Mathematical Model for the Home Health Care Routing and Scheduling Problem with Multiple Treatments and Time Windows

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1 Mathematcal Metho n Scence an Engneerng Mathematcal Moel for the Home Health Care Routng an Scheulng Problem wth Multple Treatment an Tme Wnow Anré Felpe Torre-Ramo, Egar Hernán Alfono-Lzarazo, Lorena Slvana Reye-Rubano, Carlo Leonaro Quntero-Araújo Abtract Home health care prove to patent wth pecal conton n whch the atance requre n ther home. Depenng on the pathology, each patent receve pecfc home care ervce from pecalt, manly octor, therapt an nure. In th context the Home Health Care Routng an Scheulng Problem (HHCRSP) relate wth routng an cheulng of the qualfe peronnel. It ntegrate the Nure Roterng Problem (NRP) an the Vehcle Routng Problem (VRP). The HHCRSP coner contrant relate wth tme wnow, workloa an attenton capacty among other lmtaton aocate wth patent an taff. Due the cot an qualty mplcaton that th kn of ervce generate n health care compane, th artcle preent a mxe nteger lnear programmng moel for plannng the peroc cheule of mecal taff an the route plannng for to patent vt. Keywor Home health care, mxe nteger lnear programmng, qualfe taff cheulng, taff routng. H I. INTRODUCTION OME health care a ervce that mecal nttuton prove to patent who, ue to ther health conton, can be treate n ther home, n other cae t' an trategy to ncreang the capacty of room n hoptal. Takng nto account the avalablty of qualfe peronnel (octor, nure, an therapt), the health ector compane offer a varety of treatment requre by patent n whch tme, cot an qualty of the ervce are crucal; therefore peronnel cheulng an the routng of vt ha great mportance. The optmzaton of home health care ha long been a fel of nteret for the operaton reearch, n th context the Th work wa upporte by the Mater n Operaton Management an the Internatonal School of Economc an Management Scence (EICEA) of the Unvera e La Sabana, Chía, Colomba. Anré Felpe Torre-Ramo wth the Internatonal School of Economc an Management Scence (EICEA), Unvera e La Sabana, Chía, Colomba (correponng author to prove phone: ext.: 25109; e-mal: [email protected]). Egar Hernán Alfono-Lzarazo wth the Engneerng Department, Unvera e La Sabana, Chía, Colomba (e-mal: [email protected]). Lorena Slvana Reye-Rubano wth the Internatonal School of Economc an Management Scence (EICEA), Unvera e La Sabana, Chía, Colomba (e-mal: [email protected]). Carlo Leonaro Quntero-Araújo wth the Internatonal School of Economc an Management Scence (EICEA), Unvera e La Sabana, Chía, Colomba (e-mal: [email protected]). hghlghte area of tuy are Home Health Care Routng an Scheulng Problem (HHCRSP), whch cover apect of programmng an plannng of route for the mecal taff are the Roterng Problem (RP), e.g. Nure Roterng Problem (NRP), an the plannng of route for vt to patent conere a Vehcle Routng Problem (VRP). Thee problem are conere a NP-Har [1], [2]. In th paper peronnel cheulng am to allocate mecal taff member to patent; th cheulng mut be performe accorng to the type of pathology of each patent an the avalablty of tme for patent an taff. In th paper the man treatment tue are relate wth pathologe of pallatve type, chronc care, bloo antcoagulaton an omcle for woun care. Depenng on the treatment there are three type of pecalt who can prove thee treatment: octor, nure, an therapt. Each patent requre a level of peronalze mecal care, gven that ome patent are n evere conton an requre fewer nterval between treatment, a oppoe to patent preentng better health. Th generate a peroc plannng of each pecalt accorng to the type of treatment an health conton of each patent. In orer to cheule an plan the route of patent vt t neceary to coner each patent' tme wnow, whch a tme lot n the ay whch the patent efne or requre the mecal care. Another mportant apect n the plannng of route the tartng an en pont of each taff member aly, for th paper a mult-epot problem conere, n whch cae the home of each pecalt (octor, nure, an therapt) the begnnng an en of each route, th apect ncreae the complexty of the moel epenng on the number of taff member. The outlne of the artcle a follow. Secton II preent a lterature revew for the HHCRSP. The charactertc on taff an patent for home care are preente n ecton III. A mathematcal moel for the HHCRSP preente n ecton IV. The reult of the mathematcal moel are hown n ecton V. Concluon an recommenaton for future reearch are preente n ecton VI. II. LITERATURE REVIEW The home health care routng an cheulng problem (HHCRSP), a mentone above, make up two problem ISBN:

