32 research outputs found
A model-free approach for solving choice-based competitive facility location problems using simulation and submodularity
This paper considers facility location problems in which a firm entering a
market seeks to open a set of available locations so as to maximize its
expected market share, assuming that customers choose the alternative that
maximizes a random utility function. We introduce a novel deterministic
equivalent reformulation of this probabilistic model and, extending the results
of previous studies, show that its objective function is submodular under any
random utility maximization model. This reformulation characterizes the demand
based on a finite set of preference profiles. Estimating their prevalence
through simulation generalizes a sample average approximation method from the
literature and results in a maximum covering problem for which we develop a new
branch-and-cut algorithm. The proposed method takes advantage of the
submodularity of the objective value to replace the least influential
preference profiles by an auxiliary variable that is bounded by submodular
cuts. This set of profiles is selected by a knee detection method. We provide a
theoretical analysis of our approach and show that its computational
performance, the solution quality it provides, and the efficiency of the knee
detection method it exploits are directly connected to the entropy of the
preference profiles in the population. Computational experiments on existing
and new benchmark sets indicate that our approach dominates the classical
sample average approximation method on large instances, can outperform the best
heuristic method from the literature under the multinomial logit model, and
achieves state-of-the-art results under the mixed multinomial logit model.Comment: 36 pages, 6 figures, 6 table
A multicut outer-approximation approach for competitive facility location under random utilities
fals
The Cooperative Maximum Capture Facility Location Problem
In the Maximum Capture Facility Location (MCFL) problem with a binary choice
rule, a company intends to locate a series of facilities to maximize the
captured demand, and customers patronize the facility that maximizes their
utility. In this work, we generalize the MCFL problem assuming that the
facilities of the decision maker act cooperatively to increase the customers'
utility over the company. We propose a utility maximization rule between the
captured utility of the decision maker and the opt-out utility of a competitor
already installed in the market. Furthermore, we model the captured utility by
means of an Ordered Median function (OMf) of the partial utilities of newly
open facilities. We name this problem "the Cooperative Maximum Capture Facility
Location problem" (CMCFL). The OMf serves as a means to compute the utility of
each customer towards the company as an aggregation of ordered partial
utilities, and constitutes a unifying framework for CMCFL models. We introduce
a multiperiod non-linear bilevel formulation for the CMCFL with an embedded
assignment problem characterizing the captured utilities. For this model, two
exact resolution approaches are presented: a MILP reformulation with valid
inequalities and an effective approach based on Benders' decomposition.
Extensive computational experiments are provided to test our results with
randomly generated data and an application to the location of charging stations
for electric vehicles in the city of Trois-Rivi\`eres, Qu\`ebec, is addressed.Comment: 32 pages, 8 tables, 2 algorithms, 8 figure
Mixed-Integer Programming Approaches to Generalized Submodular Optimization and its Applications
Submodularity is an important concept in integer and combinatorial
optimization. A classical submodular set function models the utility of
selecting homogenous items from a single ground set, and such selections can be
represented by binary variables. In practice, many problem contexts involve
choosing heterogenous items from more than one ground set or selecting multiple
copies of homogenous items, which call for extensions of submodularity. We
refer to the optimization problems associated with such generalized notions of
submodularity as Generalized Submodular Optimization (GSO). GSO is found in
wide-ranging applications, including infrastructure design, healthcare, online
marketing, and machine learning. Due to the often highly nonlinear (even
non-convex and non-concave) objective function and the mixed-integer decision
space, GSO is a broad subclass of challenging mixed-integer nonlinear
programming problems. In this tutorial, we first provide an overview of
classical submodularity. Then we introduce two subclasses of GSO, for which we
present polyhedral theory for the mixed-integer set structures that arise from
these problem classes. Our theoretical results lead to efficient and versatile
exact solution methods that demonstrate their effectiveness in practical
problems using real-world datasets
Binary-Continuous Sum-of-ratios Optimization: Discretization, Approximations, and Convex Reformulations
We study a class of non-convex sum-of-ratios programs which can be used for
decision-making in prominent areas such as product assortment and price
optimization, facility location, and security games. Such an optimization
problem involves both continuous and binary decision variables and is known to
be highly non-convex and intractable to solve. We explore a discretization
approach to approximate the optimization problem and show that the approximate
program can be reformulated as mixed-integer linear or second-order cone
programs, which can be conveniently handled by an off-the-shelf solver (e.g.,
CPLEX or GUROBI). We further establish (mild) conditions under which solutions
to the approximate problem converge to optimal solutions as the number of
discretization points increases. We also provide approximation abounds for
solutions obtained from the approximated problem. We show how our approach
applies to product assortment and price optimization, maximum covering facility
location, and Bayesian Stackelberg security games and provide experimental
results to evaluate the efficiency of our approach
A submodular representation for hub networkdesign problems with profits and single assignments
Hub network design problems (HNDPs) lie at the heart of network design planning in transportation and telecommunication systems. They constitute a challenging class of optimization problems that focus on the design of a hub network. In this work, we study a class of HNDPs, named hub network design problems with profits and single assignments, which forces each node to be assigned to exactly one hub facility.
