Proceedings of the IX International Workshop on Locational Analysis and Related Problems
Marta Baldomero-Naranjo, Inmaculada Espejo-Miranda, Luisa I. Martínez-Merino, Juan Manuel Muñoz-Ocaña, Antonio M. Rodríguez-Chía
WWednesday Jan 30th Thursday Jan 31st Friday Feb 1st9:00 Session 3:Continuous Location Session 9:Applications/ Routing/ Hub Location10:40 Coffee break Coffee break11:10 Invited Speaker:Francisco Saldanha da Gama Invited Speaker:Ivana Ljubic12:30 Session 4: Session 10:Bilevel Location13:50 Routing14:10 Location Network Meeting14:30 Lunch Lunch15:3016:00 Registration Session 5:Networks Design16:30 Opening Break16:40 Session16:45 Session 1: Session 6:Networks Design II17:40 Discrete Coffee break18:00 Location18:25 Coffee break18:45 Session 7:Applications19:00 Break19:10 Session 2:Networks20:05 Session 8:Discrete Location II20:1021:30 Welcome Reception Dinner onference Map
Conference Venue
Restaurante Arsenio Manila -Conference DinnerRestaurante Arteserrano -LunchLa Calle del Libre Albedrío - Welcome ReceptionHotel Spa Cádiz Plaza- Spanish Location NetworkHotelFacultad de Enfermería yFisioterapia - ConferenceVenue
ROCEEDINGS OFTHE IX INTERNATIONAL WORKSHOPON LOCATIONAL ANALYSIS ANDRELATED PROBLEMS (2019)
Edited by
Marta Baldomero-NaranjoInmaculada Espejo-MirandaLuisa I. Martínez-MerinoJuan Manuel Muñoz-OcañaAntonio M. Rodríguez-Chía
ISBN: 978-84-09-07742-7 reface
The International Workshop on Locational Analysis and Related Problemswill take place during January 30-February 1, 2019 in Cádiz (Spain). It is or-ganized by the Spanish Location Network and Location Group GELOCA(SEIO). GELOCA is a working group on location belonging to the Statisticsand Operations Research Spanish Society. The Spanish Location Networkis a group of more than 140 researchers distributed into 16 nodes corre-sponding to several Spanish universities. The Network has been fundedby the Spanish Government.Every year, the Network organizes a meeting to promote the communi-cation between its members and between them and other researchers, andto contribute to the development of the location field and related problems.Previous meetings took place in Segovia (September 27-29, 2017), Málaga(September 14-16, 2016), Barcelona (November 25-28, 2015), Sevilla (Oc-tober 1-3, 2014), Torremolinos (Málaga, June 19-21, 2013), Granada (May10-12, 2012), Las Palmas de Gran Canaria (February 2-5, 2011) and Sevilla(February 1-3, 2010).The topics of interest are location analysis and related problems. It in-cludes location models, networks, transportation, logistics, exact and heuris-tic solution methods, and computational geometry, among others.The organizing committee. i Preface
Scientific committee:
María Albareda Sambola (U. Politécnica de Cataluña, España)Giuseppe Bruno (Università degli Studi di Napoli Federico II, Italia)Ángel Corberán (U. de Valencia, España)Elena Fernández Aréizaga (U. de Cádiz, España)Joerg Kalcsics (University of Edinburgh, UK)Martine Labbé (Université Libre de Bruxelles, Belgique)Alfredo Marín Pérez (U. de Murcia, España)Juan A. Mesa (U. de Sevilla, España)Stefan Nickel (Karlsrhue Institute of Technology, Germany)Justo Puerto Albandoz (U. de Sevilla, España)Antonio M. Rodríguez-Chía (U. de Cádiz, España)
Organizing committee:
Marta Baldomero Naranjo (U. de Cádiz)Inmaculada Espejo Miranda (U. de Cádiz)Luisa Isabel Martínez Merino (U. de Cádiz)Manuel Muñoz Márquez (U. de Cádiz, España)Juan Manuel Muñoz Ocaña (U. de Cádiz, España)Antonio M. Rodríguez-Chía (U. de Cádiz, España)Dolores Rosa Santos Peñate (U. de Las Palmas de Gran Canaria)Concepción Valero Franco (U. de Cádiz, España) ontents
Preface v
Program 1Invited Speakers 9
Logistics Network Design and Facility Location: The value ofmulti-period stochastic solutions 11
F. Saldanha da Gama
Solving Very Large Scale Covering Location Problems using Branch-and Benders-Cuts 13
I. Ljubic
Abstracts 15
An exact algorithm for the Interval Transportation Problem 17
M. Albareda-Sambola, M. Landete, and G. Laporte
Exact algorithm for the Reliability Fixed-Charge Location Prob-lem with Capacity constraints 19
M. Albareda-Sambola, M. Landete, J.F. Monge, and J.L. Sainz-Pardo
An extension of the p -center problem considering stratified de-mand 21 M. Albareda-Sambola, L.I. Martínez-Merino, and A.M. Rodríguez-Chía
Non-dominated solutions for the bi-objective MST problem 23
L. Amorosi and J. Puerto
A Kernel Search for the Inventory Routing Problem 25 vii iii
CONTENTSC. Archetti, G. Guastaroba, D.L. Huerta-Muñoz. and M.G. Speranza
Time dependent continuous optimisation in solar power towerplants 27
T.Ashley, E. Carrizosa, and E. Fernández-Cara
Minmax Regret Maximal Covering on Networks with Edge De-mands 29
M. Baldomero-Naranjo, J. Kalcsics, and A.M. Rodríguez-Chía
The impact of pharmacy deregulation process on market compe-tition and users’ accessibility. Insights from two Spanish casestudies. 31
I. Barbarisi, G. Bruno, M. Cavola, A. Diglio, J. Elizalde Blasco, and C. Piccolo
The selective traveling salesman problem with time-dependentprofits 33
E. Barrena, D. Canca, L.C. Coelho, and G. Laporte
Solidarity behavior for optimizing the waste selective collection 35
E. Barrena, D. Canca, F. A. Ortega, and R. Piedra-de-la-Cuadra
Dealing with Symmetry in a Multi-period Sales Districting Prob-lem 37
M. Bender, J. Kalcsics, A. Meyer, and M. Pouls
A Mixed Integer Linear Formulation for the Maximum CoveringLocation Problem with Ellipses 39
V. Blanco and S. García
Locating Hyperplanes for Multiclass Classification 41
V. Blanco, A. Japón, and J. Puerto
Minimum covering polyellipses 43
V. Blanco and J. Puerto
The One-Round Voronoi Game Played on the Rectilinear Plane 45
T. Byrne, S. P. Fekete, and J. Kalcsics
Drone Arc Routing Problems 47
J. F. Campbell, Á. Corberán, I. Plana, and J. M. SanchisONTENTS ix The Railway Rapid Transit Network Construction SchedulingProblem 49
D. Canca, A. De-Los-Santos, G. Laporte, and J. A. Mesa
Heuristic Framework to Reduce Aggregation Error on Large Clas-sical Location Models 51
C. Castañeda and D. Serra
Rationalizing capacities in the facility location problem 53
Á. Corberán, M. Landete, J. Peiró, and F. Saldanha-da-Gama
Bilevel programming models for multi-product location prob-lems 55
S. Dávila, M. Labbé, F. Ordoñez, F. Semet, and V. Marianov
The Urban Transit Network Design Problem 57
A. De-Los-Santos, D. Canca, A. G. Hernández-Díaz, and E. Barrena
Minimum distance regulation and entry deterrence through lo-cation decisions 59
J. Elizalde Blasco and I. Rodríguez Carreño
A multi-period bilevel approach for stochastic equilibrium innetwork expansion planning under uncertainty 61
L. F. Escudero, J. F. Monge, and A. M. Rodríguez-Chía
Capacitated Discrete Ordered Median Problems 63
I. Espejo, J. Puerto, and A. M. Rodríguez-Chía
A branch-and-price algorithm for the Vehicle Routing Problemwith Stochastic Demands, Probabilistic Duration Constraint,and Optimal Restocking Policy 65
A. M. Florio, R. F. Hartl, S. Minner, and J. J. Salazar-González
Infrastructure Rapid Transit Network Design Model solved byBenders Decomposition 67
N. González-Blanco and J. A. Mesa
On computational Dynamic Programming for minimizing en-ergy in an electric vehicle 69
E. M.T. Hendrix and I. Garcia
CONTENTS
New bilevel programming approaches to the location of contro-versial facilities 71
M. Labbé, M. Leal, and J. Puerto
Locating a new station/stop in a network based on trip coverageand times 73
M. C. López-de-los-Mozos and J. A. Mesa
On location-allocation problems for dimensional facilities 75
L. Mallozzi, J. Puerto, and M. Rodríguez-Madrena
Robust feasible rail timetable 77
Á. Marín, M. A. Ruiz-Sánchez, E. Codina
Wildfire Location Model: A new proposal 79
J. A. Mesa and M. Marcos-Pérez
Optimal allocation of fleet frequency for “skip-stop” strategiesin transport networks 81
J. A. Mesa, F. A. Ortega, R. Piedra-de-la-Cuadra, and M. A. Pozo
Introduction to planar location with orloca 83
M. Munoz-Marquez
Emergency Vehicle Location Model considering uncertainty andthe hierarchical structure of the resources 85
J. Nelas and J. Dias
Solving the Ordered Median Tree of Hubs Location Problem 87
M. A. Pozo, J. Puerto, and A. M. Rodríguez-Chía
Feasible solutions for the Distance Constrained Close-EnoughArc Routing Problem 89
M. Reula, Á. Corberán, I. Plana, and J. M. Sanchis
Steiner Traveling Salesman Problems: when not all vertices havedemand 91
J. Rodríguez-Pereira, E. Benavent, E. Fernández, G. Laporte, and A. Martínez-Sykora
Addressing locational complexity: network design and networkrationalisation 93
D. Ruiz-Hernandez, J. M. Pinar-Pérez, and M. B.C. MenezesONTENTS xi Using a kernel search heuristic to solve a sequential competitivelocation problem in a discrete space 95
D. R. Santos-Peñate, C. M. Campos-Rodríguez, and J. A. Moreno-Pérez
ROGRAM ednesday January 30th
Exact algorithm for the Reliability Fixed-Charge Location Problemwith Capacity constraints
M. Albareda-Sambola, M. Landete, J.F. Monge, and J.L. Sainz-Pardo
An extension of the p-center problem considering stratified demand
M. Albareda-Sambola, L.I. Martínez-Merino, and A.M. Rodríguez-Chía
Capacitated Discrete Ordered Median Problems
I. Espejo, J. Puerto, and A. M. Rodríguez-Chía
Rationalizing capacities in the facility location problem
Á. Corberán, M. Landete, J. Peiró, and F. Saldanha-da-Gama
An exact algorithm for the Interval Transportation Problem
M. Albareda-Sambola, M.Landete, and G. Laporte
Optimal allocation of fleet frequency for “skip-stop” strategies in trans-port networks.
J. A. Mesa, F. A. Ortega, R. Piedra-de-la-Cuadra, and M. A. Pozo
Dealing with Symmetry in a Multi-period Sales Districting Problem
M. Bender, J. Kalcsics, A. Meyer, and M. Pouls
Minmax Regret Maximal Covering on Networks with Edge Demands
M. Baldomero-Naranjo, J. Kalcsics, and A. Rodríguez-Chía
Non-dominated solutions for the bi-objective MST problem
L. Amorosi and J. Puerto
Thursday January 31st
A Mixed Integer Linear Formulation for the Maximum Covering Lo-cation Problem with Ellipses
V. Blanco and S. García
Minimum covering polyellipses
V. Blanco and J. Puerto
The One-Round Voronoi Game Played on the Rectilinear Plane
T. Byrne, S. P. Fekete, and J. Kalcsics
Locating Hyperplanes for Multiclass Classification
V. Blanco, A. Japón, and J. Puerto
Introduction to planar location with orloca
M. Muñoz-Márquez
Logistics Network Design and Facility Location: The value of multi-period stochastic solutions
The selective traveling salesman problem with time-dependent prof-its
E. Barrena, D. Canca, L.C. Coelho, and G. Laporte
Solidarity behavior for optimizing the waste selective collection
E. Barrena, D. Canca, F. A. Ortega, and R. Piedra-de-la-Cuadra
Steiner Traveling Salesman Problems: when not all vertices have de-mand
J. Rodríguez-Pereira, E. Benavent, E. Fernández, G. Laporte, and A. Martínez-Sykora
A Kernel Search for the Inventory Routing Problem
C. Archetti, G. Guastaroba, D.L. Huerta-Muñoz, and M.G. Speranza
A branch-and-price algorithm for the Vehicle Routing Problem withStochastic Demands, Probabilistic Duration Constraint, and OptimalRestocking Policy
A. M. Florio, R. F. Hartl, S. Minner, and J.J. Salazar-González
Robust feasible rail timetable
Á. Marín, M. A. Ruiz-Sánchez, and E. Codina
Addressing locational complexity: network design and network ra-tionalisation
D. Ruiz-Hernandez, J. M. Pinar-Pérez, and M. B.C. Menezes
Locating a new station/stop in a network based on trip coverage andtimes
M. C. López-de-los-Mozos and J. A. Mesa
The Urban Transit Network Design Problem
A. De-los-Santos, D. Canca, A. G. Hernández-Díaz, and E. Barrena
Infrastructure Rapid Transit Network Design Model solved by Ben-ders Decomposition
N. González-Blanco and J. A. Mesa
The Railway Rapid Transit Network Construction Scheduling Prob-lem
D. Canca, A. de los Santos, G. Laporte, and J. A. Mesa
Wildfire Location Model: A new proposal
J. A. Mesa and M. Marcos-Pérez
Time dependent continuous optimisation in solar power tower plants
T.Ashley, E. Carrizosa, and E. Fernández-Cara
On computational Dynamic Programming for minimizing energy inan electric vehicle
E. M.T. Hendrix and I. Garcia
Heuristic Framework to Reduce Aggregation Error on Large Classi-cal Location Models
C. Castañeda and D. Serra
Emergency Vehicle Location Model considering uncertainty and thehierarchical structure of the resources
J. Nelas and J. Dias
Using a kernel search heuristic to solve a sequential competitive lo-cation problem in a discrete space
D. R. Santos-Peñate, C. M. Campos-Rodríguez, and J. A. Moreno-Pérez
Friday February 1st
Minimum distance regulation and entry deterrence through locationdecisions
J. Elizalde Blasco and I. Rodríguez Carreño
The impact of pharmacy deregulation process on market competitionand users’ accessibility. Insights from two Spanish case studies.
I. Barbarisi, G. Bruno, M. Cavola, A. Diglio, J. Elizalde Blasco, and C. Pic-colo
Solving the Ordered Median Tree of Hubs Location Problem
M. A. Pozo, J. Puerto, and A. M. Rodríguez-Chía
Feasible solutions for the Distance Constrained Close-Enough ArcRouting Problem
M. Reula, Á. Corberán, I. Plana, and J. M. Sanchis
Drone Arc Routing Problems
J. F. Campbell, Á. Corberán, I. Plana, and J. M. Sanchis
Solving Very Large Scale Covering Location Problems using Branch-and Benders-Cuts
On location-allocation problems for dimensional facilities
L. Mallozzi, J. Puerto, and M. Rodríguez-Madrena
New bilevel programming approaches to the location of controver-sial facilities
M. Labbé, M. Leal, and J. Puerto
A multi-period bilevel approach for stochastic equilibrium in net-work expansion planning under uncertainty
L. F. Escudero, J. F. Monge, and A. M. Rodríguez-Chía
Bilevel programming models for multi-product location problems
S. Dávila, M. Labbé, F. Ordoñez, F. Semet, and V. Marianov
NVITED SPEAKERS
X Workshop on Locational Analysis and Related Problems 2019 11
Logistics Network Design and FacilityLocation: The value of multi-periodstochastic solutions
Francisco Saldanha da Gama, Universidade de Lisboa, Centro de Matemática, Aplicações Fundamentais e InvestigaçãoOperacional, Lisboa,Portugal [email protected]
In the past decades logistics network design has been a very active researchfield. This is an area where facility location and logistics are strongly in-tertwined, which is explained by the fact that many researchers workingin Logistics address very often location problems as part of the strate-gic/tactical logistics decisions. Despite all the work done, the economicglobalization together with the emergence of new technologies and com-munication paradigms are posing new challenges when it comes to devel-oping optimization models for supporting decision making in this area.Dealing with time and uncertainty has become unavoidable in many situ-ations.In this presentation, different modeling aspects related with the inclu-sion of time and uncertainty in facility location problems are discussed.The goal is to better understand problems that are at the core of more com-prehensive ones in logistics network design. By considering time explic-itly in the models it becomes possible to capture some features of practicalrelevance that cannot be appropriately captured in a static setting; by con-sidering a stochastic modeling framework it is possible to build risk-awaremodels. Unfortunately, the resulting models are often too large and thushard to tackle even when using specially tailored procedures. This raisesa question: is there a clear gain when considering a more involved modelinstead of a simplified one (e.g. deterministic or static)? In search for ananswer to this question, several measures are discussed that include thevalue of a multi-period solution and the value of a risk-aware solution.