2 Mathematcal Metho n Scence an Engneerng aocate wth each operaton nvolve. In term of taff plannng or the allocaton of mecal taff to patent referre to the Nure Roterng Problem (NRP) [3], an route of vt to patent have been conere uner fferent varaton of the Vehcle Routng Problem (VRP) [2], [4]. The mot tue VRP varaton n the HHCRSP the Vehcle Routng Problem wth Tme Wnow (VRPTW) [1], [5] [8], whch nclue the aly tme lot that the patent ha to receve mecal attenton. Other varaton of the VRP apple to the HHCRSP have been tue nepenently are the Mult Travelng Saleman Problem wth Tme Wnow (MTSPTW), the Vehcle Routng Problem wth Mult-Depot (VRPMD) an the Vehcle Routng Problem wth Mult-Pero (VRPMP), whch nten to characterze multple taff an multple pont n whch the taff tart an en each route repectvely. [9] [11]. Dfferent methoologe have been ue to olve the HHCRSP wthn operaton reearch. Wthn the exact metho the tuy of Y. Kergoen, C. Lenté an J-C Bllaut [9], whch eek to etermne the route of the mecal taff vtng patent' home. In orer to o o they etermne a whole lnear programmng moel. Another exact metho ue the Branch-an-Prce algorthm, n the paper [6] the author ue th algorthm to agn taff an etermne route by conerng vt per group of patent. Smlarly heurtc metho have been ue to olve the HHCRSP, a n the artcle of D. Mankowka, F. Meel y C. Berwrth [10], n whch the author evelop a heurtc that etermne vt to patent through ervce nterconnecte by heterogeneou taff. A. Copp, P. Dett an J. Raffaell [12] evelop a heurtc bae on a local earch to etermne the peronnel plannng an routng of vt. Depte the goo reult that generate the exact an heurtc metho, fferent author have ue metaheurtc, whch allow the problem to be evelopment wth more ata n a reaonably hort tme. In the paper [8] the author preent the applcaton of the metaheurtc calle Partcle Swarm Optmzaton (PSO) n the programmng of the houe mecal taff. In the paper [13] the author apply Genetc Algorthm (GA) an Tabu Search (TS) metaheurtc to the elvery of rug an the collecton of bologcal ample. Smulate Annealng (SA) an Tabu Search (TS) metaheurtc are propoal n the paper [14] to etermne the cheule of therapt wthn the mecal treatment of patent home. On the other han, the author have focue ther reearch on varou objectve functon, the mot common the mnmzaton of the cot of operaton, where agnment, overtme an reagnment of taff cot are conere [7], [15], [16], an cot aocate to taff an tranport [3], [6], [11], [13], [17]. Another objectve manly aocate wth the routng of the taff the mnmzaton of tme an tance travele from the operaton [8], [10], [18] [20], whch nclue mnmzng the total journey unertaken by taff to make vt to the patent. Th artcle focue on mnmzng the total tme of operaton, a a component of the level of patent atfacton. Atonally, the tuy of the cheulng of peronal an the plannng of route of patent vt ntegratng the MTSPTW, the VRPMD an the VRPMP n a ngle problem: Mult- Travelng Saleman Problem wth Tme Wnow, Mult- Depot an Mult-Pero (MTSPTWMDMP). III. CHARACTERISTICS OF STAFF AND PATIENTS IN HOME HEALTH CARE In the mot of the reve artcle only one type of taff conere. The home care ytem tue n th paper, the ervce can be prove by nure, octor an therapt. On the other han the HHCRSP coner the attenton of fferent ervce or pathologe of the patent. In th artcle four type of ervce are conere (omcle, bloo antcoagulaton, chronc care, pallatve care), an accorng to the treatment of thee pathologe patent requre more than one type of taff. The legal an economc apect relate wth the workng tme of the mecal taff are conere [5]. A ummary of the apect conere n our moel are hown n Fgure 1, aapte from Bertel an Fahle [18]. Fg. 1 Charactertc for the HHCRSP Patent have charactertc that, n aton to the charactertc of the mecal taff, elmt the operaton; one of the mot mportant the tme wnow, whch repreent the tme lot that each patent efne or requre for the home vt. There are alo feature aocate wth the pathology of each patent. One of them the eman for peronnel, a mentone above, the type of pathology etermne the type of taff requre an the frequency between vt, whch epenent on the conton of each patent' health. IV. MATHEMATICAL FORMULATION FOR THE HHCRSP Th artcle propoe a mathematcal moel for the HHCRSP wth fferent type of pecalze peronnel (octor, nure an therapt), whch tart an en every route n ther own home (mult-epot), an performe n a ISBN:

3 Mathematcal Metho n Scence an Engneerng horzon of tme (mult-pero). The problem efne a a recte graph G=(V,A) wth a et V = CM CE CT PM of noe, whch refer to the et of noe correponng to the epot repreente by the home of octor (CM), home of nure (CE) an home of therapt (CT), an noe of patent (PM). An the et of arc A = {(, j):, j ϵ V, j}. Every patent ϵ PM uffer from a unque pathology, whch clafe nto four ervce: omcle, bloo antcoagulaton, chronc care an pallatve care. Each ervce ϵ S erve by the type of peronal p ϵ P requre accorng to the matrx. Atonally, each patent ha a eman for vt accorng to the type of taff an ervce ( DM, DE, DT ), thee vt are performe wthn a tme horzon n ay ( ϵ D) wth a perocty accorng to the patent an the type of peronnel ( KM, KE, KT ). On the other han each patent' tme wnow frame wthn a length of tme per ay ( e, l ), n whch taff mut reach the houe of the patent n e mnmum an maxmum n l. A mentone earler, each taff member tart an en t route n ther repectve home an they have a maxmum workng p tme per ay TM. In aton the travel tme ( TV ) ffer accorng to the type of peronnel, nce octor are moblze by mean of prvate tranport that fater than publc tranport by whch nure an therapt are moblze. The parameter an econ varable ue from moellng the HHCRSP are hown below n Table I. Parameter Table I Notaton ue from moellng the HHCRSP p TVj Travel tme of peronal p from the patent to the patent j. p TS Tme of treatment requrng the patent of the peronal p n ervce. DM Number of vt requre by the patent of octor n ervce. DE Number of vt requre by the patent of nure n ervce. DT Number of vt requre by the patent of therapt n ervce. SP p Peronal p attenng the ervce. TM Maxmum workng tme of the ay of the taff. e Start tme of the tme wnow of patent on the ay. l Clong tme of the tme wnow of patent on the ay. M Large number. N Inex ze. KM Pero of tme between octor vt requre by the patent. KE Pero of tme between nure vt requre by the patent. KT Pero of tme between therapt vt requre by the patent. Plannng horzon. H Decon varable X j Bnary: 1. If the peronal p vt the patent an then the patent j on the ay. 0. On the contrary. Y Tme of arrval of the peronal p vt the patent on the ay. U Auxlary varable to avo ubtour to vt each patent. j The propoe mxe nteger lnear programmng moel below to olve the HHCRSP. Objectve functon p p j j p (1) Mnmze Z X TV TS * SP jv Subject to V jv pp D S p X j TS TM, V, p P, D S (2) e Y l, PM, p P, D (3) 1,,,,, (4) Y TS TV Y M X j V j p P D p p j j j S X j X j, PM, p P, D (5) jpm jpm jv, j jpm PM X 0, V, p P, D j X 1, V / PM, p P, D j X 1, j V / PM, p P, D j jv / PM, j p pp, p j KM jv, j pm f X 0, V, D j pf X j 1, PM, H KM (10) KE pf X j 1, PM, H KE (11) jv, j pe f KT pf X j 1, PM, H KT (12) jv, j pt f jv D pm jv D pe jv D pt jv pm jv pe jv pt X SP DM, PM, S j p X SP DE, PM, S j p X SP DT, PM, S j p X 1, PM, D j X 1, PM, D j X 1, PM, D j j j 1,,,, (6) (7) (8) (9) (13) (14) (15) (16) (17) (18) U U X N N j PM p P D (19) X 0,1,, j V, j, p P, D (20) j Y 0, V, p P, D (21) U 0, V (22) The moel preent the routng an cheulng of the home mecal taff, mnmzng the total tme of operaton (tranportaton an ervce) (1). Contrant (2) etermne the maxmum workng loa per ay for each taff. Contrant (3) ISBN:

