study a facility location problem within a competitive market context, where customer demand is predicted by a random utility choice model. Unlike prior research, which primarily focuses on simple constraints such as a cardinality constraint on the number of selected locations, we introduce routing constraints that necessitate the selection of locations in a manner that guarantees the existence of a tour visiting all chosen locations while adhering to a specified tour length upper bound. Such routing constraints find crucial applications in various real-world scenarios. The problem at hand features a non-linear objective function, resulting from the utilization of random utilities, together with complex routing constraints, making it computationally challenging. To tackle this problem, we explore three types of valid cuts, namely, outer-approximation and submodular cuts to handle the nonlinear objective function, as well as sub-tour elimination cuts to address the complex routing constraints. These lead to the development of two exact solution methods: a nested cutting plane and nested branch-and-cut algorithms, where these valid cuts are iteratively added to a master problem through two nested loops. We also prove that our nested cutting plane method always converges to optimality after a finite number of iterations. Furthermore, we develop a local search-based metaheuristic tailored for solving large-scale instances and show its pros and cons compared to exact methods. Extensive experiments are conducted on problem instances of varying sizes, demonstrating that our approach excels in terms of solution quality and computation time when compared to other baseline approaches.

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arXiv2023

Joint Location and Cost Planning in Maximum Capture Facility Location under Multiplicative Random Utility Maximization

We study a joint facility location and cost planning problem in a competitive market under random utility maximization (RUM) models. The objective is to locate new facilities and make decisions on the costs (or budgets) to spend on the new facilities, aiming to maximize an expected captured customer demand, assuming that customers choose a facility among all available facilities according to a RUM model. We examine two RUM frameworks in the discrete choice literature, namely, the additive and multiplicative RUM. While the former has been widely used in facility location problems, we are the first to explore the latter in the context. We numerically show that the two RUM frameworks can well approximate each other in the context of the cost optimization problem. In addition, we show that, under the additive RUM framework, the resultant cost optimization problem becomes highly non-convex and may have several local optima. In contrast, the use of the multiplicative RUM brings several advantages to the competitive facility location problem. For instance, the cost optimization problem under the multiplicative RUM can be solved efficiently by a general convex optimization solver or can be reformulated as a conic quadratic program and handled by a conic solver available in some off-the-shelf solvers such as CPLEX or GUROBI. Furthermore, we consider a joint location and cost optimization problem under the multiplicative RUM and propose three approaches to solve the problem, namely, an equivalent conic reformulation, a multi-cut outer-approximation algorithm, and a local search heuristic. We provide numerical experiments based on synthetic instances of various sizes to evaluate the performances of the proposed algorithms in solving the cost optimization, and the joint location and cost optimization problems.

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