X Workshop on Locational Analysis and Related Problems 2019 13
Solving Very Large Scale CoveringLocation Problems using Branch-andBenders-Cuts
Ivana Ljubic ESSEC Business School of Paris, Cergy Pontoise Cedex, France [email protected]
Covering problems constitute an important family of facility location prob-lems with widespread applications. These problems embed a notion ofproximity (or coverage radius) that specifies whether a given demand pointcan be served or “covered” by a potential facility location. Proximity is of-ten defined in terms of distance or travel time between points. A demandpoint is then said to be covered by a facility if it lies within the coverageradius of this facility. Location problems with covering objectives or con-straints are commonplace in the service sector (schools, hospitals, libraries,restaurants, retail outlets, bank branches) as well as in the location of emer-gency facilities or vehicles (fire stations, ambulances, oil spill equipments).They also find applications in the location of access points for the wirelesscommunication in the smart grid deployments.Many of these applications involve a relatively small number of poten-tial facility locations while the number of demand points can run in thethousands or even millions. Such very large scale problem instances re-main out of reach for modern MIP solvers.In this talk we address the maximal covering location problem (MCLP)which requires choosing a subset of facilities that maximize the demandcovered while respecting a budget constraint on the cost of the facilitiesand the partial set covering location problem (PSCLP) which minimizesthe cost of the open facilities while forcing a certain amount of demand tobe covered. We propose an effective decomposition approach based on thebranch-and-Benders-cut reformulation. We also exploit the submodularityof the covering function and derive a formulation based on submodularcuts. A connection between Benders and submodular cuts is drawn as well. The results of our computational study demonstrate that, thanks to de-composition, optimal solutions can be found very quickly, even for bench-mark instances involving up to twenty million demand points.The talk is based on a joint work with S. Coniglio, J.F. Cordeau and F.Furini.
BSTRACTS
X Workshop on Locational Analysis and Related Problems 2019 17
An exact algorithm for theInterval Transportation Problem
Maria Albareda-Sambola, Mercedes Landete, and Gilbert Laporte , Universitat Politècnica de Catalunya.BarcelonaTech, Terrassa, Spain, [email protected] Centro de Investigación Operativa,Universidad Miguel Hernández de Elche, Spain, [email protected] Canada Research Chair in Distribution Management, HEC Montréal, Montréal H3T2A7, Canada
This work focuses on the Interval Transportation Problem. For this prob-lem, we propose two variants of an exact algorithm. Their efficiency is com-pared on a set of instances from the literature.
1. Problem definition
Given a set of customers j ∈ J , each with demand d j > , a set of fa-cilities i ∈ I , each with capacity q i > and unit transportation costs c ij for i ∈ I, j ∈ J , the well known Transportation Problem (TP) consists indetermining the amount of product to send from each facility to each cus-tomer so that each customer receives exactly his demand, the total amountdelivered from each facility does not exceed its capacity, and the total trans-portation cost is minimized.In the Interval Transportation Problem (ITP) it is assumed that, in fact,each demand can lie in a given interval: d j ∈ [ D j , ¯ D j ] and each capacityalso: q i ∈ [ Q i , ¯ Q i ] . Let D = [ D , ¯ D ] × . . . × [ D m , ¯ D m ] , Q = [ Q , ¯ Q ] × . . . × [ Q n , ¯ Q n ] , and R = { ( d, q ) ∈ D × Q : P i ∈ I q i ≥ P j ∈ J d j } (Set of ( d, q ) pairs withall demands and capacities within their intervals, that define a feasible TPinstance) . Then, the goal of the ITP is to find U ∗ = max ( d,q ) ∈ R z ( d, q ) , (1) where z ( d, q ) is the optimal solution of the TP defined by demands d andcapacities q . That is, the ITP aims at finding the feasible TP instance definedby parameters in R with the most expensive optimal solution.
2. Solution Algorithm
As already observed in [1], the ITP is in fact a bilevel optimization problem.For this reason, we propose to adapt to the ITP the algorithm presentedin [2] for bilevel optimization problems among others. Roughly speaking,the authors propose a bisection line search on an interval containing theoptimal value U ∗ ; at each iteration a sequence of subproblems is solved todetermine wether a given candidate value u can be attained (there existsan instance defined by some ( d, q ) ∈ R with z ( d, q ) ≥ u ) or not.The work [2] considers a very general setting. For this reason, in thatpaper subproblems are solved by means of very general algorithms. In thiswork, we propose to take advantage of the structture of these subproblemsin the particular case of the ITP to solve them more efficiently by suitablystating them as a linear or mixed integer linear problems.Two alternative adaptations are proposed, that differ in the way howalready visited TP solutions are considered. Their behavior is comparedby means of a computational experience on a set of instances taken fromthe literature. References [1] S.-T. Liu. The total cost bounds of the transportation problem with varyingdemand and supply.
Omega , 31,4, 247-251, 2003.[2] A. Tsoukalas, Berç Rustem and Efstratios N. Pistikopoulos. A global optimiza-tion algorithm for generalized semi-infinite, continuous minimax with coupledconstraints and bi-level problems.
Journal of Global Optimization , 44,2, 235–250,2009.X Workshop on Locational Analysis and Related Problems 2019 19
Exact algorithm for the ReliabilityFixed-Charge Location Problemwith Capacity constraints
Maria Albareda-Sambola , Mercedes Landete , Juan FranciscoMonge , Jose Luis Sainz-Pardo Departament de Estadística i Investigació Operativa,Technical University of Catalonia-Barcelona, Terrassa, Spain,[email protected] Centro de Investigación Operativa,Universidad Miguel Hernández de Elche, Spain,[email protected], [email protected], [email protected]
This work addresses the exact solution of the Reliability Fixed-Charge Lo-cation Problem with Capacity Constraints (RFLPCC). The proposed methodis based on a formulation that considers all possible scenarios. Therefore,directly solving this model is computationally hard and it is not availableto solve it even in small instances. We propose a dynamic approach in or-der to exactly bound the expected overload in the Reliability Fixed-ChargeLocation Problem. Too, we analyse by an exhaustive computational studythe quality of the solutions according to several criteria.
1. Dynamic approach
The dynamic approach proposed is based on the usual philosophy of master-slave problems. From the facilities that have been opened in the solutionreturned by the master problem, the slave problem solves a model in orderto make a new assignment with expected overload under the given bound B . Both solutions are iteratively used to introduce different constraints inthe master problem until the optimal value is obtained.
2. Computational experience
We compare by several computational experiments the dynamic approachversus the following three models proposed in [1] which can be seen asmatheuristics for the RFLPCC:QRFLP: this model does not manage the overload, it only keeps thedemand below the capacity of the facilities in the scenario in whichno facility fails,CRFLP-B1: this model constrains an upper bound for the expectedoverload,CRFLP-LR: this model constrains a linear estimation of the expectedoverload, then it is an approximated modelOur intention is to analyse the performance of the proposed approachbut not only in terms of optimal cost, also in terms of expected overload,overload probability, non-served demand, number of open facilities, com-putational time and instances solved before the limit time. Table 1 respec-tively contains the cited average values for one of the most representa-tive computational set of instances of our study with a bound for the ex-pected overload fixed to . Here, RFLP-EX stands for the exact algorithmpresented in this work. v ∗ E ( X, Y ) P(overload) Non-served
Table 1.
Average values for a requested overload of B =3 References [1] M. Albareda-Sambola, M. Landete, J.F. Monge, J.L. Sainz-Pardo, Introducingcapacities in the location of unreliable facilities. Europen Journal of OperationalResearch 259 (2017) 175–188.[2] L.V. Snyder, M.S. Daskin, Reliability models for facility location: the expectedfailure cost case. Transportation Science 39 (2005) 400–416.X Workshop on Locational Analysis and Related Problems 2019 21
An extension of the p -center problemconsidering stratified demand ∗ Maria Albareda-Sambola, Luisa I. Martínez-Merino, and AntonioM. Rodríguez-Chía Universitat Politècnica de Catalunya.BarcelonaTech, Terrassa, Spain, [email protected] Departamento de Estadística e Investigación Operativa, Universidad de Cádiz, Cádiz,Spain, [email protected] Departamento de Estadística e Investigación Operativa, Universidad de Cádiz, Cádiz,Spain, [email protected]
This work introduces an extension of the classical discrete p -center prob-lem ( pCP ), called the stratified p -center problem ( SpCP ). In this extensionit is assumed that the population of each demand site is divided into dif-ferent categories or strata depending on the kind of service that they re-quire. The objective of the proposed model is to locate p centers minimiz-ing the weighted sum of the largest assignment distances associated witheach stratum. This model could be applied in humanitarian relief planningwhere centers of humanitarian assistance cover different kind of needs andthe demand of each need may be distributed in a spatially different way.
1. Introduction to the problem
Given a set of potential locations J = { , . . . , n } to open p centers, a set ofdemand sites I = { , . . . , m } and a set of strata S , the goal of the SpCP ∗ Thanks to Agencia Estatal de Investigación (AEI) and the European Regional Development’sfunds (FEDER), projects MTM2013-46962-C2-2-P and MTM2016-74983-C2-2-R, Universidadde Cádiz, PhD grant UCA/REC02VIT/2014. could be expressed as follows, min P ⊆ J | P | = p X s ∈S w s d ( I s , P ) . In this expression, w s is the weight related to stratum s , I s ⊆ I is the subsetof demand sites where stratum s is present and d ( I s , P ) = max i ∈ I s min j ∈ P d ij .In addition, note that for a given site i ∈ I , we will refer to min j ∈ P d ij asthe allocation distance of site i . Consequently, d ( I s , P ) is the maximumallocation distance among the sites with presence of stratum s .In this work, we propose different formulations to address the SpCP .One of them is based on the approach proposed in [1] for the classical pCP .Besides, several formulations based on covering variables are introduced.They are inspired in formulations for classical discrete location problemsas the ones appearing in [2] and [4].In addition, we use the formulations and improvements developed forthe
SpCP to obtain a heuristic approach for another extension of the pCP :the probabilistic p -center problem ( P pCP ), see [5]. This heuristic is basedon the Sample Average Approximation (SAA), which is usually applied indiscrete stochastic problems, see [3].
References [1] H. Calik and B.C. Tansel. Double bound method for solving the p -center loca-tion problem. Computers & Operations Research , 40:2991–2999, 2013.[2] S. García, M. Labbé, and A. Marín. Solving large p-median problems with aradius formulation.
INFORMS Journal on Computing , 23(4):546–556, 2011.[3] A.J. Kleywegt, A. Shapiro, and T. Homem-de-Mello. The sample average ap-proximation method for stochastic discrete optimization.
SIAM Journal on Op-timization , 12(2):479–502, 2002.[4] A. Marín, S. Nickel, J. Puerto, and S. Velten. A flexible model and efficientsolution strategies for discrete location problems.
Discrete Applied Mathematics ,157(5):1128–1145, 2009.[5] L.I. Martínez-Merino, M. Albareda-Sambola, and A.M. Rodríguez-Chía. Theprobabilistic p-center problem: Planning service for potential customers.
Euro-pean Journal of Operational Research , 262(2):509–520, 2017.X Workshop on Locational Analysis and Related Problems 2019 23
Non-dominated solutions for thebi-objective MST problem ∗ Lavinia Amorosi and Justo Puerto Dep. Statistical Sciences, Sapienza University of Rome, Italy, [email protected] IMUS, Universidad de Sevilla, Sevilla, Spain, [email protected]
This paper presents a new two phase algorithms for the computation of theentire set of non-dominated solutions of the bi-objective minimum span-ning tree problem.
1. The model
In this work we focus on a particularly appealing problem: the bi-objectiveminimum spanning tree (BMST) problem that has applications in differentcontexts [4]. For example, in the energy industry, for planning efficient dis-tribution systems or in the telecommunication sector. The BMST problemin its basic form can be formulated as follows.Let G = ( N, E ) be an undirected graph with node set N and edge set E .Let c and c be two different cost vectors on the edge set. The bi-objectiveminimum spanning tree problem is defined as: min Cx = min( X e ∈ E c e x e , X e ∈ E c e x e ) (1) s.t. X e ∈ E x e = n − (2) X e ∈ E ( S ) x e ≤| S | − ∀ S ⊆ N, S = ∅ (3) x e ∈ { , } ∀ e ∈ E (4) ∗ This research has been partially supported by Spanish Ministry of Economía and Competi-tividad/FEDER grants number MTM2016-74983-C02-01. In the multi-objective context, and thus in the bi-objective case, the feasibleset in the decision space (or decision set) X = { x ∈ R n : Ax = b, x ≥ } is distinguished from the feasible set in the objective space (or outcomeset) Y = { Cx : x ∈ X } , containing the points associated with the feasiblesolutions by means of the linear mapping defined by the problem criteria.Among the feasible solutions, we search for the ones which correspondto points in the outcome set for which it is not possible to improve onecomponent without deteriorating another one.The main contributions of this paper can be summarized as follows: 1)it provides a new approach to solve the bi-objective MST problem, basedon a generic two-phase algorithm applicable to many bi-objective combi-natorial optimization problems defined on graphs, 2) it gives a compari-son between two alternative methods and open source solvers adopted forimplementing the first phase: the dual variant of Benson’s algorithm, [3],by means of BENSOLVE [5] and the weighted sum method by means ofPolySCIP [2]; 3) it proposes a new enumerative recursive procedure basedon the analysis of reduced costs, first introduced in [1] for the bi-objectiveinteger min cost flow problem, able to generate all the spanning trees ofa connected graph and it reports extensive computational results obtainedtesting the algorithm on different problem instances, including completeand grid graphs. References [1] Amorosi L (2018)
Bi-criteria network optimization: problems and algorithms . Ph.D.thesis, PhD Program in Automatica, Bioengineering and Operations Research.[2] Borndörfer R, Schenker S, Skutella M, Strunk T (2016) Polyscip.
MathematicalSoftware – ICMS 2016, 5th International Conference, Berlin, Germany, July 11-14,2016, Proceedings
Annals of Operations Research
Vol. 147:343–360.[4] Fernández E, Pozo MA, Puerto J, Scozzari A (2017) Ordered weighted aver-age optimization in multiobjective spanning tree problem.
European Journal ofOperational Research http://bensolve.org .X Workshop on Locational Analysis and Related Problems 2019 25
A Kernel Search for the Inventory RoutingProblem
C. Archetti , G. Guastaroba , D.L. Huerta-Muñoz , and M.G. Speranza Department of Economics and ManagementUniversità degli Studi di Brescia. Brescia, Italy. [email protected], [email protected]@unibs.it, [email protected]
In this talk, we propose a Kernel Search heuristic to solve the InventoryRouting Problem (IRP). The idea behind Kernel Search is to iterativelysolve small restricted MILPs by selecting an initial subset of promisingvariables, called Kernel, and by adding sequentially subsets of the remain-ing variables, which are divided in groups of specific size, called Buckets.Preliminary results on a small set of benchmark instances show that the al-gorithm is able to improve, on average, the best-known solutions availablein the literature.
1. The Kernel Search for the IRP
The
Inventory Routing Problem [2] includes periodic demands, inventorymanagement, and delivering–scheduling decisions over a given time hori-zon. The objective is to determine the best distribution plan over the timehorizon to serve customers taking into account their consumption rate andinventory levels. The IRP variant we have focused this work takes into ac-count a finite time horizon, a single depot that serves the customers, a fleetof homogeneous vehicles, a maximum-level policy, and the prohibition ofstockouts.The main contribution of this work is the development of an effective solu-tion method, called
Kernel Search (KS), to solve the IRP. KS is a general pur-pose scheme that has been proposed for the solution of Mixed-Integer Lin-ear Programming (MILP) problems [4]. The idea is to identify a subset (or
Kernel ) of promising variables of the original problem and iteratively solve restricted MILPs by adding to this Kernel the remaining variables, whichare grouped in small subsets called Buckets . To the best of our knowledge,KS has never been applied to IRPs.Some preliminary experiments were run to analyze the performance ofthe KS in comparison with CPLEX and two state-of-the-art metaheuristics,M1 [1] and M2 [3], on a set of 72 benchmark IRP instances with a timehorizon of six periods. In Table 1, we can observe that the KS obtained, onaverage, better results in considerably shorter computing times in most ofthe cases. The numbers inside the parentheses correspond to the numberof solutions where KS outperforms the solution value found by the corre-sponding solution method.
Table 1.
Preliminary results of the KS performance.
Gap KS vs Time(s)Size
We expect these results can be significantly improved by carefully tuningthe KS parameters (size of the initial kernel and buckets), which are a cru-cial issue as quality of the solution and computing times are strongly re-lated to them.
References [1] Archetti C., Boland N. and Speranza M. G.
A Matheuristic for the MultivehicleInventory Routing Problem . INFORMS Journal on Computing, 29(3), pages 377–387, 2017.[2] Coelho L. C., Cordeau J.-F. and Laporte G.
Thirty Years of Inventory Routing .Transportation Science, 48(1), pages 1–19, 2013.[3] Chitsaz M., Cordeau J.-F. and Jans R.
A unified decomposition matheuristic forassembly, production and inventory routing . Submitted.[4] Guastaroba G., Savelsbergh M. and Speranza M. G.
Adaptive Kernel Search: Aheuristic for solving Mixed Integer linear Programs . European Journal of Opera-tional Research, 263(3), pages 789–804, 2017.X Workshop on Locational Analysis and Related Problems 2019 27
Time dependent continuous optimisationin solar power tower plants
Thomas Ashley, Emilio Carrizosa, and Enrique Fernández-Cara Instituto de Matemáticas de la Universidad de Sevilla, Spain, [email protected] Instituto de Matemáticas de la Universidad de Sevilla, Spain, [email protected] Dep. EDAN and IMUS, Universidad de Sevilla, Spain, [email protected]
Research into renewable energy sources has continued to increase in re-cent years, and in particular the research and application of solar energysystems. Concentrated Solar Power (CSP) used by a Solar Power Tower(SPT) plant is one technology that continues to be a promising researchtopic for advancement.The chosen aiming point for the heliostats on the receiver surface willhave an effect on the production of energy, as well as an effect on the life-time of the materials used in the receiver surface, due to thermal stresses.Therefore, the aiming strategy used by an SPT plant is of importance whenseeking to achieve the optimal energy production, whilst minimising riskof damage to components.The aiming strategy used in recent research into the optimisation of SPTsassumes that all heliostats in the field aim at the centre of the receiver, see[5]. This assumption allows for easier computation of the flux distributionacross the receiver surface and reduces complexity of the adjustment of theheliostats. Some research has been conducted where more complex aimingstrategies are considered for different receiver types [4].In previous research [2] the authors considered the optimal aiming strat-egy for a SPT plant, assuming a fixed grid of aim points on the receiversurface, with run times low enough to allow for near real-time updates tothe aiming strategy over time. This was extended in [3] to consider a con-tinuous optimisation technique, whereby the aiming strategy for a partic-ular time point was optimised without restricting the location or numberof aiming points on the receiver. In this work, the method from [3] is extended to consider the optimalaiming strategy across time, using a dynamic optimisation algorithm withan objective function of the form:Maximise J ( p ) = Z T G ( t, p ( t )) dt (1)Subject to p ∈ P ad where P ad is the subset of a Hilbert space P definedby time dependent inequality constraints of the form: p ( t ) ∈ R a.e.p ∈ Q, M ( p ) ≤ e. (2)Here, R is a bounded and closed convex set in an Euclidean space, Q ⊂ P is a second Hilbert space, M : Q E is a regular mapping with valuesin the Euclidean space E and e ∈ E .In this work, we discuss the existence of solutions and their optimal-ity conditions and develop two possible algorithms to solve the problem.These algorithms are implemented in Python, and their functionality demon-strated with numerical examples for the SPT plant in Sanlucar la Mayor,Seville [1]. References [1] Abengoa. Abengoa PS10 SPT Plant. [Date Accessed: 22/02/2017], 2017.[2] T. Ashley, E. Carrizosa, and E. Fernández-Cara. Optimisation of aiming strate-gies in Solar Power Tower plants.
Energy , 137, 2017.[3] T. Ashley, E. Carrizosa, and E. Fernández-Cara. Continuous OptimisationTechniques for Optimal Aiming Strategies in Solar Power Tower Plants (Sub-mitted). 2018.[4] Marco Astolfi, Marco Binotti, Simone Mazzola, Luca Zanellato, and GiampaoloManzolini. Heliostat aiming point optimization for external tower receiver.
Solar Energy , pages 1–16, 2016.[5] E. Carrizosa, C. Domínguez-Bravo, E. Fernández-Cara, and M. Quero. Aheuristic method for simultaneous tower and pattern-free field optimizationon solar power systems.
Computers & Operations Research , 57:109–122, 2015.X Workshop on Locational Analysis and Related Problems 2019 29
Minmax Regret Maximal Covering onNetworks with Edge Demands ∗ Marta Baldomero-Naranjo, Jörg Kalcsics, and Antonio M.Rodríguez-Chía Departamento de Estadística e Investigación Operativa, Universidad de Cádiz, Spain, [email protected] [email protected] School of Mathematics, University of Edinburgh, United Kingdom, [email protected]
In this work, we focus our research on covering location problems. Al-most all models analyzed in the literature assume that demand only oc-curs at the nodes of the network. However, there are some applicationswhere this assumption is not realistic; e.g. the location of emergency fa-cilities where the coverage areas are extremely distance-dependent. Thus,assuming that the demand is concentrated at nodes may lead to gaps inservice levels that are not acceptable in some situations, rendering the solu-tions useless. Hence, our goal is to solve the single-facility location problemtrying to cover the maximum demand on a network where the demand isdistributed along the edges.In the literature, see [1], some models are proposed to solve the deter-ministic version of this problem. Nevertheless, one of the big challengesis that the demand for a specific service is often not known exactly, butonly approximately. Hence, we have to find locations for those facilitiesthat provide an adequate level of service even under changing and un-known service demands. For this reason, we will treat demands as beingunknown. However, we usually have a good idea of what the minimalor maximal demand will be, so that we can at least assume demand to liewithin a known range. In the face of this situation of total uncertainty in thedemand, we propose to employ concepts from robust optimization, more ∗ Thanks to the support of Agencia Estatal de Investigación (AEI) and the European RegionalDevelopment’s funds (FEDER): project MTM2016-74983-C2-2-R, Universidad de Cádiz: PhDgrant UCA/REC01VI/2017, Telefónica and the BritishSpanish Society Grant. concretely minimizing the maximal regret, a well-known criterion used bymany researchers, see e.g. [2].Our first aim is to provide mathematical models considering that de-mand is uncertain and distributed along the edges of a network and thatthe service facilities can, essentially, be located anywhere along the net-work. Furthermore, we will propose polynomial time algorithms for find-ing the location that minimizes the maximal regret assuming that the de-mand lies within a known range and it is constant or linear on each edge. References [1] O. Berman, J. Kalcsics, and D. Krass. On covering location problems on net-works with edge demand.
Computers & Operations Research , 74:214–227, 2016.[2] J. Puerto, A. M. Rodríguez-Chía, and A. Tamir. Minimax regret single-facilityordered median location problems on networks.
INFORMS Journal on Comput-ing , 21(1):77–87, 2009.X Workshop on Locational Analysis and Related Problems 2019 31
The impact of pharmacy deregulationprocess on market competition andusers’ accessibility. Insights from twoSpanish case studies.
Ilaria Barbarisi, Giuseppe Bruno, Manuel Cavola, Antonio Diglio, Javier Elizalde Blasco, and Carmela Piccolo Department of Industrial Engineering (DII), Università di Napoli Federico IIP.le Tecchio, 80 - 80125, Napoli, Italy [email protected], (giuseppe.bruno, manuel.cavola, antonio.diglio, carmela.piccolo)@unina.it Facultad de Ciencias Económicas y Empresariales, Universidad de Navarra, Spain [email protected]
Most European countries adopt regulations of the retail pharmacy mar-ket with the aim of guaranteeing some objectives in terms of accessibility,equity, efficiency and quality of services. In this regard, one critical aspectconcerns the conditions for opening new pharmacies in a given area [1].These conditions typically combine demographic (e.g. maximum number ofpharmacies per inhabitant) and geographic (e.g. minimum distance amongpharmacies) criteria, in order to ensure accessibility to medicine productsfor the entire population while preserving adequate market niches to thepharmacists. In the last years, many countries introduced policies aimed atpromoting the competition in this sector. In particular, the restrictions forthe release of licenses for new openings were progressively relaxed witha consequent increase in the number of opened pharmacies. In Spain, theDecree-Law 11/1996 established new threshold values valid at nationallevel (i.e., minimum distance of 250 meters among pharmacies and onepharmacy per 2,800 inhabitants) but it transferred the right to the Au-tonomous Communities to modify such rules in order to better take intoaccount the specificities of their competence areas. Some communities, likeCatalunia, just implemented the indications fixed at national level whilesome others further relaxed the above values. The most relevant deregu- lation episode took place in 2000 in the region of Navarre, as the spatialand demographic criteria were respectively reduced to 150 meters and toone pharmacy per 700 inhabitants. These changes induced a dramatic en-try process, almost doubling the overall number of pharmacies. [2].In this context, the first aim of the present work is to evaluate the effectsproduced by this phenomenon, in terms of users’ accessibility and canni-balization of potential customers among pharmacies. To this end, we se-lected two case studies, i.e. two cities belonging to different AutonomousCommunities, and we performed an in-depth spatial analysis with the sup-port of Geographic Information Systems (GIS) to represent demand points,facilities and to study their interaction.Our analysis shows that the deregulation process produced effects thatshould be better addressed. On one side, the location of new pharmaciesproduced an overall increase of accessibility, but it has not contributed tomake the access more equitable, as it has not improved the condition of theleast well served users. On the other side, the cannibalization effect pro-duced unbalanced situations, in which old pharmacies have not been ableto maintain an adequate market niche. In this context, policy-makers arerecommended to take actions to ensure equitable accessibility [3] and sus-tainable competition in a more deregulated environment [4]. To this end,more effective regulation mechanisms should be defined. We propose amathematical model, aimed at generating alternative scenarios, with theobjective of providing users with more equitable accessibility conditionsto the service and, at same time, of mitigating the cannibalization effectamong drugstores. The model is tested on the two selected real case stud-ies. Obtained results show that the model is able to produce good scenar-ios, that can be evaluated by the Local Authorities to guide an informed-process for the definition of alternative regulation mechanisms. References [1] Vogler, S., Habimana, K., Arts, D. (2014).
Does deregulation in community phar-macy impact accessibility of medicines, quality of pharmacy services and costs? Evi-dence from nine European countries . Health policy, 117(3), 311-327.[2] Elizalde, J., Kinateder, M., Rodríguez-Carreño, I. (2015).
Entry regulation, firm’sbehaviour and social welfare . European Journal of Law and Economics, 40(1), 13-31[3] Barbati, M., Piccolo, C. (2016).
Equality measures properties for location problems.
Optimization Letters, 10(5), 903-920.[4] Plastria, F. (2001).
Static competitive facility location: an overview of optimisationapproaches . European Journal of Operational Research, 129(3), 461-470.X Workshop on Locational Analysis and Related Problems 2019 33
The selective traveling salesman problemwith time-dependent profits ∗ Eva Barrena , David Canca , Leandro C. Coelho and Gilbert Laporte University Pablo de Olavide, Seville, Spain [email protected] University of Seville, Seville, Spain [email protected] Université Laval, Québec, Canada,
[email protected] Canada Research Chair in Distribution Management and HEC Montréal, Montréal, Canada, [email protected]
Based on the definition of the selective traveling salesman problem (STSP),we define and analyze the selective travelling salesman problem with time-dependent profits (STSP-TDP). Given a weighted graph with time-dependentprofits associated with the vertices, the STSP-TDP consists of selecting asimple circuit of maximal total profit, whose length does not exceed a pre-specified bound and whose starting and ending time must lie within a pre-specified planning horizon. The length of the planning horizon is biggerthan the length of the circuit, thus being the starting and ending times ofthe circuit variables of the problem. This problem arises for example inthe planning of tourist itineraries and in the collection of letters from mail-boxes. We analyze several variants of the problem depending on the shapeof the time-dependent profit functions. If these functions are not monotone,it may be worth visiting a site more than once. We propose a formulationfor the case of multiple visits which reduces the problem to an STSP. Wealso propose three mathematical formulations for the single-visit case andcompute optimal solutions for some benchmark instances. ∗ This research work was partially supported Ministerio de Economía y Competitividad(Spain)/FEDER under grant MTM2015-67706-P and by the Natural Sciences and Engineer-ing Research Council of Canada (NSERC) under grant 2015-06189
X Workshop on Locational Analysis and Related Problems 2019 35
Solidarity behavior for optimizing thewaste selective collection
Eva Barrena, David Canca, Francisco A. Ortega and Ramón Piedra-de-la-Cuadra Universidad Pablo de Olavide, Sevilla, Spain, [email protected] Universidad de Sevilla, Sevilla, Spain, [email protected] Universidad de Sevilla, Sevilla, Spain, [email protected] Universidad de Sevilla, Sevilla, Spain, [email protected]
The problem of managing selective collection of waste within containersinside historic centers can be performed in three sequential phases: first,the location of containers along the streets; then, the determination of theminimum fleet size required to perform all collecting services; and finally,a model devoted to identify the optimal routes, in terms of total and equi-librated number of kilometers travelled by the trucks, is required. Obvi-ously, the result of the first phase (location of the containers) highly influ-ences the procedure since this will determine the decision to be taken forthe subsequent phases (route of collection vehicles and service program-ming).The main contribution of this paper focuses on this first phase: the loca-tion of collecting facilities (waste containers), where facility-customer dis-tances must be considered in the collecting design system, as well as otherconsiderations such as the size of container groups, their capacities in ac-cordance with the closest population and the installation cost of those con-tainers in specific sites along the streets.On the other hand, we assume that customers are willing to have a sol-idarity behavior when they bring their trash bags. This behavior consistsof using the container assigned to them within a pre-established proximityradius, although that container is not necessarily the closest to their placeof residence. In this scenario, we show that a more efficient distribution ofthe containers can be obtained. The proposed methodology for the deployment of containers for selec-tive collection of urban solid waste can be identified as a version of thePartial Set Covering problem, whose computational complexity motivatesthe use of heuristics to face large real-life scenarios. Following that rec-ommendation, a greedy algorithm of overflowing deviated to the immedi-ate neighborhood has been developed to solve the proposed mathematicalprogramming model.To illustrate the performance of the developed methodology, a compu-tational experience has been carried out on a network with randomizeddata inspired in a zone belonging to city of Seville (Spain).
X Workshop on Locational Analysis and Related Problems 2019 37
Dealing with Symmetry in a Multi-periodSales Districting Problem
Matthias Bender , Jörg Kalcsics , Anne Meyer and Martin Pouls Department of Logistics and Supply Chain Optimization, Research Center for InformationTechnology (FZI), Haid-und-Neu-Str. 10–14, 76131 Karlsruhe, Germany School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, TheKings Buildings, Edinburgh, EH9 3FD, Scotland, United Kingdom Faculty of Mechanical Engineering, TU Dortmund University, Leonhard-Euler-Straße 5,44227 Dortmund, Germany
In sales districting, the task is to assign a given set of customer accounts,each with a fixed market potential, to the individual members of a salesforce such that each customer has a unique representative, each sales per-son faces an equitable workload and has an equal income opportunity, andtravel times are minimal. Concerning the latter, if a sales person visits eachcustomer every day, then the travel time is proportional to the length of aTSP tour. However, the workload of districts is usually balanced over sev-eral weeks and some customers may have to be visited only once duringthis time whereas others require weekly service. Moreover, customers mayhave time windows, tours may include overnight stays, and so on, whichprohibits the computation of the actual travel times for practical problemsizes. Hence, in most cases one has to rely on estimates. The most commonestimate is to compute either the sum of distances between a sales person’slocation and his assigned customers or the sum of pairwise distances be-tween all customers assigned to the sales person.One important, but only recently addressed aspect of sales districting isthat customers often require service with different frequencies. Some cus-tomers have to be visited weekly, while others require service only once permonth. As a result, planners not only have to design the districts, but alsoschedule visits to customers within the planning horizon. For example, ifthe planning horizon is divided into weeks and days, then we also haveto decide which customers should be visited in which week and on whichday of that week. This introduces a scheduling component to the district- ing problem. The criteria for scheduling customer visits are very similarto the ones for designing the sales territories. The total workload incurredby all customers served in each time period should be the same across allperiods and the set of all customers visited in the same time period shouldbe as compact as possible to reduce travel times during each period.In this talk, we review the mixed-integer programming formulation forthe problem that was derived in [1, 2]). Unfortunately, only very small in-stances can be solved to optimality within a reasonable amount of timewith this formulation. One of the main factors contributing to that is, ap-parently, the high amount of symmetry that comes from the schedulingcomponent. Therefore, we will present in this talk a characterisation of(some of) the symmetries arising in the problem and try to find some tech-niques to counter them. All findings will be supported by numerical tests. References [1] M. Bender, A. Meyer, J. Kalcsics, S. Nickel. The multi-period service territorydesign problem – An introduction, a model and a heuristic approach.
Trans-portation Research E , 96:135–57, 2016[2] M. Bender, J. Kalcsics, S. Nickel, M. Pouls. A branch-and-price algorithm forthe scheduling of customer visits in the context of multi-period service territorydesign.
European Journal of Opererational Research , 269:382–396, 2018X Workshop on Locational Analysis and Related Problems 2019 39
A Mixed Integer Linear Formulation forthe Maximum Covering Location Problemwith Ellipses
Víctor Blanco and Sergio García Facultad de Ciencias Económicas y Empresariales, Universidad de Granada, Spain, [email protected] School of Mathematics, University of Edinburgh, United Kingdom, [email protected]
In a covering location problem, there is a set of demand points and thereis a set of potential sites for locating facilities. A point can be covered bya facility only if it is within a certain distance from this facility. Coveringlocation problems have many applications in different areas like locationof emergency services, analysis or markets, nature reserve selection, etc. Inthe Maximal Covering Location Problem (MCLP) introduced in [1], a fixednumber of facilities must be located so that the amount of covered demandis maximized. As Euclidean distances on the plane are used in the MCLP,the geometric shape used to cover the demand point is a circle.A much less studied variant of this problem is to use not circles butellipses to cover the points, which has applications to wireless telecommu-nications networks as shown in [2]. There is a finite set of demand pointsand there is a finite catalogue of ellipses. The centers of these ellipses canbe located anywhere on the plane. There are profits for serving the de-mand points and costs for locating the ellipses. The goal is to maximizethe net profit by not using more than a certain number of ellipses givenbeforehand. As the formulation that introduced in [2] is nonlinear, the au-thors propose a simulated annealing heuristic that can solve very smallinstances. This method is outperformed by the exact and heuristic algo-rithms proposed in [3], although their formulation is still nonlinear. Ourcontribution in this paper is to use some geometric properties of this prob-lem to give for the first time a mixed integer linear formulation that can solver much larger instances much more efficiently, as it will be shownwith a computational study. References [1] Church, Richard L and ReVelle, Charles (1974). The maximal covering locationproblem. Papers of the Regional Science Association 32(1):101–118[2] Canbolat, Mustafa S and von Massow, Michael (2009). Planar maximal cover-ing with ellipses. Computers & Industrial Engineering, 57(1), 201-208.[3] Andretta, Marina and Birgin, Ernesto G. (2013). Deterministic and stochasticglobal optimization techniques for planar covering with ellipses problems. Eu-ropean Journal of Operational Research, 224(1), 23-40.X Workshop on Locational Analysis and Related Problems 2019 41
Locating Hyperplanes for MulticlassClassification
Víctor Blanco, Alberto Japón, and Justo Puerto IEMath-Granada, Universidad de Granada, Granada, Spain [email protected] IMUS, Universidad de Sevilla, Sevilla, Spain [email protected] [email protected]
In this work we present a novel approach to construct multiclass clasiffiersby means of arrangements of hyperplanes. We propose different mixedinteger non linear programming formulations for the problem by usingextensions of widely measures for misclassifying observations.
General description
Given a training sample { ( x , y ) , . . . , ( x n , y n ) } ⊆ R p × { , . . . , k } the goalof supervised classification is to find a separation rule to assign labels ( y ) to data ( x ) , in order to be applied out of sample. We assume that a givennumber of linear separators (hyperpanes in R p ) have to be built to obtaina partition of the space into polyhedral cells. Each of the subdivisions ob-tained with such an arrangement of hyperplanes will be then assigned toa label in { , . . . , k } , see [5]. In Figure 1, where colors represent the dif-ferent labels of ( y ) , we can see two examples with 5 hyperplanes parti-tioning R reaching a perfect classification. The formulations are based onthe Support Vector Machines paradigm in which a maximum separationbetween classes is desired and in which different measures for the misclas-sifying errors are considered. Also, for the sake of solving larger instances,different strategies are proposed for the dimensionality reduction of theMINLP problems. We have run a series of experiments over some wellknown multiclass datasets from UCI machine learning repository [4]. Inthose we have tried four different versions of our model, using hinge-lossand ramp-loss measures for evaluating errors, and combining these withthe ℓ and ℓ norms for measuring distances. We compare the results ob- tained with some of the most popular multiclass SVM techniques: One VsOne [1], Weston-Watkins [2], and Crammer-Singer [3]. Figure 1.
References [1] Cortes, C., Vapnik, V.: Support-vector networks. Machine learning (3), 273–297 (1995)[2] Weston, J., Watkins, C.: Support vector machines for multi-class pattern recog-nition. In: European Symposium on Artificial Neural Networks, pp. 219–224(1999)[3] Crammer, K., Singer, Y.: On the algorithmic implementation of multiclasskernel-based vector machines. Journal of Machine Learning Research , 265–292 (2001)[4] Lichman, M.: UCI machine learning repository (2013). http://archive.ics.uci.edu/ml [5] Blanco, V., Japón, A. and Puerto, J. (2018). Optimal arrangements of hyper-planes for multiclass classification. Preprint available at https://arxiv.org/abs/1810.09167 .X Workshop on Locational Analysis and Related Problems 2019 43 Minimum covering polyellipses
Víctor Blanco and Justo Puerto IEMath-GR, Universidad de Granada, Spain, [email protected] IMUS, Universidad de Sevilla, Spain, [email protected]
In this work we study different continuous location problems usingpolyellipses. These problems allow to model situations in which one orseveral facilities are to be located in which the transportation costs are com-puted by means of the sums of distances to a set of sub-facilities.
1. Introduction
Given a sets of demand points on the plane, Continuous Facility LocationProblems (CFLP) deal with the determination of optimal positions by min-imizing certain measures of the distances to the points. The most popularCFLP is the Weber problem [5], in which a finite set of demands pointsis provided,
U ⊆ R , and a point x ∈ R is to be located that minimizesthe function Φ( x ) = P u ∈U ω i k x − u k , for some weights ω , . . . , ω n (here k · k stands for the Euclidean norm). If the data points are not collinear, Φ is strictly convex, and therefore has a unique optimum, which can be ob-tained with the Weiszfeld’s Algorithm [6]. The levels curves of Φ are givenby the following sets: E r ( U ) = n x ∈ R : X u ∈U ω j k x − u k = r o for a radius r ≥ . The set E r ( U ) is called a polyellipse with foci U andradius r . The region bounded by E r ( U ) is clearly a nonempty convex setprovided that r is greater than the optimal value of the Weber problem.These convex bodies have been partially analyzed from geometric or al-grebraic viewpoints (see [1, 3, 4])In this work, we analyze full covering problems by using these sets. Onthe one hand, we analyze covering problems, in which the goal is to find, for a given set of foci, U the smallest radius r for which the polyellipse E r ( U ) contains a set of given demand points, A . A direct implication ofthis problem in facility location consists of the placement of the facilitiesin U (whose relative positions on the plane are given) such that the max-imum sum of the distances from a demand point to the facilities is mini-mized. The problem extends the classical planar center problem [2, 7] sincepolyellipses with a single foci are circles. Several approaches are providedto efficiently solve the problem: a Second Order Cone Programming for-mulation, a primal-dual approach, and a Elzinga-Hearn based algorithm.On the other hand, instead of providing the set of foci, U , we propose amodel and solution approaches to select, among the set of demand points,the relative positions of the foci to adequately cover the set of points. InFigure 1 we show the solutions of minimum radius polyellipses for differ-ent number of foci for the same dataset. Figure 1.
Full Covering Polyellipses for different number of foci.
References [1] Erdös, P., & Vincze, I. (1982). On the Approximation of Convex, Closed PlaneCurves by Multifocal Ellipses. J. App. Probability, 19, 89-96.[2] Elzinga, D. & Hearn, D. (1972) Geometric solutions for some minimax locationproblems. Transp Sci 6:379–394[3] Maxwell, J.C. (1846). Paper on the Description of Oval Curves. The ScientificLetters and Papers of James Clerk Maxwell. 1846-1862. 23rd IEEE Sympos.Found. Comput. Sci 329–338.[4] Nie, J., Parrilo, P.A., & Sturmfels, B. (2008) Semidefinite Representation of thek-Ellipse. IMA Volumes in Math. and its App. 146.[5] Weber, A. (1957). Theory of the location of industries , Univ. Chicago Press.[6] Weiszfeld, E. (1937). Sur le point lequel la somme des distances de n pointsdonnés est minimum. Tôhoku Math. J. 43, 355–386[7] Welz, E. (1991). Smallest enclosing disks (balls and ellipsoids). Lecture NotesComput. Sci. 555, 359–370.X Workshop on Locational Analysis and Related Problems 2019 45 The One-Round Voronoi GamePlayed on the Rectilinear Plane
Thomas Byrne, Sándor P. Fekete, and Jörg Kalcsics University of Edinburgh, School of Mathematics, James Clerk Maxwell Building,King’s Buildings, Edinburgh, EH9 3FD, United Kingdom [email protected] | [email protected] TU Braunschweig, Mühlenpfordtstraße 23, 38106 Braunschweig, Germany [email protected]
Location is undoubtedly one of the most important issues when de-termining the success or failure of an operation. The distance between aproposed facility placement and its potential customer sites is perhaps themost natural way to discern the value of this position, and the need foreffective facility locations becomes vital in competitive situations whereincustomers will be gained or lost depending on whichever facility is closest.This importance of good location strategies is epitomised in the Voronoigame, a simple geometric model proposed in [1].We consider this competitive facility location problem with two players.Players alternate placing points into the playing arena, until each of themhas placed n points. The arena is then subdivided according to the nearest-neighbour rule, and the player whose points control the larger area wins.While some literature on this problem exists, there is a noticeable ab-sence in the presentation of the game using the l norm. A winning strat-egy for the second player, where the arena is a circle or a line segment, ispresented in [1] for both variations where players can play more than onepoint at a time. There it is shown that the first player can ensure that thesecond player wins by an arbitrarily small margin. Optimal strategies forboth players were found for a rectangular arena with Euclidean distancein [2] and it was ascertained that the particular values of n and the aspectratio of the arena determine which player wins. We start with a definition of the game. There are two players, White andBlack, each having n points to play, where n > . The players alternateplacing points on a rectangular playing area P . As in chess, White startsthe game, placing their first batch of points within P , and Black the secondbatch of points, White the third batch, etc., until all n points are played.We assume that points cannot lie upon each other. Let W be the set of whitepoints and B be the set of black ones. After all of the n points have beenplayed, the arena is partitioned into the Voronoi diagram of W ∩ B using the l metric and each player receives a score equal to the area of the Voronoicells of their points, or rather their total market share.The question we ask is what is each player’s best strategy? We answer this question in the one-round game, determining whetherit is still chivalrous to play last, or is first the worst, second the best.
References [1] Ahn, H.-K., Cheng, S.-W., Cheong, O., Golin, M., and van Oostrum, R.. “Com-petitive facility location: the Voronoi game,” Theoretical Computer Science 310(2004) 457–467.[2] Fekete, S. P., and Meijer, H.. “The one-round Voronoi game replayed,” Compu-tational Geometry 30 (2005) 81–94.X Workshop on Locational Analysis and Related Problems 2019 47
Drone Arc Routing Problems
James F. Campbell, Ángel Corberán, Isaac Plana and José M.Sanchis University of Missouri-St. Louis, USA, [email protected] Universitat de València, Spain, [email protected] Universitat de València, Spain, [email protected] Universidad Politécnica de Valencia, Spain, [email protected]
In this talk we present some drone arc routing problems (Drone ARPs) andstudy their relation with well-known postman arc routing problems. Ap-plications for Drone ARPs include traffic monitoring by flying over road-ways, infrastructure inspection such as by flying along power transmis-sion lines, pipelines or fences, and surveillance along linear features suchas coastlines or territorial borders. Unlike the postmen in traditional arcrouting problems, drones can travel directly between any two points inthe plane without following the edges of the network. As a consequence, adrone route may service only part of an edge, with multiple routes beingused to cover the entire edge. Thus the Drone ARPs are continuous opti-mization problems with an infinite number of feasible solutions. In orderto solve them as a discrete optimization problem, we approximate eachcurve in the plane by a polygonal chain, thus allowing the vehicle to en-ter and leave each curve only at the points of the polygonal chain. If thecapacity of the vehicles is unlimited, the resulting problem is a Rural Post-man Problem (RPP). We propose an algorithm that iteratively solves RPPinstances with an increasing number of points of the polygonal chain andpresent results on several sets of instances. We also briefly discuss the casein which the drones have limited capacity and several drones are needed.
X Workshop on Locational Analysis and Related Problems 2019 49
The Railway Rapid Transit NetworkConstruction Scheduling Problem
David Canca, Alicia de los Santos, Gilbert Laporte and Juan A.Mesa Department of Industrial Engineering and Management Science, University of Seville,Spain, [email protected] Department of Statistics, Econometrics, Operational Research, Management Science andApplied Economics, University of Cordoba, Spain, [email protected] CIRRELT and HEC Montréal, Montréal, Canada, [email protected] Department of Applied Mathematics II, University of Seville, Spain, [email protected]
In this work we face up the problem of scheduling the different activitiesconcerning the construction of a railway rapid transit transportation net-work. Supposing the network topology has been determined in an earlystage, the problem consists on defining the best sequence of constructiontasks in order to maximize the long term profit of the project. The prob-lem can be viewed as a resource-constrained project scheduling problem,where both, construction budget and tunnel boring machines act as re-sources influencing the schedule. Since lines can be put into operation asthey are finished, both, costs and revenues are dependent on the definedschedule.
1. Introduction
The Resource-Constrained Project Scheduling Problem (RCPSP) is identi-fied as the determination of the time required to implement the activitiesof a project to achieve a certain objective. It was assumed that the activitiesof a project are described by their processing times. Activities are relatedtrough precedence relationships, which are commonly represented by setsof immediate predecessors. A certain amount of resources is required for each activity to be performed. It is assumed that, for each resource, a con-stant amount is available before the start of each period. The aim is to findthe start time for all the activities accordingly to different objective func-tions: The minimization of the completion time [1], [5], the maximizationof the net present value [2], [3] or the penalization of earliness-tardiness ofthe total completion time [4].
2. Contributions
The problem we face here presents some important differences with re-spect to the RCPSP. First, since transportation services can be put into op-eration as part of the lines are being finalized, the revenue is obtained asthe project is being executed. A second difference is that there is not prece-dence constraints, instead, the temporal construction project is governedby connectivity constraints since, in normal circumstances, the transporta-tion agency is interested in developing a connected network, allowing topassengers the possibility of transferring among lines. Finally, there is nota predefined initial task (or a set of initial tasks). The construction orderwill be a consequence of the city transportation demand patterns, whichcan vary along the time.
References [1] Creemers, S. (2015). “Minimizing the expected makespan of a project withstochastic activity durations under resource constraints,” Journal of Schedul-ing, 18(3):263–273.[2] Khalili, S., Najafi, A. A., and Niaki, S. T. A. (2013). “ Bi-objective resource con-strained project scheduling problem with makespan and net present value cri-teria: two meta-heuristic algorithms, ”. The International Journal of AdvancedManufacturing Technology, 69(1-4):617–626.[3] Leyman, P. and Vanhoucke, M. (2017). “Capital and resource-constrainedproject scheduling with net present value optimization,” European Journal ofOperational Research, 256(3):757–776.[4] Rajeev, S., Kurian, S., and Paul, B. (2015). “A modified serial scheduling schemefor resource constrained project scheduling weighted earliness tardiness prob-lem,” International Journal of Information and Decision Sciences, 7(3):241–254.[5] Xiao, J., Wu, Z., Hong, X. X., Tang, J. C., and Tang, Y. (2016). “Integration of elec-tromagnetism with multi-objective evolutionary algorithms for RCPSP,” Euro-pean Journal of Operational Research, 251(1):22–35.X Workshop on Locational Analysis and Related Problems 2019 51
Heuristic Framework to ReduceAggregation Error on Large ClassicalLocation Models
Carolina Castañeda P. and Daniel Serra Department of Economics and Business, Universitat Pompeu Fabra, Barcelona, Spain, [email protected], [email protected]
Location analysis has a wide range of applications in the context of manyreal-world systems in public [2, 6] and private sectors [3], also it is a veryinteresting research field because its interaction with other disciplines [5,7]. Facility location problems lie at the core of location analysis and areconcerned with determining the location of a set of facilities satisfying oneor more objective functions and constraints, regarding the demand for theservice provided from the facilities.Solving large discrete location problems may be time consuming or in-tractable due to the presence of many demand points, which are usuallyaggregated, thereby inducing error in the solution. According to [4], ag-gregation decreases costs of data modeling, collection and computing butalso increases the errors incurred when solving location models to optimal-ity because these are approximated models where the solution is optimalfor the aggregated data but not necessarily for the non-aggregated data.Aggregation error was first formally defined by [1]. Based on this defini-tion the main classification of aggregation errors is the commonly knownas ABC type errors [4]. Type A error appears when the distance betweenan aggregated demand point, instead of a real point, and a facility is usedto solve a location problem. Type B error occurs when a facility is locatedat an aggregated demand point instead of a real point and type C errorhappens when a real demand point is assigned to the wrong facility.We propose a heuristic framework to reduce error type C caused byaggregated demand points in classical location models on networks. Ourframework integrates a solution method for large location problems withan algorithm to find a suitable demand aggregation for them. The framework contains four stages. In the first stage, we obtain an ini-tial aggregation of the original demand points through a heuristic basedon the k-means algorithm. In the second stage, we calculate the centroid ofeach group, using the concept of a centroid in a minimum expansion tree.These centroids become the candidate locations to be selected by a locationmodel solved in stage three. In the fourth stage, we evaluate the qualityof the solution, calculating the improvement in the objective function withthe current aggregation and then we repair the solution considering theaggregation error measure.In this work we are avoiding type B error because facilities are locatedin the centroids that are real demand points. However, errors A and Care present. For measuring type C error, after finding the facility locationsamong centroids, we measure the dispersion of real demand points respectto the assigned facility, those points that have the largest dispersion are as-signed to other facility that decrease the value of the total dispersion. ErrorA is not addressed in the current version of the framework.The framework follows the four stages iteratively. After the first itera-tion, in the aggregation algorithm, we use a Greedy Randomized Adap-tive Search Procedure (GRASP) in order to obtain new and diverse ag-gregated demand configurations. The algorithm stops when the solutionquality does not improve marginally.
References [1] Hillsman, E. L., and Rhoda, R. (1978). Errors in measuring distances from pop-ulations to service centers. The Annals of Regional Science, 12(3), 74–88.[2] Marianov, V., and Serra, D. (2004). Location Problems in the Public Sector. InZ. Drezner and H. W. Hamacher (Eds.), Facility Location: Applications andTheory (pp. 119–150). Springer Science and Business Media.[3] Church, R. L., and Murray, A. T. (2008). Business Site Selection, Location Anal-ysis and GIS. John Wiley and Sons.[4] Francis, R. L., Lowe, T. J., Rayco, M. B., and Tamir, A. (2009). Aggregation errorfor location models: survey and analysis. Ann Oper Res, 167, 171–208.[5] Murray, A. T. (2010). Advances in location modeling: GIS linkages and contri-butions. Journal of Geographical Systems, 12(3), 335–354.[6] Marianov, V., and Serra, D. (2011). Location of Multiple-Server Common Ser-vice Centers or Facilities, for Minimizing General Congestion and Travel CostFunctions. International Regional Science Review, 34(3).[7] Laporte, G., and Nickel, S. (2015). Introduction to Location Science. In G. La-porte, S. Nickel, and F. Saldanha da Gama (Eds.), Location Science (pp. 1–21).Springer.X Workshop on Locational Analysis and Related Problems 2019 53
Rationalizing capacities in the facilitylocation problem
Ángel Corberán, Mercedes Landete, Juanjo Peiró and FranciscoSaldanha-da-Gama Departament d’Estadística i Investigació Operativa. Universitat de València, Spain, [email protected],[email protected] Departamento de Estadística, Matemáticas e Informática. Instituto Centro de Investi-gación Operativa. Universidad Miguel Hernández de Elche, Spain [email protected] Departamento de Estatística e Investigação Operacional. Centro de Matemática, Apli-cações Fundamentais e Investigação Operacional. Universidade de Lisboa, Portugal, [email protected]
The capacitated facility location problem is a core problem in LocationScience (see [3] for a survey of fixed-charge location problems). In this workwe consider a variant of this problem in which facilities may cooperate inorder to adapt their capacities to the demand of their customers. In par-ticular, we consider the situation in which there may be capacity transferbetween facilities. The existence of a potential flow between facilities leadsto a redefinition of the capacity of a facility, and the actual capacity of afacility results from the original one plus the amount received from otherfacilities minus the amount sent to other facilities. This problem has mul-tiple applications in real markets, see [1] and [2]. In this work we intro-duce the problem and give a mixed-integer linear formulation. Then, weenhance the model by using several families of valid inequalities. Some ofthe valid inequalities imitate classical valid inequalities of the capacitatedfacility location problem while some other are specific for this problem.Computational results illustrate the benefits of both models and their im-provements.
References [1] Correia, I. and Melo, T. (2016a). Multi-period capacitated facility location underdelayed demand satisfaction. European Journal of Operational Research , 255: 729–746.[2] Correia, I. and Melo, T. (2016b). A multi-period facility location problem withmodular capacity adjustments and flexible demand fulfilment
Computers & In-dustrial Engineering , 110: 307–321.[3] Fernández, E. and Landete, M. (2015). Fixed-charge facility location problems.In Laporte, G., Nickel, S., and Saldanha-da-Gama, F., editors,
Location Science,chapter 3 , pages 47–77. Springer.X Workshop on Locational Analysis and Related Problems 2019 55
Bilevel programming models formulti-product location problems ∗ Sebastián Dávila, Martine Labbé, Fernando Ordoñez, Frédéric Semet, and Vladimir Marianov, Department of Industrial Engineering , Universidad de Chile, Chile, [email protected] Computer Science Department, Université Libre de Bruxelles, Belgium and INRIA, Lille,France, [email protected] Department of Industrial Engineering , Universidad de Chile, Chile, [email protected] CRIStAL Centre de Recherche en Informatique Signal et Automatique de Lille, France,and INRIA, Lille, France, [email protected] Department of Electrical Engineering, Pontificia Universidad Católica de Chile, Santiago,Chile, [email protected]
We consider a retail firm that owns several malls with a known location.A particular product, e.g., food processor, comes in p types, which differby shapes, brands and features. The set of all p products is P . Each mall j has a limited capacity c j of products in P to be sold at that location, sothe firm has to choose what products to sold at what mall. Furthermore,the firm can apply discrete levels of discount on the products, e.g., 5% and10% over the price π k of product k . The objective of the firm is to find whatproducts to sell at which mall, with what level of discount, so that its profitis maximized.Consumers are located in points of the region. Each consumer or group ofconsumers i has a different set P i ⊆ P of acceptable products, and will ∗ Supported by INRIA Associated Team BIPLOS purchase one of these, or none if it is not convenient for her. Consumersmaximize their utility, defined as u ijkl = r ik − α jkl · π jk − d ij (1)where r ik is the maximum expenditure that customer i is willing to maketo acquire product k ; α jkl is (100 - discount level l in percent) of the prod-uct k in mall j ; π jk is the price of the product k in the mall j and d ij isthe distance between consumer i ’s origin and mall j . Whenever this utilityis negative for product k at all malls, the consumer does not purchase theproduct.The agents (firm and consumers) play a Stackelberg game, in which thefirm is the leader and the customers the follower. Once the firm decidesthe products to sell at each mall and the possible discounts, consumerspurchase (or not) one of their acceptable products wherever their utility ismaximized. We model the problem using first a bilevel formulation, andwe further replace the follower problem by the primal constraints and op-timality restrictions, to obtain a compact formulation. We also present astrong and a weak formulation. X Workshop on Locational Analysis and Related Problems 2019 57
The Urban Transit Network DesignProblem
Alicia De-Los-Santos, David Canca, Alfredo G. Hernández-Díaz and Eva Barrena Department of Statistics, Econometrics, Operational Research, Management Science andApplied Economics, University of Cordoba, Spain, [email protected] Department of Industrial Engineering and Management Science, University of Seville,Spain, [email protected] Deparment of Economic, Quantitative Methods and Economic History, University ofPablo de Olavide, Spain, [email protected] Deparment of Economic, Quantitative Methods and Economic History, University ofPablo de Olavide, Spain, [email protected]
In this work we consider the problem of simultaneously designing the in-frastructure of a urban bus transportation network and its set of lines whileminimizing the total travel time of all passenger willing to travel in thenetwork. As main differences with respect to other works in the bus trans-portation design field, we do not consider an a priori line pool, but wedesign the set of lines from square one, presenting a detailed descriptionof the travel time (which incorporates the time spent in transferring alongthe passengers paths) and we jointly determine the transit assignment ac-cordingly to the users’ minimum trip time.The transfer time plays an important role in the passenger decisionssince transfers represent discomfort for the passenger, i.e., an extra-timeto perform a trip. Most authors incorporate transfers into the computationof the travel time as a penalty term considering only the number of trans-fers. In this work we are interested in introducing a detailed description ofthe transfer time. To this end, we consider two layers: the first one affectingthe off-board passengers movement (pedestrian layer) and the second onecorresponding to the road infrastructure over which buses can run along(road-infrastructure layer). In a realistic way, we can distinguish two types of transfers: transfers atthe same stop and transfer between different stops. Obviously, the secondtype requires an extra-time to walk between stops over the pedestrian layerand therefore, a greater discomfort for passengers. We present a mathemat-ical programming model for solving the problem on the directed graphthat results when superimposing both layers.We illustrate the problem with some computational experiments overseveral network.
X Workshop on Locational Analysis and Related Problems 2019 59
Minimum distance regulation and entrydeterrence through location decisions
Javier Elizalde, and Ignacio Rodríguez Carreño, University of Navarra, Pamplona, Spain, [email protected] University of Navarra, Pamplona, Spain, [email protected]
This paper analyses the location strategies and the resulting market struc-ture in a model of spatial competition, illustrating location in two dimen-sions, when there is a restriction of minimum distance between plants.Such a regulation exists in some retail markets, such a drugstores, aim-ing to avoid agglomeration and provide accessibility for all consumers,and may change the optimal location decisions of managers with the pur-pose of reducing the eventual number of competitors. The latter becomesendogenously determined by the size of the market and the distance ruleand we evaluate the welfare consequences of the firms’ location strategieswhen they take their decisions with the purpose of deterring additionalentry. We describe a theoretical model of spatial competition in two di-mensions and solve for the equilibria through algorithmic simulations. Theeventual number of active firms becomes endogenously determined by thesize of the market and the distance rule. In a sequential entry game, weobtain a location equilibrium for a wide range of the binding distance ruleand compare the equilibria reached under two types of firm behavior: witha simple maximum capture behavior and with entry deterrence strategies.We then discuss the results in terms of welfare in order to assess the effectof such regulatory policies and the distortions in firm’s location decisionsthat they imply. The main finding of the paper is that, with a minimumdistance constraint, location equilibrium exists for each level of minimumdistance which allows for two or more firms. Even though entry deterrenceactivities by incumbent firms tend reduce the level of consumers’ welfareas it tends to reduce the number of firms for some levels of minimum dis-tance, it may in some cases be welfare enhancing as it may lead to a more even distribution of firms in the plane reducing the distance travelled bythe average consumer. X Workshop on Locational Analysis and Related Problems 2019 61
A multi-period bilevel approach forstochastic equilibrium in networkexpansion planning under uncertainty
Laureano F. Escudero, Juan Francisco Monge, and Antonio M.Rodríguez-Chía Area de Estadística e Investigación Operativa, Universidad Rey Juan Carlos, Spain, [email protected] Centro de Investigación Operativa, Universidad Miguel Hernández, Spain, [email protected] Departamento de Estadística e Investigación Operativa, Universidad de Cádiz, Spain, [email protected]
This study focuses on the development of a mixed 0-1 bilinear model-ing stochastic framework for the multi-period network expansion planningproblem under uncertainty, where stochastic equilibrium-based strategicdecisions are to be made. The problem addressed here involves strategicdecisions to be taken for a stochastic equilibrium-based network expansionplanning problem (SE-NEP) in a multi-period time horizon under uncer-tainty, where it is required that the optimization models for the upper andlower levels have an equilibrium at the nodes of the scenario tree.Some important features distinguish this work from other ones in theliterature, they are as follows:Topological decisions are now considered as dynamic decisions taken atdifferent periods along a time horizon.Several sources of uncertainty are considered at the periods, namely, in-vestment cost for building network links, commodities volume to be trans-ported by using network links as well as alternative ones, and commodi-ties transport unit cost through alternative links, among others. The uncer-tainty is assumed to be captured by a finite set of scenarios.A multi-period SE-NEP is dealt with, where some extensions of the clas-sical static deterministic TAP (Toll Assignment Problem) are taken as a pi- lot case. On one hand, an upper level multi-period stochastic model forexpansion planning is considered for determining the network links in-vestment as well as the unit tariff for commodities transportation in orderto maximize the expected profit in the scenarios along the time horizon.On the other hand, a lower-level single-period deterministic model is con-sidered at each scenario node for minimizing the commodities transportcost in a mix of available network links and alternative ones. So, an equi-librium is sought for obtained on upper level profit and lower level costat each scenario node. That equilibrium is obtained via the optimization ofa single model. That type of modeling has been preferred to the also clas-sical KKT constraint system due to computational reasons baswed on themodel’s difficulty for problem solving.The upper level investment-oriented 0-1 step variables modeling objectsallow that the state variables in the model only link two consecutive peri-ods. A new feature of the problem that is considered consists of allowingupper bounded freedom for considering tariffs in the commodities trans-portation through the already available network links. As a consequencea related mixed 0-1 bilinear term is considered for each commodity trans-portation through the network links at each scenario node. Those terms areequivalently replaced with mixed 0-1 linear constraint systems.Given the huge problem’s dimensions (due to the network size of real-istic instances as well as the cardinality of the scenario tree), it is unrealisticto seek for an optimal solution. As an illustrative example, for an instancewith 20 network nodes, 12 network links, 25 links by other means, 4 pe-riods, 156 scenario nodes and 125 scenarios to represent the uncertainty,the mathematically equivalent mixed 0-1 deterministic model has the fol-lowing dimensions 38,1108 constraints, 84,396 continuous variables and156,624 0-1 variables. This fact motivates the development of a matheuris-tic algorithm based on a Nested Stochastic Decomposition methodologyfor determining the appropriate network expansion planning along thetime horizon, where a solution goodness quality is guaranteed. Some co-mutational results are shown.This research has been partially supported by the projects MTM2015-63710-P (L.F. Escudero), MTM2016-79765-P (Juan F. Monge) and MTM2016-74983-C2-2-R (Antonio Rodríguez-Chía) from the Spanish Ministry of Econ-omy, Industry and Competitiveness and the European Regional Develop-ment Fund (AEI/FEDER, UE). X Workshop on Locational Analysis and Related Problems 2019 63
Capacitated Discrete OrderedMedian Problems
Inmaculada Espejo, Justo Puerto, and Antonio M. Rodríguez-Chía Departamento de Estadística e Investigación Operativa, Universidad de Cádiz, Spain Departamento de Estadística e Investigación Operativa, Universidad de Sevilla, Spain
Flexible discrete location problems, or the so-called discrete ordered me-dian problems, have been widely studied in combinatorial optimization.In this paper we deal with the capacitated version of this problem. Differ-ent formulations of the capacitated discrete ordered median problem arepresented as well as some preprocessing phases for fixing variables. In ad-dition, some strategies to generate incomplete formulations and increasethe size of the instances that we are able to solve are also studied.
1. Introduction
Discrete location problems have been intensively studied over the last de-cades. Numerous surveys and textbooks give evidence of this fact (see[1, 2]). The need for location models that better fit different real worldsituations has made necessary to develop new and flexible location mod-els. Nickel in [5] proposed the Discrete Ordered Median Problem (DOMP)which is used to model different discrete locations problems. It is a flexi-ble formulation that introduce a type of objective function called orderedmedian function. This objective function is based on applying an orderedweighted averaging operator to the costs as they appear in the solutionand taking them into account with a suitable vector λ . Hence, handlingthe most important objective functions in Location Analysis is possiblewith one unique model and also new ones may be created by adaptingthe parameters λ adequately. Important references in location problemsare [6,9], which summarize and review modeling and solution approachespublished for continuous, network and discrete location problems. This paper deal with the capacitated discrete ordered median problem(CDOMP). Flexible models using capacity constraints can be found in [3,4, 7, 8]. In [8] provided formulations for the case of hubs. To the best of ourknowledge, the first paper dealing with the capacitated discrete orderedmedian problem is [7]. It considers a new formulation based on a coverageapproach and compare its performance with respect to previously knownformulations for the uncapacitated problem. However, it is able to solveonly small instances. In [4] describe three different points of view of a lo-cation problem in a logistics system. These models are extensions of thebasic DOMP but the demands can be split. [3] can be considered as a firstbuilding block in the analysis of capacitated strategic location problemswith order requirements. Here the number of facilities to be located is notgiven in advance. This is an important difference to the CDOMP, where thenumber of new facilities is fixed a priori.
References [1] Daskin, M.S. (1995)
Network and discrete location: Models, algorithms and applica-tions , Wiley Interscience, New York.[2] Mirchandani, P.B., and Francis, R.L. (1990).
Discrete Location Theory . Wiley, NewYork.[3] Kalcsics, J., Nickel, S., Puerto, J., and Rodríguez-Chía, A.M. (2010).
The orderedcapacitated facility location problem , TOP, 18(1): 203–222.[4] Kalcsics, J., Nickel, S., Puerto, J., and Rodríguez-Chía, A.M. (2010).
Distributionsystems design with role dependent objectives , European Journal of OperationalResearch, 202: 491–501.[5] Nickel, S. (2001).
Discrete ordered Weber problems . In: Fleischmann B, Lasch R,Derigs U, Domschke W, Rieder U, editors. Operations Research Proceedings2000. Berlin: Springer. 71–76.[6] Nickel, S., and Puerto, J. (2005).
Location Theory. A Unified Approach , Springer.[7] Puerto, J. (2008).
A New Formulation of the Capacitated Discrete Ordered MedianProblems with {0, 1} Assignment , In: Kalcsics J., Nickel S. (eds) Operations Re-search Proceedings 2007, Springer, Berlin, Heidelberg.[8] Puerto, J., Ramos, A.B., Rodríguez-Chía, A.M., and Sánchez-Gil, M.C.(2016).
Ordered median hub location problems with capacity constraints , Transportation Re-search Part C, 70: 142–156.[9] Puerto, J., and Rodríguez-Chía, A.M.(2015).
Ordered Median Location Problems .In: Laporte G., Nickel S., Saldanha da Gama F. (eds) Location Science. Springer,Cham.X Workshop on Locational Analysis and Related Problems 2019 65
A branch-and-price algorithm for theVehicle Routing Problem withStochastic Demands,Probabilistic Duration Constraint, andOptimal Restocking Policy ∗ Alexandre M. Florio , Richard F. Hartl , Stefan Minner , andJuan-José Salazar-González University of Vienna, 1090 Vienna, Austria Technical University of Munich, 80333 Munich, Germany Universidad de La Laguna, 38271 Tenerife, Spain [email protected]
When customers’ demands are stochastic, the duration of the routes arealso stochastic. We consider the vehicle routing problem with stochastic de-mands and probabilistic duration constraints where we must design a setof routes with minimal total expected cost, visiting all customers, and suchthat the duration of each route, with some high probability, does not ex-ceed some prescribed limit. We assume that the "Optimal Restocking Pol-icy" is applied, which means that, before starting a route, the drive is givenwith a sequence of customers to serve, and with threshold values to checkwhether the vehicle must continue directly to the next customer in the se-quence, or restock at the depot before. This is a more convenient and so-phisticated policy than the "detour-to-depot policy", commonly used in theliterature and where the driver is only given with the customer sequenceand must keep following the route until it failures or ends at the depot. Theproblem, without duration constraint, has been recently studied in [1–3].We solve the problem to optimality for the first time with a novel branch-and-price algorithm. An orienteering-based completion bound is proposedto control the growth of labels in the pricing algorithm. New procedures ∗ Research partially supported by MTM2015-63680-R (MINECO/FEDER, UE) are developed to keep track of the variance of the duration of a route. Thefeasibility of an a-priori route is verified either by applying Chebyshev’sbounds, or by Monte Carlo simulation and statistical inference. Consis-tency checks are incorporated into the branch-and-price framework to de-tect statistical errors. Computational experiments are performed with de-mands following binomial, Poisson, or negative binomial probability dis-tributions, and with duration constraints enforced at levels of 90%, 95%and 98%. The vehicle capacity is considered in the objective function toforce the vehicle going to the depot when convenient. Then, optimal so-lutions may contain a-priori routes that serve an expected demand largerthan the capacity of the vehicle. These solutions actively employ optimalrestocking to reduce traveling costs and the number of required vehicleswhen compared to the detour-to-depot policy solutions. Sensitivity anal-yses indicate that over-dispersed demands and strict duration constraintsnegatively impact the solution, both in terms of total expected cost andnumber of routes employed. References [1] Florio A.M., Hartl R.F., Minner S. (2018) “New exact algorithm and solutionproperties for the vehicle routing problem with stochastic demands”. Technicalreport, University of Vienna.[2] Louveaux F.V., Salazar-González J.J. (2018) “Exact approach for the vehiclerouting problem with stochastic demands and preventive returns”. Transporta-tion Science.[3] Salavati-Khoshghalb M., Gendreau M., Jabali O., Rei W. (2019) “An exact al-gorithm to solve the vehicle routing problem with stochastic demands un-der an optimal restocking policy”. European Journal of Operational Research273(1):175–189.X Workshop on Locational Analysis and Related Problems 2019 67
Infrastructure Rapid Transit NetworkDesign Model solved byBenders Decomposition
Natividad González-Blanco and Juan A. Mesa University of Seville, Seville, Spain, [email protected] University of Seville, Seville, Spain, [email protected]
In recent years, citizens mobility patterns are increasing due to longertrips which are caused by some factors as house spreading, enlargementof the urbanised areas, traffic problems in the city centres or in entrancesof cities and the reduction of average ground traffic speed. These are somereasons why new rail transit systems have been constructed or expandedin determined agglomerations, or are being planned for construction. Thisinvestment is motivated by the necessity of energy saving and for pollu-tion reduction too. Besides, in cases in which exist an infrastructure this ismotivated by the increase in travel demand. The Rapid Transit NetworkDesign Problem consists of locating alignments and stations covering asmuch as possible, knowing that the demand makes its own decisions aboutthe transportation mode.The infrastructure design problem has been treated in some papers asin [1].We propose some modifications of the model in order to improve thecomputational time for medium size networks. It is known that the modelconsists of maximizing trip coverage taking into account some consider-ations as budget limits and routing demand conservation. It is assumedthat the mobility patterns in a metropolitan area are known and also, thelocations of potential stations and potential links between each pair of sta-tions. In addition, there already exists a different mode of transportation(for example, a bus network or private cars) competing with the railwayto be built. According to other papers, we allocates the demand by using travel time.As in previous models, this has a budget constraint, an alignment lo-cation constraint and a set of routing demand conservation constraints. Inaddition, it has an location-allocation constraint and an splitting demandconstraint.Actually this problem is difficult to solve because it has a lot of binaryvariables and, of course, constraints. Branch&Bound does not get to solve itin a efficient way. This reason has motivated us to applied Benders Decom-position algorithm and modifications of it. Subsequently, Branch&Boundand Benders Decompositions has been compared computationally. References [1] García-Archilla, B., Lozano, A.J., Mesa, J.A. et al. (2013)
GRASP algorithms forthe robust railway network design problem . J Heuristics 19: 399.X Workshop on Locational Analysis and Related Problems 2019 69
On computational Dynamic Programmingfor minimizing energy in an electric vehicle ∗ Eligius M.T. Hendrix , Inmaculada Garcia Computer Architecture, Universidad de Málaga [email protected] [email protected]
In literature, one can find a branch and bound approach for the control ofelectric vehicles was published. Using that model, we create a DP imple-mentation to obtain similar results.
1. Modelling energy consumption
Literature on control typicially focuses on continuous control using theoryabout Hamiltonians and co-states. In contrast, [1, 2] applied a completelydifferent approach based on branch and bound (B&B). Given our experi-ence applying dynamic programming (DP), our hypothesis is that compu-tational dynamic programming can reach the same result. This providespotential for further investigating the generation of control tables. More-over, for the dynamics of the model, we will apply difference equationsrather than differential equations based on a step size of 0.1 seconds and atime index t = 0 , . . . , T . For parameter values we use captial letters and forvariable lower case symbols. Parameter values: F : Final control horizon in seconds; T = Fδ : Number of periods inthe horizon; P : Target position to be reached in control horizon; R : Ra-dius of the wheels in m, B = . Ohm: Resistance of the battery; S = 150 volts: Voltage of power supply, T r = 10 : Transmission coefficient motorto wheels; C = . : resistance depending on air density, surface car andaerodynamics; L = . : Inductance rotor; I = . Ohm: Inductor resistance; ∗ This paper has been supported by The Spanish Ministry (TIN2015-66680-C2-2-R) in partfinanced by the European Regional Development Fund (ERDF). Q = . : Coefficient motor torque; M = 250 kg: Mass vehicle; G = 9 . :Gravity constant; F r = . : Friction of the wheels; J : Summarizing con-stant J = F r MR Variables i t ∈ [ − , Induction of the engine ω t radial speed (radius/second), i.e. velocity v t = . RT r ω t p t ∈ [0 , P ] position of the vehicle u t ∈ {− , } Control, switch.As, one can switch very frequently, u t can also be considered continuous.We will make use of that to limit its value such that i t ∈ [ − , . Theobjective is given by E = T − X t =0 Su t i t + Bu t i t . (1) The dynamics is given by difference equations taking the time step size δ into account. Position: p t = p t − + δv t . (2) Induction: i t = i t − + δ Su t − Ii t − − Qω t − L . (3)
To keep the induction into boundaries, we limit the control to u t ∈ { max { ∆ t , − } , min { ∆ t , }} with ∆ t = 150 L + ( δI − L ) i t − + δQω t − δS Radial speed ω t = ω t − + δJ ( Qi m − − RT r ( MGF r + Cv t − )) . (4) The idea is to find the trajectory u t of control in order to minimise the totalenergy consumption. We will show how this can be realised applying DP. References [1] Abdelkader Merakeb, Frédéric Messine, and Mohamed Aidéne. A branch andbound algorithm for minimizing the energy consumption of an electrical vehi-cle. , 12(3):261–283, 2014.[2] Sebastian Sager, Mathieu Claeys, and Frédéric Messine. Efficient upper andlower bounds for global mixed-integer optimal control.
Journal of Global Opti-mization , 61(4):721–743, 2015.X Workshop on Locational Analysis and Related Problems 2019 71
New bilevel programming approachesto the location of controversial facilities ∗ Martine Labbé, Marina Leal, and Justo Puerto Computer Sciences Department of the Université Libre de Bruxelles. [email protected] IMUS, Universidad de Sevilla. [email protected] [email protected]
We propose a novel bilevel model for the location of facilities whose place-ment generates disagreement among users with different, non-aligned oropposite interests. We develop the bilevel location model for one followerand for any polyhedral distance, and we extend it for several followers andany ℓ p -norm, p ∈ Q , p ≥ .
1. The models
Motivated by recent realworld applications in Location Theory in whichthe location decisions generate controversy, we propose a novel bilevellocation model in which, on the one hand, there is a leader who wantsto locate some primary facilities and must choose among a fixed num-ber of potential locations where to establish them. Next, on the secondhand, there is one follower that, once the primary facilities have been set,chooses the placement of a secondary facility, in a continuous framework.The leader and the follower have opposite targets; the leader’s and fol-lower’s goal is to maximize and minimize, respectively, some proxy of theoverall weighted distance between the primary and secondary facilities.We assume that the distances are measured via polyhedral distances. Weprove then the NP-hardness of the models.Later, we extend the model to the cases in which several followers areinvolved in the decision process and/or the considered distances are not ∗ This work has been partially supported by MINECO Spanish/FEDER grants numberMTM2016-74983-C02-01. polyhedral but more general distances induced by ℓ p -norms with p ∈ Q , p ≥ . Examples of this controversial location can be found in the literature,for example, in areas of semiobnoxious facilities, [1, 3, 4], or in the locationand protection of critical infrastructures or facilities sensitive to intentionalattacks, [2, 5].
2. Mathematical programming formulationsand resolution algorithms
In order to deal with the model with one follower and polyhedral distanceswe develop two different procedures: one based on the evaluation of thenorm through its primal expression, and other based on the evaluationof the norm through its dual expression. For each of the procedures wedevelop Mixed Integer Linear Programming formulations, using duality,and also alternative Benders decomposition algorithms.We conduct a computational study that shows the very-good perfor-mance of the Benders algorithms, being able to solve instances with possibilities for the primary facilities from dimension until dimension .For the extension to several followers and ℓ p -norms we use replicas ofthe follower problem and conic duality, respectively. References [1] Brimberg, J., and Juel, H. (2008) A bi-criteria model for locating a semi-desirable facility in the plane. European Journal of Operational Research 106(1), 144–151.[2] Church, R.L., and Scaparra M.P. (2007). Protecting critical assets: the r-interdiction median problem with fortification Geographical Analysis 39 (2),129-146[3] Erkut, E., and Neuman, S. (1989) Analytical models for locating undesirablefacilities. European Journal of Operational Research 40 (3), 275-291.[4] Melachrinoudis,E., and Xanthopulos, Z. (2003) Semi-obnoxious single facilitylocation in Euclidean space. Computers & Operations Research 30, 2191–2209.[5] Scaparra M.P., and Church, R.L. (2008). A bilevel mixed-integer program forcritical infrastructure protection planning. Computers & Operations Research35 (6), 1905-1923X Workshop on Locational Analysis and Related Problems 2019 73
Locating a new station/stop in a networkbased on trip coverage and times
María Cruz López-de-los-Mozos, and Juan A. Mesa, Dpto. Matemática Aplicada I. Universidad de Sevilla, Spain, [email protected] Dpto. Matemática Aplicada II. Universidad de Sevilla, Spain, [email protected]
There are in the literature several covering location problems in a planar-network context (see a review in [2], and references therein). Some of themare devoted to cover origin-destination pairs (OD-pairs) instead of singlepoints [1, 3]. This work is also focused on covering OD-pairs in a mixedtransportation mode context, in which traveling times are a combinationof planar and network times.More specifically, we consider a network embedded in the plane repre-senting a rapid transportation system, such that the nodes are either junc-tions or stations/stops already located, and we assume a set of existingfacilities in the plane (not necessarily belonging to the network), such thattraveling along the network is faster than traveling within the plane withsome planar metric (in this work, the Euclidean metric). An OD-pair is saidto be covered if the time spent in the combined plane-network mode islower than in the planar mode. Within this context, the problem of locatinga new station on the network is studied. The aim is to maximize the tripcoverage when heterogeneous dwell times at the stations of the networkare considered.In order to incorporate considerations on trip times to the problem,some insights must be taken into account. First, locating a new station in-creases the accessibility of the network, with a possible increasing of theOD-pairs covered. However, the travel time of the OD-pairs initially cov-ered could be increased due to the additional dwell time at the new station,and some of such pairs could not be covered by the modified network. Thatis, a new station leads to an opposite effect since simultaneously the objec-tive value increases with new pairs covered, and decreases with the pairswhich are lost. In the second place, a new station penalizes the travel time of those trips which maintain their combined plane-network mode withthe new station, that is, those trips already covered by the initial network,and which continue covered after adding the new station. We say perma-nent trips to such trips.For avoiding an excessive penalization on the travel time of permanenttrips, we introduce a constraint on the increasing of their traveling time.With these considerations we formulate a trip covering location problemon a tree network, and propose a solution approach based on decomposingthe problem in a collection of subproblems and, for each of them, identify-ing a subquadratic in the number of OD-pairs Finite Dominating Set. References [1] Körner, M-C., Mesa J.A., Perea, F., Schöbel, A., & Scholz, D. (2014).
A Maximumtrip covering location problem with alternative mode of transportation on tree networksand segments . TOP , 227-253.[2] Laporte, G. and Mesa, J.A. (2015). The Design of Rapid Transit Networks . InG. Laporte, S. Nickel, F. Saldanha da Gama (Eds.), Location Science 241-255,Springer.[3] López-de-los-Mozos, M.C., Mesa, J. A., & Schöbel, A. (2017).
A general approachfor the location of transfer points on a network with a trip covering criterion and mixeddistances . European Journal of Operational Research 260, 108-121.X Workshop on Locational Analysis and Related Problems 2019 75
On location-allocation problems fordimensional facilities ∗ Lina Mallozzi, Justo Puerto, and Moisés Rodríguez-Madrena University of Naples Federico II, Naples, Italy, [email protected] IMUS, Universidad de Sevilla, Sevilla, Spain, [email protected]@us.es
This work deals with a bilevel approach of the location-allocation problem( [2]) with dimensional facilities. We present a general model that allows usto consider very general shapes of domains for the dimensional facilities.
1. The problem
We are given a number of dimensional facilities that are shaped like generaldomains and the customers are distributed in the demand region accordingto a demand density that is an absolutely continuous probability measure.The goal is to locate them in a general planar demand region.For a feasible location of the facilities, a partition of the demand regionthat determines the allocation of the customers to the facilities has to bedone minimizing the total social cost. See e.g. [1] for similar partitions in adifferent context.We are interesting in finding the best location of the facilities in sucha way that the summation of certain realistic costs over all the facilitiesmust be the cheapest possible, knowing that the partition of the customersis done as it is explained above. Among that costs, the congestion costis computed once the partition of the customers in the demand region isdone. These assumptions impose to our problem a hierarchical structureof bilevel problem. ∗ This research has been partially supported by Spanish Ministry of Economia and Competi-tividad/FEDER grants number MTM2016-74983-C02-01. Different particular applications, which are discussed in our work, fitwithin this general problem. We prove the existence of optimal solutionsfor the bilevel problem under mild, natural assumptions. To achieve theseresults we borrow tools from optimal transport mass theory that allow usto give an explicit solution structure of the considered lower level problem.
2. Solution approaches
We propose a discretization approach that can approximate, up to any de-gree of accuracy, the optimal solution of the original problem. This dis-crete approximation can be optimally solved via a mixed-integer linearprogram. To address very large instance sizes we also provide a GRASPheuristic that performs rather well according to our experimental results.The graphical output of our algorithms for some illustrative test examplescan be seen in Figure 1.(a) (b) (c)
Figure 1.
Graphical output of the proposed algorithms for three different test ex-amples.
References [1] Mallozzi, L., Puerto, J.: The geometry of optimal partitions in location prob-lems. Optimization Letters 12(1), 203-220 (2018)[2] Nickel S., Puerto J.: Facility Location - A Unified Approach. Springer, Berlin(2005)X Workshop on Locational Analysis and Related Problems 2019 77
Robust feasible rail timetable
Ángel Marín, Miguel A. Ruiz-Sánchez , Esteve Codina Instituto Matemática Interdisciplinar, Universidad Complutense Madrid, Spain Universitat Politécnica de Catalunya, Barcelona, Spain
The railway timetable problem consists in selecting the optimal train routesand schedules to minimize the rail traffic service time. The growth in de-mand encourages rail managers to improve the effective use of the infras-tructure occupations, but keeping the passenger service quality (i.e., traveltime). Upgrading the infrastructure helps to achieve these objectives but in-creasing resource investment. The design of effective timetables may alsohelp.The timetable planning problem may be studied by means of two ap-proaches: micro and macro. The macro approach simplifies the represen-tation of the railway infrastructure: it considers railway segments linkingconsecutive control areas (stations, junctions, et.) as a single network node.This approach is suitable for tackling problems from the passengers’ pointof view. The micro approach makes use of a detailed description of therailway control infrastructure and considers the potential traffic conflictsin the control area.
1. Macro Timetable
The macro railway timetable will be defined under the passengers or oper-ative point of view. A set of platforms joint with the entrée and leave tracksis considered a station, and a set of multiple tracks crossing in a node is ajunction. Both will be considered in the micro model as control zones. Themacro model tries to minimize the optimal train travel time, consideringthe passenger travel time and the delays at destination. The macro doesnot consider the conflicts at the control zones, so the solution will be notfeasible considering those.A timetable to be feasible needs to be conflict-free: trains must be able totravel at their planned velocity without ever having to stop or slow down due to restrictive signals. To ensure feasibility timetable, the micro modelallow longer stays at platforms or special tracks, where the trains haveplanned stops.
2. Micro Timetable
The micro model is defined in terms of track-circuits, they are track seg-ments on which the presence of a train is automatically detected. Sequencesof track-circuits are grouped into block sections, which access is controlledby signals.
Figure 1.
Micro vision services Madrid-Barcelona.
We define the routes in control areas in terms of a sequence of track-circuits and by intermediate stops. A sequence of track-circuits can be trans-versed performing or not intermediate stops defining different routes. Theobjective function of the micro models in each control area is to minimizethe shifted time between the arrival and departure times given by themacro optimization model, which need to obtain micro-feasible solutions.In short, the micro constrains may be described by: the train occupation,reservation and stay management, plus the train routing and schedulingin the control areas.An integrated model may tackle both approaches at once. The macromodel plan the arrival and departure timetable using a tentative train time,providing it at micro level. The micro model makes feasible and robust thistimetable by computing the optimal routing and schedule of the train atthe control areas, minimizing the shifted arrival and departure at each ofthem. The essential micro inputs are the arrival and departing of each trainat each control area. The micro outputs are the shifted arrival and departof each train at each control area, so that the train route and sequence free-conflicts in the area.The essential macro inputs are the shifted arrival and departure traintimes at the control area. The macro outputs are the arrival and departureof the trains, joint with the generalized costs perceived by the passengersand the scheduled delay of the trains being early and lateness at the controlareas.
X Workshop on Locational Analysis and Related Problems 2019 79
Wildfire Location Model:A new proposal
Juan A. Mesa, Mariano Marcos-Pérez. Escuela Técnica Superior de Ingenieros. Departamento de Matemática Aplicada II, Uni-versidad de Sevilla, Camino de los Descubrimientos s/n, 41092 Seville, Spain
Nowadays there are an increasing awareness of environmental problems.Wildfires pose a serious threat to communities and ecosystems throughoutthe world. Wildfires containment and locations of limited resources to mit-igate the impact of natural disasters are important but challenging tasks.The location of a set of resources when fighting fire is proposed it is oneof the many problems we need to improve in order to get a better solutionor at least more useful. Wildfire suppression combines multiple objectives,the main objective of forest-fire management is to minimize the damagecaused by forest fires. Empirical studies have identified several factors thataffect the development of a forest fire. The prevailing meteorological con-ditions of the area under consideration, that evidently play an importantrole in the location of firefighting resources, have been completely ignoredor inadequately considered.There are many models which can be improved adding a wind restriction.Karkazis (1992) [1] presented the most recent wind-discrete models andsolution methods to locate facilities causing airbone pollution. Hodgsonand Newstead (1978) [2] developed two location-allocation models for as-signing a limited number of airtanker. Dimopoulou and Giannikos (2001)[3], using information provided by a GIS have determined the optimal lo-cation of fire-fighting resources. Belval et al. (2015) [4] have presented amixed integer program to model spatial wildfire behaviour and suppres-sion placement decisions. Alvelos (2018) [5] presents the location of a setof resources when fighting fire is proposed. In this work, we present a wildfire location model including a wind-discrete model. This model is intended to be a future supplement in re-source planning to get a better results when we fight against fires. Thepurpose of investigations such as this is to improve forest science usingOperation Research and Optimization, the future of the forest science islargely a question of cooperation between different branches of study, inthis case, mathematics.
References [1] Karkazis J., Boffey T.B., Malevris N. (1992). Location of Facilities ProducingAirborne Pollution. Journal of The Operational Research Society - J OPER RESSOC. 43. 313-320..[2] Hodgson M.J., Newstead R.G. (1978) Location-allocation models for one-strikeinitial attack of forest fires by airtankers .Canadian Journal of Forest Research8(2), 145-154.[3] Dimopoulou M., Giannikos I. (2001) Spatial optimization of resources deploy-ment for forest-fire management. International Transactions in Operational Re-search 8(5), 523- 534.[4] Belval E.J., Wei Y; Bevers M. (2015) A mixed integer program to model spatialwildfire behavior and suppression placement decisions. Canadian Journal ofForest Research 45, 384-393.[5] Alvelos F. (2018). Mixed Integer Programming Models for Fire Fighting. Com-putational Science and Its Applications. ICCSA 2018. Lecture Notes in Com-puter Science Vol 10961. 637-652.X Workshop on Locational Analysis and Related Problems 2019 81
Optimal allocation of fleet frequency for“skip-stop” strategies in transport networks
Juan A. Mesa, Francisco A. Ortega, Ramón Piedra-de-la-Cuadra and Miguel A. Pozo Universidad de Sevilla, Sevilla, Spain, [email protected] Universidad de Sevilla, Sevilla, Spain, [email protected] Universidad de Sevilla, Sevilla, Spain, [email protected] Universidad de Sevilla, Sevilla, Spain, [email protected]
The planning of the public transport systems includes the design of thelines and the frequency of the services of transport. The occasional inci-dents in the functioning of the system generally are not considered in theinitial planning. To reduce the disturbing effect, the operator of a service ofpassengers’ transport must be able to implement some strategy of controlto fit the schedules to the conditions of the real time traffic. The strategiesmore commonly used are the express service (certain stations of the corri-dor skip due to little passengers’ flow), short-turning, or deadheading andthe combination of different actions of control [3].In this research work a methodology is developed to implement a redis-tribution of services along a line of railway traffic, which must be carriedout by the operator choosing new train schedules within a series of feasiblespace-time windows, previously established by the railway infrastructuremanager. The objective is to minimize the loss of users, who could perceivea worsening in the quality of the service that until now they had been re-ceiving. In addition, the result obtained in that first phase will conditionthe subsequent decision of the transport operator, consisting of classifyingthe fleet of its trains into two types (A and B) and the stations, at the sametime, in three types of stops: type A, where will stop the trains identifiedwith the distinction A; type B, where the trains identified as mode B willstop; Type AB, where both kinds of trains will stop. This double allocation,both for trains and for stations, is called the Skip-Stop strategy.
1. Skip-Stop Strategy and KnapsackProblem
The Skip-Stop mechanism consists of privileging a larger number of pas-sengers offering shorter travel times, as a result of having previously se-lected a group of low-activity stations, where trains wouldn’t stop to pickup or let off passengersThe travel time between stations in a railway line consists of five compo-nents, identified as phases of acceleration, constant speed, inertia, brakingand downtime, in [2] can find the algebraic expressions habitually usedto calculate these times. In consequence, the operation of omitting stopsreduces the travel time for the users within the vehicle and increases thespeed of operation in the provision of the service. However, some userswill experience a longer time of waiting, accessing, exiting and, possibly,transferring. Therefore, there is no guarantee that any skipping operationwill decrease the total travel time of the potential travellers. The selectionand coordination of stops must be made by using a criterion according toan objective function.We propose, in this work, to model an existing problem (Skip-Stop prob-lem) through the Knapsack Problem (KP) taking advantage of the largeamount of material available from the KP. The solution of the Skip-Stopproblem will consist of two phases; in the first, we find the optimal strat-egy of skipping stops for a given train fleet and, in the second phase, wedetermine, by means of a heuristic, the optimal allocation for train types.For this last purpose, we will develop the concept of proximity betweenthe railway routes and, in accordance with the Hall method [1], design aMatheuristic that optimizes the Skip-Stop Strategy.
References [1] Hall, K. M. (1970).
An r-Dimensional Quadratic Placement Algorithm . Manage-ment Science.[2] Lee, Y.-J. (2012).
Mathematical modeling for optimizing skip-stop rail transit oper-ation strategy using genetic algorithm . National Transportation Center ResearchReport 26, Morgan State University.[3] Mesa, J. A., Ortega, F. A. and Pozo, M. A. (2009).
Effective allocation of fleet fre-quencies by reducing intermediate stops and short turning in transit systems.
LectureNotes in Computer Science 5868(8), pp. 293 - 309.X Workshop on Locational Analysis and Related Problems 2019 83
Introduction to planar location with orloca
Manuel Munoz-Marquez, Cadiz University, Cadiz, Spain [email protected]
The
RcmdrPlugin.orloca package devoted to solve the planar continuouslocation problem is presented. It has been developed for R as free software.It is intended to be used as an easy way to introduce the planar pointlocation problems to the students. This is done providing a GUI and anon-line interactive application to handle and solve such problem.
1. Introduction
In a location problem, we seek the optimal location of a service. Examplesof localization problems are: find the optimal location of the central ware-house or an ambulance that must attend to the patients.The package solves the problem of locating a single point in the plane,minimizing of the sum of the weighted distances to the demand points.New versions of the package will include new location models. The pack-age
RcmdrPlugin.UCA [1] provides a GUI to do that.
2. Package features
A new class of objects, designated as loca.p , has been defined to handleinstances of the problem.From the menu one can create new instances of loca.p objects, generatenew random instances and make summaries of the data. One can also eval-uate the weighted average distance, in the following distsum and calculatethe gradient of it.There is also an option to find the minimum of distsum using the Weiszfeldalgorithm, see [2] or [3] , or a global optimization one. The Weiszfeld algo-rithm includes a test for optimality for demand points. Four options has been provided to make some plots as the figures show.The help menu provides several options to get more information aboutthe use of the package and provides several examples.
Figure 1.
Demand point set
Figure 2.
Contour plot
Figure 3.
Demand and contour plot
Figure 4.
3D plot
3. Conclusions
The
RcmdrPlugin.orloca package solves the planar location problem of asingle service using an user friendly menus, so it is a good way to introducethis problem to students. More info in http://knuth.uca.es/orloca andan on-line interactive demo in http://knuth.uca.es/shiny/orloca/ . References [1] Manuel Munoz-Marquez.
A GUI for Planar Location Problems . Cadiz University,4.6 edition, 2018.[2] E. Weiszfeld. Sur le point pour lequel la Somme des distances de n pointsdonnés est minimum.
Tohoku Mathematical Journal, First Series , 43:355–386, 1937.[3] E. Weiszfeld and Frank Plastria. On the point for which the sum of the distancesto n given points is minimum.
Annals of Operations Research , 167(1):7–41, Mar2009.X Workshop on Locational Analysis and Related Problems 2019 85
Emergency Vehicle Location Modelconsidering uncertainty and thehierarchical structure of the resources
José Nelas, Joana Dias , Faculdade de Economia, Universidade de Coimbra, Portugal CeBER and Inesc-Coimbra, Universidade de Coimbra, Portugal, [email protected]
The main goal of emergency services is to guarantee that help arrives topopulations where and when it is needed. Depending on the severity ofthe emergency episodes, it is possible to define maximum time limits thatshould be respected to assure a proper and timely assistance. The locationof emergency vehicles is crucial for guaranteeing that this goal is achieved.In this work we present a mathematical model that considers the loca-tion of emergency vehicles under uncertainty. Moreover, the model takesexplicitly into account the hierarchical features of these vehicles, namelyconsidering that some types of vehicles can substitute others that are notavailable when the episode occurs.
1. Introduction
The location of emergency resources has been a subject studied by sev-eral different researchers in the last decades. One of the most used type ofmodel is the covering location model: the objective is to guarantee that allthe population is covered, considering a given distance or time limit be-tween each population and the nearest available resource. When full cov-erage is not possible, due to budget restrictions, for instance, then one ofthe approaches is to consider the maximization of the covered population.This choice can have as consequence severe inequalities, especially consid-ering the most distance and depopulated regions. While the initial modelswere deterministic and static, recent developments have recognized theimportance of considering the inherently stochastic nature of the problem, as well as time. When dealing with emergency resources, it is very impor-tant to acknowledge that not all the resources will be available at all times,namely because they can already be assigned to an occurrence.
2. The Proposed Model
In the developed model, uncertainty is represented through a set of sce-narios. Each scenario is characterized by having a given set of differentemergency episodes, that occur in different locations, at different time pe-riods, and that have different levels of severity (thus requiring differenttypes and number of vehicles).It is necessary to decide where to locate the vehicles (this decision is notscenario dependent), and the way in which vehicles are assigned to emer-gency occurrences (these decisions are scenario dependent). An occurrenceis considered "covered" if it is possible to assign to it all the required ve-hicles, within the defined time limits. A coverage matrix determines, foreach level of assistance, which vehicles are within the established time limitfrom the emergency occurrences. The objective function considers the max-imization of the covered occurrences.Time is considered in the model through the use of an incompatibility ma-trix that, for each scenario, each pair of emergency occurrences and eachtype of vehicle determines whether these occurrences have or have notoverlapping time periods. If there is no overlapping time periods, then thesame vehicle can be assigned to both occurrences. If there is at least oneoverlapping time period, this is not possible anymore (if a vehicle is as-signed to one of the occurrences it cannot be assigned to the other). Thereis also information about the possibility of a given type of vehicle beingable to substitute another one, if necessary. A vehicle capable of providingmore differentiated emergency care can also be assigned to an occurrencethat would only need a less differentiated care, for instance. When decidingthe assignment of vehicles to occurrences, this flexibility is also considered.The developed mathematical model will be presented, as well as some il-lustrative examples and preliminary computational results.
X Workshop on Locational Analysis and Related Problems 2019 87
Solving the Ordered Median Tree of HubsLocation Problem
Miguel A. Pozo, Justo Puerto, and Antonio M. Rodríguez-Chía, Universidad de Sevilla, Spain., [email protected] Universidad de Sevilla, Spain., [email protected] Universidad de Cádiz, Spain., [email protected]
Standard Hub Locations Problems assume that inter-hub connectionsbetween an origin-destination pair can be routed through one or at mosttwo hubs. However, it has been observed by several authors that in manyapplications the backbone network is not fully interconnected [1,2] or it caneven be not necessarily connected.It is of special interest the case where theunderlying interconnection network is connected by means of a tree. Suchproblem is called the Tree of Hubs Location Problem and was introducedby Contreras et al ( [3, 4]).Another feature, namely weighted averaging objective functions, hasalso been incorporated to the analysis of Hub Locations Problems [5, 6].It has been recognized as a powerful tool from a modeling point of viewbecause its use allows to distinguish the roles played by the different en-tities participating in a hub-and-spoke network inducing new type of dis-tribution patterns. Each one of the components of any origin-destinationdelivery path gives rise to a cost that is weighted by different compensa-tion factors depending on the role of the entity that supports the cost. Thisadds a “sorting”-problem to the underlying hub location problem. The ob-jective is to minimize the total transportation cost of the flows betweeneach origin-destination pair after applying rank dependent compensationfactors on the transportation costs.In this paper, we study the Ordered Median Tree of Hub Location Prob-lem (OMTHLP). The OMTHLP is a single allocation hub location prob-lem where p hubs must be placed on a network and connected by a non-directed tree. Each non-hub node is assigned to a single hub and all the flow between origin-destination pairs must circulate using the links con-necting the hubs. The objective is to minimize the sum of the orderedweighted averaged assignment costs plus the sum of the circulating flowcosts. We will present different MILP mathematical formulations for theOMTHLP based on the properties of the Minimum Spanning Tree Prob-lem and the Ordered Median optimization. We establish theoretical andempirical comparisons between these new formulations and we also pro-vide reinforcements that together with a proper formulation are able tosolve medium size instances on general graphs. References [1] J.F. Campbell, A. Ernst, and M. Krishnamoorthy. Hub arc location problems.II: Formulations and optimal algorithms.
Manage. Sci. p -hub center allocation problem. European Journal of Operational Research,
European Journal of Operational Research,
Computers & Operations Research,
Computers and Operations Research , 38:559-570,2011.[6] Justo Puerto, A. B. Ramos and A. M. Rodríguez-Chía, A specialized branch& bound & cut for Single-Allocation Ordered Median Hub Location problems.Discrete Applied Mathematics, 161:16-17, 2624-2646, 2013.X Workshop on Locational Analysis and Related Problems 2019 89
Feasible solutions for the DistanceConstrained Close-Enough Arc RoutingProblem
Miguel Reula ∗ , Ángel Corberán , Isaac Plana and José Maria San-chis Dept. d’Estadística i Investigació Operativa, Universitat de València, Spain Dept. de Matemáticas para la Economía y la Empresa, Universitat de València, Spain Dept. de Matemática Aplicada, Universidad Politécnica de Valencia, Spain
The Close-Enough Arc Routing Problem (CEARP), also known as Gener-alized Directed Rural Postman Problem, is an arc routing problem withinteresting real-life applications, such as routing for meter reading. In thisapplication, a vehicle with a receiver travels through a series of neighbor-hoods. If the vehicle gets within a certain distance of a meter, the receiveris able to record the gas, water, or electricity consumption. Therefore, thevehicle does not need to traverse every street, but only a few, in order tobe close enough to each meter. We deal with an extension of this problem,the Distance-Constrained Close Enough ARP, in which a fleet of vehicleswith distance constraints is available. The vehicles have to leave from andreturn to the depot, and the length of their routes must not exceed a maxi-mum distance (or time). Several formulations and exact algorithms for thisproblem were proposed in [1]. Since the size of the instances solved to op-timality is far from those arising in real-life problems, we propose here amulti-start heuristic algorithm with an improvement phase that incorpo-rates an effective exact procedure to optimize the routes obtained. In orderto assess the relative efficiency of our algorithm, extensive computationalexperiments have been carried out. The results show the good performanceof the proposed heuristic, even in the instances with a very tight maximumdistance. ∗ [email protected] References [1] T. Ávila, Á. Corberán, I. Plana, and J.M. Sanchis, (2017). Formulations and ex-act algorithms for the distance-constrained generalized directed rural postmanproblem.
EURO Journal on Computational Optimization , 5: 339-365.X Workshop on Locational Analysis and Related Problems 2019 91
Steiner Traveling Salesman Problems:when not all vertices have demand
Jessica Rodríguez-Pereira, , Enrique Benavent, Elena Fernández, Gilbert Laporte, and Antonio Martínez-Sykora Canada Research Chair in Distribution Management, HEC Montréal, Canada Department of Statistics and Operation Research, Universitat Politècnica de Catalunya-BcnTech, Spain Department of Statistics and Operation Research, Universitat de Valencia, Spain Southampton Business School, University of Southampton, United Kingdom
The purpose of this work is to present a new compact formulation and effi-cient exact solution algorithm for the Steiner Traveling Salesman Problem(STSP) on an undirected network and its location extension. The STSP isan uncapacitated node-routing problem looking for a minimum-cost routethat visits a known set of customers with service demand, placed at ver-tices of a given network, which is assumed to be uncomplete. Althoughnot all the vertices in the network have demand, some non-demand ver-tices may have to be visited for connecting demand vertices served consec-utively in a route. Thus, the specific set of vertices that must be traversedin feasible routes is not known in advance. The location extension, LSTSP,studies the case when several depots are allowed and their location has tobe decided as well.We propose compact mixed integer linear programming formulationsfor the STSP and the SLTSP. All formulations are defined on the originalundirected graph and use a small number of two-index decision variablesonly. They exploit the property that there is an optimal solution, both forthe STSP and the LSTSP, where no edge is traversed more than twice, anduse two sets of binary variables only, which are associated with the first andsecond traversal of edges, respectively. Feasibility of solutions is modeledwith two families of constraints of exponential size, one for the connectiv- ity with the depot and another one for the parity of the visited vertices.For the parity of the vertices we use an adaptation of co-circuit constraints,which exploits the relationship between our two sets of decision variables.While co-circuit inequalities are nowadays very often used to model parityin arc routing problems, we are not aware of any node-routing problemwhere such inequalities have been used to model the parity of visited ver-tices. Since the two-index decision variables do not associate traversals ofedges with the facilities of the routes they belong to, the LSTSP formulationrequires an additional set of constraints that involve the location decisionvariables as well, in order to guarantee that routes are well defined andreturn to their starting location.We have developed an exact branch-and-cut algorithm for the STSP thatallows us to optimally solve instances with up to 500 vertices in very mod-erate computing times. Our computing times never exceed 350 seconds forinstances with up to 250 vertices and, except for two out of 60 instances,do not exceed our time limit of 7200 seconds for the larger instances with anumber of vertices ranging in 275-500.We have also developed an efficient exact branch-and-cut algorithm tosolve the two-index formulation for the SLTSP. As could be expected, thecomputational effort required to solve the instances is now considerablylarger than for the STSP. Still, SLTSP instances with up to 500 vertices and10 potential locations were optimally solve within the maximum allowedcomputing times 7200 seconds. X Workshop on Locational Analysis and Related Problems 2019 93
Addressing locational complexity: networkdesign and network rationalisation
Diego Ruiz-Hernandez, Jesus M. Pinar-Pérez, and Mozart B.C. Menezes Operations Management and Decision Sciences Division, Sheffield University Manage-ment School, S10 1FL, Sheffield, UK d.ruiz-hernandez@sheffield.ac.uk Dept. of Quantitative Methods, University College for Financial Studies, 28040, Madrid,Spain [email protected] Operations Management and Information Systems Dep., Kedge Business School, 33405,Bordeaux, France [email protected]
Facility location problems are well known combinatorial problems wherethe objective is to minimize certain measure of the cost incurred for (or thebenefit attained from) serving customers from a set of facilities. A typicallocation problem will either aim at maximising the demand covered bystrategically locating a given number of facilities, or at finding the optimalnumber and location of facilities necessary for satisfying the total demandin a region.Our aim is to bring to the field of facility location the concept of struc-tural complexity, opening up a new research line. Broadly speaking,structuralcomplexity refers to the negative effects of the proliferation of products,distribution channels and markets. Focusing on locational complexity, themain objective of this project is to create awareness about the need of con-sidering complexity issues –and their impact on profitability- when decid-ing the location and size of a distribution network. The rationale behindour argument is that an oversized distribution network may cause hiddencosts that hinder the capacity of the supply chain for translating revenueinto bottom-line benefits.In this work, using an entropy based measure for structural complexitydeveloped by the authors in previous research [1], we propose an variantof the traditional p-median problem that includes a complexity parameterin the model’s formulation, the K-MedianPlex problem: max S ⊂ N : | S | = K Z KP lex = X k ∈ S R ( k ) (cid:16) − αC ( k ) p (cid:17) − φK with C ( k ) p = X i ∈N k w i log (cid:18) w i (cid:19) , k ∈ SR ( k ) = X i ∈N k ( r − γd ik ) W i , k ∈ Sα : αC ( k ) p < , k ∈ S where N is the set of network nodes; S ⊂ N , the set of open facilities; W i ,the weight of demand node i ∈ N ; φ , a fix facility cost; r , the revenue perunit; α , a profit loss factor due to complexity; γ , a generic transportationcost; and w i = W i / P i ∈N W i for all i ∈ N .Given the strongly combinatorial nature and non-convexity of the ob-jective function, we propose an algorithmic approach based on solving aK-median problem and sequentially reasigning demand nodes across facil-ities and solving local 1-Median problems aiming at maximising the func-tion Z P lex (cid:16) N | S ′ | , S ′ (cid:17) , where N | S ′ | represents the collection of allocationsets associated to a given solution S ′ .However, location complexity is not typically a result of network de-sign, but a problem that arises from successive network expansions aimedat capturing market share. In order to reduce complexity, firms may find itprofitable abandoning certain markets (although they are usally reluctantunder the rationale that lost sales will affect profit negatively). We pro-pose a strategy for succesively uncovering demand nodes until no profitimprovement can be further attained.Experimental results suggest that higher profits can be attained by real-locating demand nodes across facilities, relocating facilities and/or elimi-nating non-profitable demand nodes. As it may be expected, the improve-ment routines return better results for high values of the complexity costparameter α , and for larger transportation costs. References [1] Ruiz-Hernández, D. and Menezes, M.B.C., (2018)
An information-content basedmeasure of proliferation as a proxi for structural complexity . SUMS-KBS, Mimeo.X Workshop on Locational Analysis and Related Problems 2019 95
Using a kernel search heuristic to solve asequential competitive location problem ina discrete space ∗ Dolores R. Santos-Peñate, Clara M. Campos-Rodríguez, and JoséA. Moreno-Pérez Instituto de Turismo y Desarrollo Económico Sostenible/Dpto de Métodos Cuantitativosen Economía y Gestión, Universidad de Las Palmas de G.C., 35017 Las Palmas de GranCanaria. Spain, [email protected] Instituto Universitario de Desarrollo Regional. Universidad de La Laguna. 38271 La La-guna. Spain, [email protected] Instituto Universitario de Desarrollo Regional. Universidad de La Laguna. 38271 La La-guna. Spain, [email protected]
In the leader-follower, ( r | p ) -centroid or Stackelberg location problem, twoplayers sequentially enter the market and compete to provide goods orservices. This work considers this competitive facility location problem ina discrete space. The customer choice rule is defined as a possibility func-tion. For each customer, an S-shaped function is used to share the demandamong competitors when the difference in distances to the competing firmsis small. To solve the problem, the linear programming formulation for theleader and the follower are integrated into an algorithm which, in an iter-ative process, finds a solution by solving a sequence of these linear prob-lems. We propose a matheuristic procedure that provides solutions for theleader via a kernel search algorithm. The proposed solution approach is il-lustrated with some computational results obtained for the binary rule anddifferent S-shaped customer choice functions. The results obtained with anexact algorithm and the heuristic procedure are compared. ∗ This study was partially funded by Ministerio de Economía y Competitividad (Spanish Gov-ernment) with FEDER funds, through grants ECO2014-59067-P and TIN2015-70226-R, andalso by Fundación Cajacanarias (grant 2016TUR19). References [1] E. Alekseeva, Y. Kochetov, and A. Plyasunov (2015). An exact method for thediscrete ( r | p ) -centroid problem. Journal of Global Optimization ( r | p ) -centroid problem. In: Talbi E (ed), Metaheuristics for Bi-level Opti-mization
SCI 482: 189–219.[3] B. Biersinger, B. Hu, and G. Raidl (2016). Models and algorithms for compet-itive facility location problems with different customer behaviour.
Annals ofMathematics and Artificial Intellegence
76: 93-119.[4] B. Biersinger, B. Hu, and G. Raidl (2015). A hybrid genetic algorithm with so-lution archive for the discrete ( r | p ) -centroid problem. Journal of Heuristics
TOP ( r | p ) -centroid problem. Automation and RemoteControl
Journal of Heuristics
European Journal of Opera-tional Research
Discrete Location Theory , Wiley,New York, pp 439-478.[10] M.C. Roboredo and A.A. Pessoa (2013). A branch-and-bound algorithm forthe discrete ( r | p ) -centroid problem. European Journal of Operational Research uthor Index A Albareda-Sambola, MaríaUniversitat Politècnica de Catalunya, Spain, [email protected] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17, 19, 21Amorosi, LaviniaSapienza University of Rome, Italy, [email protected] . . . . . . . . 23Archetti, ClaudiaUniversità degli Studi di Brescia, Italy, [email protected] . . . . . . . . 25Ashley, ThomasUniversidad de Sevilla, Spain, [email protected] . . . . . . . . . . . . . . . . . . . . . . . . .27 B Baldomero-Naranjo, MartaUniversidad de Cádiz, Spain, [email protected] . . . . . . . . . . . . . . . . 29Barbarisi, IlariaUniversità di Napoli Federico II, Italy, [email protected] . . . . . . . . . . 31Barrena, EvaUniversidad Pablo de Olavide, Spain, [email protected] . . . . . . . .33, 35, 57Benavent, EnriqueUniversitat de Valencia, Spain, [email protected] . . . . . . . . . . . . . . . . 91Bender, MatthiasResearch Center for Information Technology (FZI), Germany, . . . . . . 37Blanco, VíctorUniversidad de Granada, Spain, [email protected] . . . . . . . . . . . . . . 39, 41, 43Bruno, Giuseppe 97niversità di Napoli Federico II, Italy, [email protected] . . . . . . . . 31Byrne, ThomasUniversity of Edinburgh, UK, [email protected] . . . . . . . . . . . . . . . . . . . . . . . 45 C Campbell, James F.University of Missouri-St. Louis, USA, [email protected] . . . . . . . . . . . . 47Campos-Rodríguez, Clara M.Universidad de La Laguna, Spain, [email protected] . . . . . . . . . . . . . . . . . . . 95Canca, DavidUniversidad de Sevilla, Spain, [email protected] . . . . . . . . . . . . . . . . . 33, 35, 49, 57Carrizosa, EmilioUniversidad de Sevilla, Spain, [email protected] . . . . . . . . . . . . . . . . . . . . . . 27Castañeda, CarolinaUniversitat Pompeu Fabra, Spain, [email protected] . . . . . . . . . . 51Cavola, ManuelUniversità di Napoli Federico II, Italy, [email protected] . . . . . . . . . 31Codina, EsteveUniversitat Politécnica de Catalunya, Spain, [email protected] . . . 77Coelho, LeandroUniversité Laval, Canada,
[email protected] . . . . . . . . 33Corberán, ÁngelUniversitat de València, Spain, [email protected] . . . . . . . . . . . 47, 53, 89 D Dávila, SebastiánUniversidad de Chile, Chile, sebastian.davilaga @gmail.com . . . . . . . . . . . . 55De Los Santos, AliciaUniversidad de Córdoba, Spain, [email protected] . . . . . . . . . . . . . . 49, 57Dias, JoanaUniversidade de Coimbra, Portugal, [email protected] . . . . . . . . . . . . . . . . . . 85Diglio, AntonioUniversità di Napoli Federico II, Italy, [email protected] . . . . . . . . . . 31 E Elizalde Blasco, JavierUniversidad de Navarra, Spain, [email protected] . . . . . . . . . . . . . . . . . 31, 59Escudero, Laureano F.Universidad Rey Juan Carlos, Spain [email protected] . . . . . . . . . 61Espejo, InmaculadaUniversidad de Cádiz, Spain, [email protected] . . . . . . . . . . . . . . . 63
UTHOR INDEX F Fekete, SándorTU Braunschweig, Germany, [email protected] . . . . . . . . . . . . . . . . . . . . . . . 45Fernández, ElenaUniversitat Politècnica de Catalunya-BcnTech, Spain, [email protected] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Fernández-Cara, EnriqueUniversidad de Sevilla, Spain, [email protected] . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Florio, Alexandre M.University of Vienna, Austria, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 G García, InmaculadaUniversidad de Málaga, Spain, [email protected] . . . . . . . . . . . . . . . . . . . . . . 69García, SergioUniversity of Edinburgh, UK, [email protected] . . . . . . . . . . . . 39González-Blanco, NatividadUniversidad de Sevilla, Spain, [email protected] . . . . . . . . . . . . . . . . . . . . . 67Guastaroba, GianfrancoUniversità degli Studi di Brescia, Italy, [email protected] . . . 25 H Hartl, Richard F.University of Vienna, Austria, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Hendrix, Eligius M.T.Universidad de Málaga, Spain, [email protected] . . . . . . . . . . . . . . . . . . . . . . . 69Hernández-Díaz, Alfredo G.Universidad Pablo de Olavide, Spain, [email protected] . . . . . . . . . . . . . . . 57Huerta-Muñoz, Diana L.Università degli Studi di Brescia, Italy, [email protected] . . . . . 25 J Japón, AlbertoUniversidad de Sevilla, Spain, [email protected] . . . . . . . . . . . . . . . . . . . . . . . . 41 K Kalcsics, JörgUniversity of Edinburgh, UK, [email protected] . . . . . . . . . . 29, 37, 45 L Labbé, MartineUniversité Libre de Bruxelles, Belgium and INRIA, France, [email protected] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55, 71 AUTHOR INDEX
Landete, MercedesUniversidad Miguel Hernández de Elche, Spain, [email protected] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17, 19, 53Laporte, GilbertCIRRELT and HEC Montréal, Canada, [email protected] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17, 33, 49, 91Leal, MarinaUniversidad de Sevilla, Spain, [email protected] . . . . . . . . . . . . . . . . . . . . . . . . . 71Ljubic, IvanaESSEC Business School of Paris, France, [email protected] . . . . . . . . 13López-de-los-Mozos, M.C.Universidad de Sevilla, Spain, [email protected] . . . . . . . . . . . . . . . . . . . . . . . 73 M Mallozzi, LinaUniversity of Naples Federico II, Italy, [email protected] . . . . . . . . . . . 75Marcos-Pérez, MarianoUniversidad de Sevilla, Spain, [email protected] . . . . . . . . . . . . . . . . . . . . . . 79Marianov, VladimirPontificia Universidad Católica de Chile, Chile, [email protected] . . 55Marín, ÁngelUniversidad Complutense Madrid, Spain, [email protected] . . . . . . . 77Martínez-Merino, Luisa I.Universidad de Cádiz, Spain, [email protected] . . . . . . . . . . . . . . . . . . . 21Martínez-Sykora, AntonioUniversity of Southampton, UK,
[email protected] . . . . . . . . 91Menezes, Mozart B.C.KEDGE Business School, France, [email protected] . . . . . . . . 93Mesa, Juan A.Universidad de Sevilla, Spain, [email protected] . . . . . . . . . . . . 49, 67, 73, 79, 81Meyer, AnneTU Dortmund University, Germany, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Minner, StefanTechnical University of Munich, Germany, . . . . . . . . . . . . . . . . . . . . . . . . . 65Monge, Juan F.Universidad Miguel Hernández de Elche, Spain, [email protected] . 19, 61Moreno-Pérez, José A.Universidad de La Laguna, Spain, [email protected] . . . . . . . . . . . . . . . . . . . 95Munoz-Marquez, ManuelUniversidad de Cádiz, Spain, [email protected] . . . . . . . . . . . . . . . . . . 83
UTHOR INDEX N Nelas, JoséUniversidade de Coimbra, Portugal, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 O Ordoñez, FernandoUniversidad de Chile, Chile, [email protected] . . . . . . . . . . . . . . . . . . . . . . 55Ortega, Francisco A.Universidad de Sevilla, Spain, [email protected] . . . . . . . . . . . . . . . . . . . . . . 35, 81 P Peiró, JuanjoUniversitat de València, Spain, [email protected] . . . . . . . . . . . . . . . . . . . . . 53Piccolo, CarmelaUniversità di Napoli Federico II, Italy, [email protected] . . . . . . . . 31Piedra-de-la-Cuadra, RamónUniversidad de Sevilla, Spain, [email protected] . . . . . . . . . . . . . . . . . . . . . 35, 81Pinar-Pérez, Jesús M.CUNEF, Spain, [email protected] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93Plana, IsaacUniversitat de València, Spain, [email protected] . . . . . . . . . . . . . . . . . 47, 89Pouls, MartinResearch Center for Information Technology (FZI), Germany, . . . . . . 37Pozo, Miguel A.Universidad de Sevilla, Spain, [email protected] . . . . . . . . . . . . . . . . . . 81, 87Puerto, JustoUniversidad de Sevilla, Spain, [email protected] . . . . . 23, 41, 43, 63, 71, 75, 87 R Reula, MiguelUniversitat de València, Spain,
[email protected] . . . . . . . . . . . . . . . . . . . 89Rodríguez Carreño, IgnacioUniversidad de Navarra, Spain, [email protected] . . . . . . . . . . . . . . . . . . 59Rodríguez-Chía, Antonio M.Universidad de Cádiz, Spain, [email protected] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21, 29, 61, 63, 87Rodríguez-Madrena, MoisésUniversidad de Sevilla, Spain, [email protected] . . . . . . . . . . . . . . . . . . . . . . . 75Rodríguez-Pereira, JessicaHEC Montréal, Canada, [email protected] . . . . . . . . . . . . . . . 91Ruiz-Hernandez, Diego INDEX
Sheffield University Management School, UK, d.ruiz-hernandez@sheffield.ac.uk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93Ruiz-Sánchez, Miguel A.Universidad Complutense Madrid, Spain, . . . . . . . . . . . . . . . . . . . . . . . . 77 S Sainz-Pardo, José L.Universidad Miguel Hernández de Elche, Spain, [email protected] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Salazar-González, Juan-JoséUniversidad de La Laguna, Spain, [email protected] . . . . . . . . . . . . . . . . . . . . 65Saldanha da Gama, FranciscoUniversidade de Lisboa, Portugal, [email protected] . . . . . . . . . . . . 11, 53Sanchis, José M.Universidad Politécnica de Valencia, Spain, [email protected] . 47, 89Santos-Peñate, Dolores R.Universidad de Las Palmas de Gran Canaria, Spain, [email protected] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95Semet, FrédéricCRIStAL Centre de Recherche en Informatique Signal et Automatiquede Lille, France, and INRIA, France, [email protected] . . . . . . . . . . 55Serra, DanielUniversitat Pompeu Fabra, Spain [email protected] . . . . . . . . . . . . . . . . 51Speranza, M. GraziaUniversità degli Studi di Brescia, Italy, [email protected]@unibs.it