4 Mathematcal Metho n Scence an Engneerng an (4) mpoe the tme wnow per each patent each ay accorng to the taff. Contrant (5) enure the flow of taff patent every ay. Contrant (6) avo fcttou route of the taff. Contrant (7), (8) an (9) etermne the mecal taff every ay goe out an return to ther repectve home. Contrant (10), (11) an (12) etermne the pero of tme between vt to each patent accorng to the type of taff an plannng horzon. Contrant (13), (14) an (15) guarantee the fulfllment of the eman for vt of each patent accorng to the type of taff requre. Contrant (16), (17) an (18) mpoe maxmum one ervce wth every vt to mecal peronnel for each patent per ay. Contrant (19) elmnate the ubtour generate n the programmng moel. Fnally, contrant (20) efne the varable X an contrant (21) j an (22) etermne non-negatvty Y an U varable. V. RESULTS A mentone n ecton II the moel propoe n th artcle complex, an bae on a Mult-Travelng Saleman Problem wth Tme Wnow, Mult-Depot an Mult-Pero (MTSPTWMDMP). For HHCRSP moel valaton tet wth nformaton from a company' nutry n Colomba, tet ha 16 patent of ervce: 1. Domcle, 2. Bloo antcoagulaton, 3. Chronc care an 4. Pallatve care. Each ervce requre attenton of fferent type of taff (octor, nure an therapt), accorng to the ervce requre of one or another peronal type a hown n the matrx SP (ee Table II). In p aton, Table II how the number of people for each type of taff, where a total of 19 taff member etermne. The 16 patent requre a total of 101 vt of all mecal taff n a two week' tme horzon, n aton to a perocty between vt a hown n Table III. Table II Servce hanle by each type of peronal Type of taff Number of Type of ervce pecalt Doctor Nure Therapt Total 19 Patent Table III Type of ervce, eman an perocty of vt requre by patent Type of ervce Perocty between vt Deman of vt Doctor KM Nure KE Therapt KT Doctor DM Nure DE Therapt DT Total The moel wa mplemente ung GAMS commercal oftware veron , wth a tme lmt of 4000 econ n a peronal computer Intel(R) Core(TM) U CPU wth 1.6 GHz wth 8 GB of RAM. The oluton of the moel etermne the route per ay neee to meet the eman of vt that patent requre. Each route carre out by a member of the mecal taff who begn an en at home. Table IV how the route to perform on ay 1, whch entfe 3 route that are performe by the nure 6, nure 8 an therapt 5 repectvely. For example, the nure 6 route tart n her home, then vt the patent 6, 5, 8 an 13 n that repectve orer, an fnally part of the lat patent (13) to her home a the en pont of the route. The moel gve total of 27 route ve nto 11 ay (Appenx: Reult of the Moel of the HHCRSP), route are carre out by a total of 12 taff member (octor 1, octor 2, octor 3, nure 2, nure 6, nure 8, therapt 1, therapt 3, therapt 4, therapt 5, therapt 6 an therapt 7), a the total number of taff member mentone above are 19, therefore complance evence wth the route wth 7 member le, provng the optmzaton of human reource an the capacty to erve a greater number of patent. ISBN:

5 Mathematcal Metho n Scence an Engneerng Table IV Rout for ay 1 Day 1 Nure 6 Nure 8 Therapt 5 From To From To From To NH6 6 NH8 11 TH TH5 13 NH6 2 NH8 The total operaton tme of the taff n all the tme horzon 8346 mnute, for whch the 64.1% of the tme correpon to the tme of travel, an the remanng 35.9% of tme correpon to the tme of ervce. I. CONCLUSION Th artcle propoe a moel for the problem of routng an cheulng of mecal taff n the home health care ytem, whch coner charactertc a fferent type of taff, fferent ervce, mult-epot, tme wnow an multpero. Thee feature facltate the applcaton of th moel n real conton relate wth the home health care ervce. The tuy of the HHCRSP can lea n many recton. Frt mplement heurtc an metaheurtc, whch allow the analy of the problem wth more ata n le tme. On the other han the ntegraton of other type of ervce a elvery an pck-up of mecne an bologcal ample an the emergency ervce, whch nvolve new contrant an coneraton aocate wth uncertanty n the eman an the avalablty of taff. APPENDIX: RESULTS OF THE MODEL OF THE HHCRSP The reult of the moel of the HHCRSP etermne a total of 27 route n 12 ay (two week) of operaton. Day 1 Nure 6 Nure 8 Therapt 5 From To From To From To NH6 6 NH8 11 TH TH5 13 NH6 2 NH8 Day 3 Therapt 3 From To TH3 5 5 TH3 Day 4 Doctor 3 Nure 2 Therapt 4 Therapt 6 From To From To From To From To DH3 10 NH2 10 TH4 8 TH TH TH4 12 DH3 1 NH2 Day 5 Doctor 1 Nure 8 From To From To DH1 11 NH DH NH8 Day 6 Nure 2 From To NH NH2 Day 7 Doctor 1 Therapt 1 Therapt 5 Therapt 7 From To From To From To From To DH1 5 TH1 7 TH5 11 TH TH DH TH TH7 Day 8 Doctor 1 Nure 2 From To From To DH1 10 NH DH NH2 ISBN:

