Models for Intra-Hospital Patient Routing

Transcription

1 Models for Intra-osptal Patent Routng Belma uran, Verena Schmd and Karl. F. Doerner Unversty of Venna, Venna, Austra Johannes Kepler Unversty Lnz, Lnz, Austra Abstract he am of ths work s to ntroduce a new model for ntra-hosptal routng of patents, consderng both clent- and management related ssues. Patents n a hosptal have a fxed appontment, such as x-rays or ultrasonc and due to medcal reasons they cannot go on ther own, so they wll be escorted by porters. In our model logstcal costs for the usage of porters and patent nconvenence wll be mnmzed. We show that the dfferent developed model varants are tractable for realstc problem nstances of medum-szed hosptals. I. INRODUCION AND LIERAURE REVIEW In the recent past transportaton, schedulng and supply chan management orented problems for health care related applcatons are ganng ncreased attenton n the scentfc communty. Decson stuatons may arse on a strategc, tactcal as well as an operatonal level. In the current paper we wll concentrate on those occurrng on the operatonal level. he underlyng problems can formally be modeled n terms of combnatoral optmzaton problems comng from schedulng (for both personnel ([1],[2]), resources and rooms such as operatng theatres ([2],[3]), transportaton routng (of ambulances [4], nurses and doctors) and supply chan management (supply, delvery, reverse logstcs of medcal waste). In large hosptals, where dfferent wards are typcally spread across the ste n so called pavlons, routng operatons come at hgh costs. hose costs typcally nclude pure routng (.e. dstance/travel tme) related costs, but may also nclude addtonal costs comng from delays, mssed appontments, neffcent usage of resources, etc. ence fndng good solutons to the underlyng routng operatons s hghly essental. In the remander of ths paper we wll focus on transportaton routng problems arsng n the feld of patent routng wthn hosptals. Patents have a fxed appontment, such as x-rays, ultrasonc, blood testng or surgery. Due to medcal reasons they cannot go on ther own, but rather they wll be escorted there (and back) by porters. ence two transportaton requests from the porters pont of vew need to be scheduled, such that ther routes are optmzed. o the best of our knowledge so far only classcal patent routng problems (such as dal-a-rde problems) have been consdered n the scentfc communty. See [4], [5], [6], [7], [8], [9] and [10] for addtonal detals. he problem belongs to the famly of pckup and delvery problems. Ones of the frst papers to address ths ssue have been developed by [5] and [7]. her optmzaton was focused on mnmzng the transportaton costs. Clent-centered aspects (such as watng tmes observed by real persons) have been ncluded by [11] and [10]. In ther approaches maxmum rde tme restrctons have been ncluded n the optmzaton. A formal descrpton closely related to real-world patent transportaton has been proposed n [12]. For a recent survey on dfferent formulatons see [6], [13] and [14]. A decson support system for real-world patent transportaton has been descrbed n [8], based on the problem and soluton technques proposed n [4]. Several varants of ths problem have been studed recently (see [15], [16] and [17]). rue Pareto optmzaton consderng the two conflctng obectves (costs vs. user nconvenence) has been proposed n [9]. he problem under consderaton s a specal case of the classcal dal-a-rde problem, arsng wthn hosptals. he capacty of the vehcle (.e. the porter) s set to one. Furthermore transportaton requests are pared n a sense that every patent trggers two transportaton requests and ther nconvenence s supposed to be mnmzed. From the patents pont of vew t would be benefcal f the very same porter could escort them on both resultng transportaton requests. Addtonal extensons under consderaton are the porters themselves. Besdes executng transportaton requests they are also bound to fulfllng other tasks beyond the scope of ths model. ence ther duty tme should be compact n a sense, that they can also be deployed for those tasks for tme slots of suffcent length.