We propose three different combinatorial representations for maximizing the total profit defined as the difference between the perceived revenues from routing a set of commodities minus the setup cost for designing a hub network, considering the single allocation assumption. We investigate whether the objective function of each representation satisfies the submodular property or not. One representation satisfies submodularity, and we use it to present an approximation algorithm with polynomial running time. We obtain worst-case bounds on the approximations’ quality and analyze some special cases where these worst-case bounds are sharper
City decision-making : optimization of the location and design of urban green spaces
Le besoin grandissant pour une planification urbaine plus durable et pour des interventions publiques visant à l'amélioration du bien-être collectif, ont grandement contribué à un engouement pour les espaces verts. Les parcs sont reconnus pour leur impact positif en zone urbaine dense, et nous sommes intéressés par l'application des concepts théoriques du domaine de la recherche opérationnelle pour assister les décideurs publics afin d'améliorer l'accessibilité, la distribution et la conception des parcs. Étant donné le contexte, nous sommes particulièrement motivés par le concept d'équité, et étudions le comportement des usagers des parcs à l'aide d'un modèle d'interaction spatiale, tel qu'appliqué dans les problèmes d'emplacement d'installations dans un marché compétitif. Dans cette recherche, nous présentons un modèle d'emplacement d'installations à deux étapes pouvant être adapté pour assister les décideurs publics à l'échelle de la ville. Nous étudions spécifiquement l'application aux espaces verts urbains, mais soulignons que des extensions du modèle peuvent permettre d'aborder d'autres problèmes d'emplacements d'installations sujets à des enjeux d'équité. La première étape de notre problème d'optimisation a pour but d'évaluer l'allocation la plus équitable du budget de la ville aux arrondissements, basé sur une somme du budget pondérée par des facteurs d'équité. Dans la deuxième étape du modèle, nous cherchons l'emplacement et la conception optimale des parcs, et l'objectif consiste à maximiser la probabilité totale que les individus visitent les parcs. Étant donné la non-linéarité de la fonction objective, nous appliquons une méthode de linéarisation et obtenons un modèle de programmation linéaire mixte en nombres entiers, pouvant être résolu avec des solveurs standards. Nous introduisons aussi une méthode de regroupement pour réduire la taille du problème, et ainsi trouver des solutions quasi optimales dans un délai raisonnable. Le modèle est testé à l'aide de l'étude de cas de la ville de Montréal, Canada, et nous présentons une analyse comparative des résultats afin de justifier la performance de notre modèle.The recent promotion of sustainable urban planning combined with a growing need for public interventions to improve well-being and health in dense urban areas have led to an increased collective interest for green spaces. Parks have proven a wide range of benefits in urban areas, and we are interested in the application of theoretical concepts from the field of Operations Research to assist decision-makers to improve parks' accessibility, distribution and design. Given the context of public decision-making, we are particularly concerned with the concept of fairness, and are focused on an advanced assessment of users' behavior using a spatial interaction model (SIM) as in competitive facility locations' frameworks. In this research, we present a two-stage fair facility location and design (2SFFLD) model, which serves as a template model to assist public decision-makers at the city-level for the urban green spaces (UGSs) planning. We study the application of the 2SFFLD model to UGSs, but emphasize the potential extension to other applications to location problems concerned with fairness and equity. The first-stage of the optimization problem is about the optimal budget allocation based on a total fair-weighted budget formula. The second-stage seeks the optimal location and design of parks, and the objective consists of maximizing the total expected probability of individuals visiting parks. Given the non-linearity of the objective function, we apply a ``Method-based Linearization'' and obtain a mixed-integer linear program that can be solved with standard solvers. We further introduce a clustering method to reduce the size of the problem and determine a close to optimal solution within reasonable time constraints. The model is tested using the case study of the city of Montreal, Canada, and comparative results are discussed in detail to justify the performance of the model