6 Mathematcal Metho n Scence an Engneerng Day 9 Doctor 3 From To DH DH3 Day 10 Nure 2 Therapt 6 Therapt 7 From To From To From To NH2 10 TH6 8 TH TH TH7 1 NH2 Day 11 Nure 8 Therapt 6 From To From To NH8 3 TH NH TH6 Day 12 Doctor 1 Doctor 2 Nure 2 Nure 8 From To From To From To From To DH1 5 DH2 4 NH2 16 NH DH NH DH1 12 NH8 ACKNOWLEDGMENT The author thank the ponorhp of th project to the Mater n Operaton Management an the Internatonal School of Economc an Management Scence (EICEA) of the Unvera e La Sabana. [4] A. Trautamweer an P. Hrch, Optmzaton of aly cheulng for home health care ervce, J. Appl. Oper. Re., vol. 3, no. 3, pp , [5] E. Cheng an J. Lynn, A Home Health Care Routng an Scheulng Problem, n Oaklan Unverty, Rce Unverty. Techncal Report. USA, [6] M. S. Ramuen, T. Juteen, A. Dohn, an J. Laren, The Home Care Crew Scheulng Problem: Preference-bae vt cluterng an temporal epenence, Eur. J. Oper. Re., vol. 219, no. 3, pp , Jun [7] P. Eveborn, P. Flberg, an M. Rönnqvt, Lap Care an operatonal ytem for taff plannng of home care, Eur. J. Oper. Re., vol. 171, no. 3, pp , Jun [8] C. Akjratkarl, P. Yenraee, an P. Drake, PSO-bae algorthm for home care worker cheulng n the UK, Comput. In. Eng., vol. 53, no. 4, pp , Nov [9] Y. Kergoen, C. Lenté, an J.-C. Bllaut, Home health care problem: An extene multple travelng aleman problem, n Proceeng of the 4th Multcplnary Internatonal Scheulng Conference: Theory an Applcaton - MISTA, 2009, pp [10] D. S. Mankowka, F. Meel, an C. Berwrth, The home health care routng an cheulng problem wth nterepenent ervce., Health Care Manag. Sc., vol. 17, no. 1, pp , Jun [11] J. F. Bar, Y. Shao, an H. Wang, Weekly cheulng moel for travelng therapt, Socoecon. Plann. Sc., vol. 47, no. 3, pp , Jul [12] A. Copp, P. Dett, an J. Raffaell, A plannng an routng moel for patent tranportaton n health care, Electron. Note Dcret. Math., vol. 41, pp , Jun [13] R. Lu, X. Xe, V. Auguto, an C. Rorguez, Heurtc algorthm for a vehcle routng problem wth multaneou elvery an pckup an tme wnow n home health care, Eur. J. Oper. Re., vol. 230, no. 3, pp , Apr [14] J. D. Grffth, J. E. Wllam, an R. M. Woo, Scheulng phyotherapy treatment n an npatent ettng, Oper. Re. Heal. Care, vol. 1, no. 4, pp , Dec [15] E. Lanzarone an A. Matta, Robut nure-to-patent agnment n home care ervce to mnmze overtme uner contnuty of care, Oper. Re. Heal. Care, Jan [16] G. Carello an E. Lanzarone, A carnalty contrane robut moel for the agnment problem n Home Care ervce, Eur. J. Oper. Re., Jan [17] P. M. Koeleman, S. Bhula, an M. van Meerbergen, Optmal patent an peronnel cheulng polce for care-at-home ervce faclte, Eur. J. Oper. Re., vol. 219, no. 3, pp , Jun [18] S. Bertel an T. Fahle, A hybr etup for a hybr cenaro: combnng heurtc for the home health care problem, Comput. Oper. Re., vol. 33, no. 10, pp , Oct [19] S. Begur, D. Mller, an J. Weaver, An ntegrate patal DSS for cheulng an routng home-health-care nure, Interface (Provence)., vol. 27, no. 4, pp , [20] E. Alfono, V. Auguto, an X. Xe, Mathematcal Programmng Moel for Annual an Weekly Bloomoble Collecton Plannng, n IEEE Tranacton on Automaton Scence an Engneerng, 2014, vol. PP, no. 99, pp REFERENCES [1] S. Nckel, M. Schröer, an J. Steeg, M-term an hort-term plannng upport for home health care ervce, Eur. J. Oper. Re., vol. 219, no. 3, pp , Jun [2] J. Steeg an M. Schröer, A hybr approach to olve the peroc home health care problem, Oper. Re. Proc., pp , [3] W. J. Gutjahr an M. S. Rauner, An ACO algorthm for a ynamc regonal nure-cheulng problem n Autra, Comput. Oper. Re., vol. 34, no. 3, pp , Mar ISBN: