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A planning and routing model for patient transportation in health care

A planning and routing model for patient transportation in health care
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  See discussions, stats, and author profiles for this publication at: A planning and routing model for patienttransportation in health care  ARTICLE   in  ELECTRONIC NOTES IN DISCRETE MATHEMATICS · JUNE 2013 DOI: 10.1016/j.endm.2013.05.084 CITATIONS 2 READS 13 3 AUTHORS , INCLUDING:Alberto CoppiAzienda Unità Sanitaria Locale 4 Prato 5   PUBLICATIONS   16   CITATIONS   SEE PROFILE Paolo DettiUniversità degli Studi di Siena 50   PUBLICATIONS   274   CITATIONS   SEE PROFILE All in-text references underlined in blue are linked to publications on ResearchGate,letting you access and read them immediately.Available from: Paolo DettiRetrieved on: 14 January 2016  This article appeared in a journal published by Elsevier. The attachedcopy is furnished to the author for internal non-commercial researchand education use, including for instruction at the authors institutionand sharing with colleagues.Other uses, including reproduction and distribution, or selling orlicensing copies, or posting to personal, institutional or third partywebsites are prohibited.In most cases authors are permitted to post their version of thearticle (e.g. in Word or Tex form) to their personal website orinstitutional repository. Authors requiring further informationregarding Elsevier’s archiving and manuscript policies areencouraged to visit:  Author's personal copy A planning and routing model for patienttransportation in health care Alberto Coppi a , 1 , Paolo Detti a , 2 , Jessica Raffaelli a , 3 a Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, University of Siena, Via Roma, 56, 53100 Siena, Italy  Abstract In this paper, a problem concerning both the planning of health care services andthe routing of vehicles, for patients transportation is addressed. An integrated ap-proach, based on the column generation technique, is proposed to solve the planningand routing problem. Preliminary results on real data show the effectiveness of theproposed approach. Keywords:  Planning and routing, health care services, heuristic. 1 Introduction and problem description In this paper, a problem concerning the planning of health care services andthe transportation of patients, in not urgent conditions, is addressed. Theaddressed problem arises from a real world context, namely the health caresystem of Tuscany, an Italian region [1]. The transportation services are per-formed by non-profit organizations, by means of heterogeneous vehicles (e.g., 1 Email: 2 Email: 3 Corresponding author, Email:  Available online at Electronic Notes in Discrete Mathematics 41 (2013) 125–1321571-0653/$ – see front matter © 2013 Elsevier B.V. All rights  Author's personal copy ambulance, bus, car, etc.), located in geographically distributed depots. Inthis context, the transportation services include the transportation of patientsfor discharges, transfers and pickup from their home addresses to health carefacilities for health care services, such as medical examinations, consultations,therapeutic treatments. The problem has a tactical and an operational dimen-sion [1]. The operational dimension is to determine, on the basis of a givenset of transportation requests and a timetable of the visits, a transportationplan for a set of patients from a set of srcins to a set of destinations (and viceversa). The aim is of minimizing the total transportation costs, satisfying aset of constraints on the quality and the timing of the service offered. Theoperational plan of transportation must take into account of agreements, con-straints and roles established between healthcare management and non-profitorganizations, which perform the transportation services and have a certaindegree of autonomy in decision-making. The tactical dimension of the problemaims to integrate the transportation planning, i.e., the routing of the vehicles,with the definition of the visits timetable, i.e., the time in which the healthcare service of each patient will start. In this case, the problem consists indetermining a timetable of the health care services of the patients, in sucha way that the transportation costs are minimized, always respecting servicequality constraints. Each transportation request corresponds to a patient thatmust be carried on by a vehicle, from a pickup location to a delivery location,satisfying capacity, feasibility and time windows constraints. The capacityconstraints are related to the number of available seats in the vehicle andto the number of seats occupied by each patient (e.g., patients may need astretcher or a wheelchair to be transported). The feasibility constraints set thevehicle typology that can be used to serve a given patient typology, depend-ing on the specific set up of the vehicle (for example, a patient may requireto be transported by an ambulance). Time windows constraints are relatedto the patients, requiring that each pickup and delivery location need to bereached within a time interval, and depend on the timetable of the visits. Inorder to guarantee quality of service requirements, constraints on the lengthsof the routes and on the travel and waiting times of each patient must be alsoconsidered.The operational dimension of the problem we address in the paper isstrictly related to a problem known in the literature as Dial-A-Ride Prob-lem (DARP), [3,6,9,11], a generalization of the Pickup and Delivery Problem with Time Windows (PDPTW)[2,4,10]. In fact, the operational dimension of the problem can be defined as a Heterogeneous Multi-Depot Dial-A-Ride-Problem. Both the DARP and PDPTW are  NP   −  hard , therefore, the de-  A. Coppi et al. / Electronic Notes in Discrete Mathematics 41 (2013) 125–132 126  Author's personal copy velopment of solution methods for these problems have mainly focused onheuristics [5]. In some cases, the column generation technique has been suc-cessfully applied to develop exact and heuristic methods for these types of problems [2,4,7,8,10,12]. The above approaches from the literature are fo- cused in solving the operational problem (i.e., considering only the routingaspects), while in this paper, we present an integrated approach for solvingthe tactical and operational dimension of the problem. More precisely, wepropose an efficient heuristic algorithms based on a set covering formulationof the problem for finding both the visits timetable and the routing of thevehicles. Column generation is employed both to generate a set of possibleroutes and to plan the timetable of the health care services. Simulation resultsbased on real world data arising from the healthcare Italian system show theeffectiveness of the proposed approach.The paper is organized as follows. In Section 2, a mathematical formu-lation and a solution approach for the tactical dimension of the problem arepresented. Preliminary results are reported in Section 3. 2 Mathematical model and formulation The addressed problem is characterized by two main sets of elements: theVehicles Types and the Transportation Requests. In the following, the maincharacteristics of these two elements are listed.Vehicle Type characteristics: depot location, capacity (i.e., the number of seats in the vehicle), type (ambulance, equipped vehicle, car or bus), numberof available vehicles.Transportation Request characteristics: pickup and delivery location, numberof seats occupied on a vehicle, sets of possible pickup time and delivery timewindows (depending on the health care service type), health care service type(e.g., transfer, treatment, etc.). 2.1 Problem formulation and solution approach  The addressed problem can be modeled as a set covering problem, as describedin the following. Let  S   be the set of health care service types. The servicetype defines both the service typology (e.g., discharge, treatment etc.) andthe health care structure where the service is performed. Let  n ts  be the max-imum number of health care services of type  s  that can be allocated to timeslot  t  (in the health care structure associated to service  s ). Let Q be the setof transportation requests and let  Q s  be the set of transportation requests  A. Coppi et al. / Electronic Notes in Discrete Mathematics 41 (2013) 125–132 127
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