2 Alternatvely the problem at hand can also be consdered a specal type of the stacker crane problem (see [1], [18] and [19]), whereas the latter s an applcaton of full-truck load movements. he stacker crane problem typcally arses n port operatons where contaners need to be moved between dfferent stacks, such that the costs assocated wth empty movements of the correspondng transportng devce are mnmzed. In the context of our applcaton contaners correspond to patents, who need to be transported between known locatons. Porters would correspond to the movng devce (.e. the crane) whose empty movements are supposed to be optmzed. he remander of the paper s organzed as follows. A detaled problem descrpton as well as a mathematcal formulaton s gven n Secton II. We start wth the formulaton of a general model and extend t accordngly n order to capture the above-mentoned features from the porters and patents pont of vew. Secton III gves a descrpton of the data that are used to test the model, as well as detaled numercal results. We conclude ths paper wth a summary of the manageral mplcatons and core fndngs of the nvestgated model n Secton IV. II. PROBLEM DESCRIPION In ths secton the patent routng and schedulng problem at a pavlon structured hosptals s gong to be descrbed. he problem focuses on the n-house transportaton of patents, were patents have to be transported between dfferent hosptal unts. Each patent has one medcal examnaton or surgery scheduled at one hosptal unt. From ths follows, that we can dstngush between two dfferent knds of transportaton requests for each patent. he frst transportaton request s an nbound transportaton request, where the patent has to be brought from her hosptal ward to the faclty where the medcal examnaton s scheduled. After the medcal examnaton ends, the patent has to be pcked up at ths faclty and be brought back to the hosptal ward. hs knd of transportaton requests wll be referred to as an outbound transportaton request of the patent. he transportaton requests of patents are done by the so called porters, who are non-medcal staff members responsble for logstcal operatons wthn a hosptal. he model presented wthn ths paper focuses on pavlon structured hosptals that are characterzed by locally dspersed hosptal unts. hese hosptal unts can be grouped nto several hosptal wards, where only patent beds are statoned, and several facltes where medcal examnatons (lke blood tests, X-rays and other) or where surgeres take place. Furthermore there are desgnated buldngs where porters are located and have to fulfll addtonal tasks whenever they are able to. hese are called home depots of porters. he porters, besdes ther other dutes, are assgned to escort patents on ther nhouse transportaton. hs duty of the porters s very mportant especally n case of elderly or dsabled persons. he patents are transported ether n stretchers or n wheelchars, dependng on ther condton. ence, the capacty of a porter s set to one,.e. a porter can only take care of at most one patent at tme. he am of ths work s to ntroduce a new model for ntrahosptal routng of patents, consderng both clent- and management related ssues. he underlyng obectves are typcally conflctng by nature. A weghted sum approach s used n order to optmze the underlyng tradeoff. Furthermore we show that the model at hand s tractable for realstc problem nstances of medum-szed hosptals. We assume that ths problem can be appled to many hosptals wth pavlon structure and help to ncrease patent convenence, reduce costs and neffcent usage of resources. A. General problem Our research has started wth the general problem. he general problem concentrates on fndng the optmal assgnment under consderaton to mnmze the travel tme between dfferent nodes at a pavlon structured hosptal and to reduce patent nconvenence mposed by long watng tmes. he total travel tmes at the hosptal consst of the two types of travel tmes: the travel tmes when porters travel wth the patent from the pckup to the delvery locaton, and when the porters travel wthout the patent (so called empty travel tmes). As the travel tmes between pckup and delvery locaton of a transportaton request are fx, they are not consdered n the optmzaton process. herefore, the travel tmes that occur n the obectve functon are only empty travel tmes. In the general problem, all medcal assstants are n ther home depots at the begnnng of the work day. As the lst wth transportaton requests that are scheduled for that day s already known n advance, the optmal assgnment can be done ahead. he assstants leave the depot and go to complete ther transportaton requests n the order that was computed before. After they have completed ther last transportaton request n the tour, they go back to ther home depots. In ths paper the problem of offlne or statc data, where the data are already known, s studed. All nformaton about transportaton requests s randomly generated. Pckup and delvery locatons for each transportaton request and the assocated tme pont when pckup or delvery should take place are already gven. As already mentoned, transportaton requests can be dvded nto two types, nbound and outbound transportaton requests, wth respect to whether the patent s beng escorted to her medcal examnaton, or beng pcked up afterwards. Dvson of the transportaton requests nto these two types was done because t s mportant to dstngush between two dfferent tme wndows. In the frst case, a patent cannot be brought to the delvery locaton after the gven tme pont,.e. the patent cannot come late to her medcal examnaton. he patent can however be brought there earler, but n ths case watng tme s mposed, whch s gong to be penalzed for each mnute the patent has to wat. In the second case, by outbound transportaton requests, an assstant should pck up the patent at the gven tme when the patent s medcal examnaton ends. If an assstant comes late, and a patent has to wat for her, a penalty for patent s watng s mposed. In ths case, however, assstant s allowed to come earler, but her watng tme s not penalzed n the general model. In our paper, all these facts are taken nto consderaton, and a model s gong to be presented that reles on these lmtatons and obectves. Pror to presentng the mathematcal model, the requred notaton s ntroduced. Dfferent sets and numbers used are lsted n able I.

3 ABLE I: NOAION FOR SES USED Abbrevaton Descrpton R I set of nbound transportaton requests R O set of outbound transportaton requests R set of all transportaton requests, R= R I U R O B set of bed statons O set of medcal examnaton rooms D set of depots P set of patents S set of porters N set of all nodes at the hosptal, N = B U O U D R set of all transportaton requests and depots, R = R U D Furthermore, as shown n able II, the followng data assocated wth tme and penaltes are used n the model: ABLE II: NOAION ASSOCIAED WI INPU PARAMEERS Abbrevaton Descrpton E D(s) A() S α β crtcal tme pont for transportaton request travel tme between nodes and (n mnutes) home depot of a porter s patent, to whom transportaton request refers to servce tme for transportaton request penalty for tme when porter travels empty penalty for watng tme (of patent) Wth E the crtcal tme pont for transportaton request s represented. In the case of an nbound transportaton request, E s the start tme of the medcal procedure, and n the case of an outbound transportaton request, E s the tme pont when the medcal procedure ends. ravel tmes between nodes at the hosptal are represented wth. he servce tme s represented wth S. he servce tme ncludes the tme that s necessary to prepare the patent for the transportaton request and also the travel tme between the pckup and the drop off locaton of her transportaton request. In our model, we have used several dfferent decson varables. In the next secton we wll ntroduce the bnary varables that stand for routng and schedulng decsons, assgnment decsons and penaltes. he routng and schedulng decsons are represented by the bnary flow varable x, whch s equal to 1 f the porter s s assgned to transportaton request after she completes the transportaton request, and equal to 0 otherwse. he bnary varable x s defned for, R and s S. he assgnment bnary varable y s serves to represent the assgnment of transportaton requests to porters. he varable s defned for R and s S and s equal to 1 f porter s s assgned to transportaton request, and s equal to 0 otherwse. Besdes the bnary varables, we have also used varables that are mostly assocated wth tme. Wth a the porter s arrval tme to the pckup node of a transportaton request s represented. After the porter has arrved to the pckup locaton she pcks up the patent, f the patent s ready (e.g. f the medcal examnaton has already ended), and the transportaton request can start. he tme when transportaton request starts s represented wth varable b. he actual travel tme between pckup and delvery node of transportaton request s ncluded n servce tme for each transportaton request. he arrval tme to the delvery locaton of transportaton request s represented wth c. After the porter has escorted the patent to her delvery locaton, she can leave the delvery node. he tme when porter leaves the delvery node of transportaton request s represented wth d. All decson varables used are lsted n able III. Abbrevaton a b c d x y s ABLE III: DECISION VARIABLES Descrpton arrval tme to the pckup locaton of request start tme of transport of transportaton request arrval tme to the delvery locaton of request tme pont when porter leaves the delvery locaton of transportaton request bnary flow varable bnary assgnment varable After all the data and ndces used are ntroduced, and the decson varables explaned, the mathematcal model can be presented: Mnmze α s subect to s x + β ( ( E ) c + ( b E ) ) (1) I O ys = 1 (2) yd s ) s = 1 s y = x, s ( (3) s (4) y R s = x, s (5) s + S c = b a b c d c E E b d + a + M 1 x ),, s d (6) (7) (8) (9) (10) ( (11) + a M 1 x ),, s ( (12) x,0,, y s,0, { } s { } s 1 (13) 1 (14) a, b, c, d 0 (15) o determne the obectve functon (1), we have used the weghted sum approach. Each of three dfferent terms that are consdered n the obectve functon has to be multpled wth a I O

4 coeffcent. owever, these terms can be dvded nto two categores, so there are only two dfferent coeffcents. he coeffcents were subectvely estmated by the authors. he total travel tme of porters s represented wth the frst term of the obectve functon. he optmal route should be created so that the travel tmes of porters are as short as possble. As already mentoned, the pckup and delvery node of one transportaton request are pared, that means that the tme travelled between a pckup locaton and ts assocated delvery locaton s fxed and cannot be changed or mproved anymore. Wth the frst term n the obectve functon, the total travel tme of porters s measured,.e. the travel tme between the delvery node of transportaton request to the pckup node of transportaton request, f the porter s s assgned to transportaton request after transportaton request. As x s defned for set R, whch ncludes depots besdes requests, the frst term then also contans travel tmes from the porter s home depot to her frst transportaton request (or rather to the pckup locaton of her frst transportaton request) and from the delvery locaton of her last transportaton request to her home depot. Wth the other group of terms t s made sure that the comfortableness of patents s also consdered n the optmal plan. he second term s used for nbound requests. Wth ths term the total watng tme for patents, from the tme pont when they arrve to faclty where ther medcal procedure s scheduled untl the tme pont when ther medcal procedure actually begns, s mnmzed. Smlarly, the thrd term serves for the outbound requests, and ensures that patents have to wat as short as possble to be pcked up after ther medcal procedure and brought back to ther bed. Both terms are multpled wth the same penalty coeffcent n the obectve functon, as t s equally mportant for us that patents don t have to wat long n both cases. In order to obtan the optmal solutons, some restrctons have to be consdered. Constrants (2) make sure that each transportaton request s served once. Wth (3) t s ensured that each porter has to vst her home depot. Constrants (4) and (5) are n and out degree constrants. Restrctons that make sure that every transportaton request s completed punctually also need to be consdered. Wth (6) the arrval tme to the target pont s calculated. Constrants (7) and (8) ensure that the start of the transportaton request cannot begn f the porter has not arrved yet and that the porter cannot be avalable for the next transportaton request before she has delvered the patent to the target locaton. A porter can be responsble for only one transportaton request at a tme. In case of an nbound request, wth (9) s guaranteed that patent cannot come late for her medcal procedure. On the other hand, n case of an outbound request, constrants (10) make sure that the begnnng of the transport s after the end pont of the surgery. If the porter comes earler to pck up the patent after the medcal procedure, the porter has to wat untl the procedure s done, and the transport cannot start untl then. Constrants (11) and (12) enable the connecton between two consequent transportaton requests, whereas (13) and (14) are bnary constrants. Constrant (15) s a non-negatvty constrant. hs mathematcal model was mplemented n XPRESS. In Secton III we are gong to ntroduce and explan the data used and the solutons we managed to obtan. here wll be more words on how the dfferent data were generated, whch types of problems were consdered and dfferent solutons wll be dscussed. B. Patent-centered extenson In ths secton the frst extenson of the model s gong to be ntroduced. For ths extenson the general model was used and the necessary changes were appled. he extenson s based on the wsh to provde the best servce for patents. he qualty of hosptal servce s measured through patents satsfacton. herefore ther convenence and well-beng should be of hgh prorty for hosptal management. In the general model the man goal was to reduce the patents watng tme. Now the queston mposes what else could be done n order to ncrease patents convenence. It was already mentoned that every patent trggers two transportaton requests, one from her hosptal ward to the medcal examnaton room and the other one from ths examnaton room back to the hosptal ward. As the medcal examnatons are stressful for the patents, and the necessary transportaton further ncreases ths stress, t would be reasonable to try to reduce ths nconvenence by assgnng the same porter to take care of the patent. If the same person pcks up the patent and escorts her to her medcal examnaton and afterwards pcks her up and delvers back to bed, the patent could develop a feelng of trust toward ths person. Knowng that there s one porter who s responsble for her could ncrease the comfort for the patent. In order to obtan the resultng assgnments and routes, the necessary changes mentoned above need to be mplemented n the model. As we have based ths extenson on our general model, only the changes wll be stated. he data used and the decson varables reman the same as n the general model. here are also no changes n the obectve functon. owever, we need addtonal constrans that wll make sure that one porter s assgned to one patent. he followng constrant y s = y s S,, R A() = A() (16) s makes sure that f the transportaton request s assgned to the same patent as the transportaton request, then porter s has to take care of both transportaton requests. In ths case, A() stands for a patent to whom transportaton request refers to. hs model was also mplemented n XPRESS and solved. At the end, all three models are compared. he analyss of the soluton obtaned s stated n Secton III. C. osptal-centered extenson In ths secton we are gong to ntroduce the second extenson of the model. So far we assumed that medcal assstants are assgned to complete transportaton requests, they start ther workday at ther home depot, and then perform the transportaton requests assgned. After they have completed all of ther transportaton requests, they go back to ther home depots. hs model however doesn t take nto consderaton the tme that porters have to wat empty between two consecutve transportaton requests. It s assumed that a porter completes one transportaton request, pcks up a patent and delvers her at the desred locaton and then goes to the

5 pckup locaton of her next transportaton request. If the porter arrves there earler than planned, she has to wat. From the managements pont of vew ths however s suboptmal. Porters are also responsble for executng addtonal tasks beyond the transportaton requests. So far (empty) watng tmes occurred somewhere n the hosptal compound and the management s not able to use ther resources effcently. ence we now want to consder ths ssue explctly. By sendng porters temporarly back home to ther home depots, porters could be assgned other tasks there. hs may lead to a deteroraton of the soluton wth respect to the dstance travelled empty by porters as they may encounter a detour va ther home depot. On the other hand ths allows to effcently use ther resources for other tasks (e.g. collecton of blood samples, delvery and supply of medcal nstruments, etc.) Porters however should only be sent back to ther home depots f the resultng tme spent there exceeds a certan mnmum tme span. akng all these nformaton nto consderaton, we have changed and extended the model accordngly. As the general model was our bass, we are now gong to ntroduce all the changes that were necessary to mplement. One of the man changes was to ntroduce new decson varables that wll force the model to send the porter back home f there s enough tme and another varable to capture the actual tme travelled empty. he decson whether the porter goes back to her home depot between her two consequent transportaton requests s represented by a bnary varable w. hs varable s defned for R and s equal to 1 f the porter temporarly goes back to her home depot after she has completed her transportaton request. here s a dfference between w and x D(s)s, where the latter one refers to the last transportaton request of the porter s, after whch completon the porter wll fnally go back to her home depot. In the frst case, f w s equal to 1, the porter wll only go to her home depot, spend some tme there, but then leave the depot n order to complete some other transportaton requests. he actual travellng tme spent empty between two transportaton requests and by porter s s no longer constant. It wll be represented by a varable t capturng an eventual detour va her depot. he new data used are lsted n able IV. ABLE IV: DECISION VARIABLES AND PARAMEERS FOR 2 nd EXENSION Abbrevaton t w w W γ Descrpton tme travelled between locatons and by porter s tme that porter spends n her home depot after transportaton request bnary varable equals 1 f porter s sent back home temporarly after transportaton request mnmal watng tme porter needs to spend at home depot penalty, f porter has to wat dle between transportaton requests As far as the mathematcal model s concerned, we are now only gong to state the changes of the general model that took places and lst the addtonal constrants. Mnmze α s t + β ( ( E c ) + ( b E ) ) + I O γ ( ( b a ) + ( d c ) ) (17) subect to w + x 1 s D s) s, D ( s) ( (18) + D ( s) + w a d + M (2 x w ) (19),, s D ( s) + D ( s) + w a d w t W w M (2 x w ),, s (20) (21) + d a + M 1 x + w ), s ( (22), d a M 1 x + w ),, + s D ( s) + D ( s) ( (23) M (2 x w ),, s t x w ),, s, 0 w (24) ( (25) { } R 1 (26) 0 (27) 0 R, s (28) w t, here have been few changes n the obectve functon. Instead of the frst sum n general model, where the total tme travelled by the porter s was only dependng on the tme travelled between her two consequent transportaton requests and, the new term (18) calculates the sum of tme travelled between two transportaton requests, and also adds up the travel tme the porter needed to go to her home depot and back, f she goes there temporarly between these transportaton requests. here was also one more group of goals added. he fourth and the ffth term n the obectve functon stand for the watng tme of the porter, ether before or after havng executed a transportaton request. hese watng tmes are penalzed wth the same coeffcent n the obectve functon, as these watng tmes are equally mportant. owever, t s mportant to say that these watng tmes refer only to watng tmes between two transportaton requests n case when the porter doesn t go back to her home depot. he watng tme n the home depot s not penalzed, as t s mportant that the porter spends as much tme there as possble, so that she could be assgned to some other dutes. he fourth term penalzes the watng tmes for porters when they arrve to the pckup locaton and have to wat untl the transportaton request starts. Smlarly, the ffth term penalzes the watng tme at the delvery locaton, when the porter has delvered a patent and wats there to be assgned to next transportaton request.

6 here were also few changes n constrants. Instead of constrants (11) and (12) n the general model, new constrants (18) to (28) are added. Wth (18) t s made sure that porter can ether temporarly go home between her two transportaton requests, spend some tme there and then leave the depot agan to complete some other transportaton requests, or the porter can fnally go to her home depot at the end of her workday. Constrants (19) and (20) make sure that f a porter goes to her home depot temporarly between two transportaton requests, there s enough tme between two successve transportaton requests. here should be enough tme to go to her home depot after one transportaton request, to spend some tme there and then go to complete her next transportaton request. me between two transportaton requests s measured as the dfference between the tme when porter s free from her frst transportaton request and the tme when she should arrve to her next transportaton request, f the porter s assgned to complete transportaton request after request. (21) enables that, f the porter goes back to her home depot after transportaton request, she should spend at least some gven tme W there. Constrants (22) and (23) make sure that there s a connecton between two transportaton requests. Wth (24) and (25) the total travel tme between two successve transportaton requests s modeled. he travel tme wll nclude the travel tme from the drop-off locaton of the porter s last transportaton request to her home depot and from her home depot to the pck-up locaton of her next transportaton request, n case that the porter goes back to her home depot after she has completed the transportaton request. If the porter goes straght to her next transportaton request wthout vstng the home depot, the travel tme wll be equal to the travel tme from the drop-off locaton of her last transportaton request to the pck-up locaton of her next transportaton request. Constrants (26) are bnary and constrants (27) and (28) are non-negatvty constrants. hese changes are also mplemented and the obtaned soluton as well as the comparson to the general model s dscussed n the Secton III. III. NUMERICAL RESULS In ths secton, the data used n the model wll be ntroduced and explaned. We tested all nstances wth XPRESS untl the optmal soluton was found, or untl the termnaton crteron was reached. For our purposes, the termnaton crteron was set to one hour of computaton tme. If the optmal soluton s not found by then, the best soluton found so far was used as an obectve value and all analyss and comparsons s made usng ths value. A. Instances Four dfferent classes of nstances are generated, that dffer n the number of transportaton requests and the number of porters that are avalable. For the frst three classes, ten dfferent nstances are created and tested. For the fourth class, the number of porters s fxed, and the number of transportaton requests s beng vared. he travel tme matrx s drawn from the real-world data. he travel tme matrx and the penalty coeffcents reman the same for all nstances used. he numbers of porters and transportaton requests, and the penalty coeffcents are set by the authors. Instances dffer from each other only n the crtcal tme ponts, when patents medcal procedures are scheduled at and the tme ponts when they fnsh,.e. when the patent should be pcked up and brought back to bed staton. hese crtcal tme ponts are generated wth random number generator. he dea was to generate dfferent tme ponts n the perod from 8 am to 1 pm, for the frst three classes, and the perod from 8 am to 5 am for the fourth class. me ponts were expressed n mnutes. We also assumed that all of these medcal procedures had a length of approxmately 20 to 30 mnutes. So addng these two randomly generated values up, we could determne the start and the end pont of the medcal procedure. he pckup and delvery nodes are randomly chosen among nodes at the hosptal that stand for bed statons and for the facltes where medcal procedures are beng done. ransportaton requests are assgned to patents n ascendng order. For the frst class, the number of porters was set to 2 and the number of patents to 3, respectvely. ence a total of 6 transportaton requests have to be consdered. en nstances were created and tested for the general verson of our model and for both extensons of the model. he soluton for the general model, and also the comparson of the general model soluton to the solutons of the two extensons s gven n the next Secton. Smlarly, for the second (thrd) class of nstances the number of porters was set to 3 (4) and the number of patents was set to 5 (7). he soluton for these two types s gven n the next Secton. For nstances belongng to class II addtonal test runs are made to nvestgate how the soluton changes f the mnmal acqured tme the porter has to spend n the home depot ( W ) vares. An overvew on the sze of problem nstances under consderaton can be found n able V. For the fourth class, the number of porters s fxed to 4 and the number of patents s vared from 8 to 20. he soluton and the senstvty analyss are gven n the next Secton. he penalty coeffcent for the tme travelled α s set to be 1, penalty coeffcent χ (watng tme of porters) s set to be 2, and the penalty coeffcent for the watng tme of patents (β) s set to be 3. hs can be nterpreted as followng: one mnute that a patent (porter) has to wat s three (two) tmes more mportant than an addtonal mnute that a porter has to travel. hese coeffcents can easly be adapted by the decson maker n order to reflect ther true preferences. ABLE V: INSANCES CLASSES Class Number of porters Number of patents I 2 3 II 3 5 III 4 7 IV B. General model he obtaned soluton wth XPRESS-Solver for the frst class s gven n the able VI. he table contans all relevant data related to the patents transportaton request, e.g. average watng tme from both porters and patents pont of vew, total tme travelled by porters and average tme needed to compute the results. he average value of the obectve value s 29.30, weghted over 10 nstances. he total tme travelled by porters s equal to the obectve functon value. As the value of the obectve functon s weghted sum of travel tme and

7 watng tmes for patents, ths leads to the concluson that patents don t have to wat at all. On the other hand, porters do have to wat, ether n case that they have to pck up the patent after the medcal procedure, where the porter wats at the pckup node, or n case that they have ust dropped off a patent, when they wat at the delvery node. he average number of patents that are assgned to one patent s 1.23, what can be nterpreted that n approxmately 2 out of 3 tmes patent wll be assgned to only one porter. he model s solved to optmalty for all ten nstances. he model could be solved n only couple of centseconds. ABLE VI: GENERAL MODEL (FIRS CLASS) N f tt w (s) w (p) No (p) GAP tme % % % % % % % % % % 0.03 avg % 0.06 he notaton used n the table VI as well as the followng tables s lsted n the table VII. Abbrevaton N C f tt w(s) w(p) No(p) GAP No (OS) tme Descrpton ABLE VII: NOAION Instance number Class of Instances Value of obectve functon otal travel tme (empty) otal porters watng tme otal patents watng tme Number of porters that are assgned to one patent Gap between best soluton found so far and the best bound n % Number of optmal solutons found Elapsed run tme untl termnaton After the frst class nstances were tested, the nstances for the class two and three were created and tested. A comparson s gven n the table VIII. In the table VIII, C stands for the class number, and wth No(OS) the number of optmal solutons found (summed over ten dfferent nstances for each class) s represented. ABLE VIII: GENERAL MODEL COMPARISON C f tt w (s) w (p) No (p) GAP No (OS) tme % % % avg % It can be concluded that the sze of the nstances tested nfluences the computaton tme. he tme needed to fnd the optmal soluton ncreases wth the ncrease n the sze of the nstances. owever, the optmal soluton stll could be found very fast. C. Patent centered extenson After the general model was tested, the two extensons were also mplemented n the XPRESS-Solver and solved. he frst extenson nvestgates the change n the obectve functon values n case when t s requred that only one porter can be assgned to patent. In ths case, the obectve functon value s slghtly ncreased n comparson to the obectve value of the general model. hs can be nterpreted as, that the costs to ncrease the patents convenence and to make sure that they are taken care of by the same porter, are not much hgher than n general case. he solutons for the frst three classes are gven n the able IX. he requrement of only usng one porter per patent has a mnor mpact on the qualty of the soluton obtaned, due to medocre addtonal empty travel tmes by porters. On average the resultng empty travel tmes ncrease by 11.7% from to 43.2 tme unts. From the patents pont of vew however the stuaton mproves. Watng tmes stll do not occur. Wth ths extenson the average number porters n use per patent s forced to one (n contrast to 1.27 n the prevous general case, where ths feature has not been addressed explctly). hs addtonal constrant has only mnor mpact on the run tmes requred. On average all nstances can be solved to optmalty wthn 0.8 seconds. ABLE IX: PAIEN-CENERED EXENSION COMPARISON C f tt w (s) w (p) No (p) GAP No (OS) tme % % % avg % D. osptal centered extenson Up to now optmzaton from the porters vew was focused on ther tme travelled empty. Watng tmes,.e. tme slots when they are off-duty, were not consdered explctly. hs, however, seems to be a waste of resources, as porters have other tasks to fulfll. Watng tmes occurred between servng consecutve transportaton requests somewhere wthn the hosptal complex. Wthn ths extenson we tred to nclude the mnmzaton of watng tmes (.e. dle tmes) spent somewhere n the compound from the porters pont of vew, by sendng them back to ther home depots, where they are supposed to fulfll other tasks. ence we now try to emphasze generatng schedules for porters where dle tmes are connected, long enough and occur at the correspondng home depots. he results for the hosptal centered extenson are gven n the able X. he mnmal tme that porters should spend at ther home depot, n the case that she goes there between two transportaton requests, was set to 15 mnutes, for all classes tested. In ths extenson, where the porters were sent to ther home depots n case that there was enough tme, a sgnfcant ncrease n the obectve functon can be notced. hs ncrease however s due to the ncrease n the tme travelled empty by porters. What can be notced s that the porters average watng tme s decreased sgnfcantly n comparson to the general model (7.16 vs tme unts). hs s due to the fact that porters now can go temporarly back to ther home depots between two consecutve transportaton requests. he tme they spend n the home depot s represented wth w and

8 the number of tmes they go to the home depot s represented wth w. On average porters are send back to ther home depots 4.70 tmes and spend tme unts there. Consderng ths feature comes at hgh costs. he soluton qualty deterorates n a three-fold way. he average number of porters n use ncreases from 1.27 to Patent nconvenence s further ncreased by addtonal watng tmes that occur before or after the transportaton requests. (hese watng tmes ncrease from 0.00 to 4.13 tme unts.) Furthermore empty travel tmes by porters ncrease from to tme unts. hs effect s not surprsng as sendng porters to ther home depots leads to ncreased travel tmes. Wth ths extenson however the underlyng combnatoral complexty ncreases dramatcally. In contrast to the prevous case ths extenson heavly nfluences the solvers capabltes of quckly fndng good (optmal) solutons. Especally for the thrd class t can be notced, that none of the nstances tested could be solved to optmalty n one hour of runnng tme. he gap between the best soluton found and the best bound after 3600 seconds s stll at 95.90%. decreases. Smultaneously the tme spent at home (w ) frst ncreases and starts to decrease startng from W = 50, as the total tme spent there s offset by the reduced number of vsts at the home depot. Smlarly the nconvenence of patents (measured n terms of ther watng tmes w(p)) frst ncreases and starts to decrease agan startng from W = 80, as t becomes less effcent to send porters back home. E. Evaluaton of the soluton qualty o show how the value of the obectve functon and the best bound change over tme, a random nstance s chosen among those that were used for the thrd class of nstances. he relaton s shown n the Fgure 1. ABLE X OSPIAL-CENERED EXENSION - COMPARISON No C f tt w (s) w (p) w w No (p) GAP (OS) tme % % % avg % For the second class, a senstvty analyss s performed n order to depct how the requred watng tme affects the value of the obectve functon. he watng tme was vared between 5 mnutes to 140 mnutes. he senstvty analyss s performed on one randomly chosen nstance among those that were used for the second class. he values obtaned are lsted n the able XI. ABLE XI: SENSIIVIY ANALYSIS W f w w w(s) w(p) he mnmal tme that s requred for porter to spend at her home depot s denoted by W. he table shows the followng thngs: wth an ncrease n the mnmum tme that needs to be spent at the home depot f porters are sent back there temporarly between two successve transportaton requests, the number of tmes porters fnally go back home (w ) Fgure 1. Obectve value and best bound vs. tme Instances wthn the thrd class where among the largest nstances under consderaton, where the number of porters was set to four and 14 transportaton requests had to be consdered. For ths partcular nstance the frst feasble soluton was found after seconds, wth an ntal gap of 3363%. Wthn the frst half of the total run tme the soluton qualty could be mproved by 86.6% to 142, the resultng gap decreased to 125%. Wthn the last 1800 seconds the soluton (gap) can only be mproved margnally. he problem under consderaton s statc and operatonal by nature. ence the optmzaton could be executed over nght n order to generate a soluton for the next day. It could be observed however that solutons of reasonable qualty already could be obtaned wthn half an hour. F. Larger Instances In order to test the performance of the solver n use the followng experment s set up: the number of patents (and the resultng transportaton requests) s gradually ncreased up to 20 (40). For the resultng fourth class of nstances, the number of porters stll s set to 4. he best solutons found wthn one hour of run tme can be found n able XII and XIII. he number of patents s represented wth p. he XPRESS- Solver could solve the problem for maxmally 40 transportaton requests n one hour of runnng tme. As the computng tmes were very large, only the general model and the frst extenson were tested for the fourth class. he solutons obtaned for the general model are depcted n table XII.

9 ABLE XII: FOUR CLASS GENERAL MODEL p f tt w (s) w (p) No (p) GAP No (OS) tme % % % % % % % % % % % % % Wth the ncrease n the number of patents, the value of the obectve functon, as well as the computatonal tme, tend to ncrease. he problems up to p = 15 (up to 30 transportaton requests) could be solved wthn less than 30 seconds. Afterwards the computatonal tme has ncreased rapdly, but the problem could stll be solved to optmalty for almost all nstances tested (excepton s a problem wth 17 patents). he fourth class was also tested wth the extended model, namely wth the patent centered extenson. he obtaned results are gven n the able XIII. Compared to the general model, ths extenson could be solved to optmalty for all nstances tested. he optmal soluton could be obtaned wthn less than 10 mnutes. he values of the obectve functon (f) deterorated up to 20% n comparson to the general model, whch means that wth slght ncrease n the travel tme, the patent convenence (.e. the same porter to take care of them) could be guaranteed. From the porters pont of vew, the soluton deterorates slghtly n the sense that the travel tme, tt, ncreases, but the porters watng tmes even decrease for few nstances tested. ABLE XIII: FOUR CLASS PAIEN CENERED EXENSION p f tt w (s) w (p) No (p) GAP No (OS) tme % % % % % % % % % % % % 0.00% Larger nstances were also tested wth XPRESS, where the computaton tme was ncreased to fve hours. Already for the problem wth 4 porters and 25 patents (.e. 50 transportaton requests) no feasble soluton could be found wthn the gven tme. IV. CONCLUSION AND OULOOK he scope of ths paper was to present a novel optmzaton model for consderng ntra-hosptal routng, a very mportant aspect prevalng n health-care modelng. Besdes consderng classcal obectves for vehcle routng (.e. mnmzng dstance or travel tme related costs) we are also focusng on the patents perspectve (.e. the so called clent-centered perspectve, consstng of mnmzng watng tmes before and after ther scheduled appontments, as well the number or porters assgned to patents). Reducng patents nconvenence by mposng a lmt on the number of porters n use per patent can easly be mplemented. he qualty of the soluton obtaned deterorates slghtly due to addtonal travel tmes by porters (on average 11.7%). he effect on run tmes for obtanng the soluton s neglectble. ence the nconvenence of the patent can easly be reduced at low costs and should be consdered whenever possble. he number of porters n use per patent can be reduced by 27%. Consderng concentrated watng tmes of porters, such that they can easly be assgned to other tasks beyond the scope of ths model s also possble. From the patents pont of vew the soluton deterorates slghtly. owever one maor drawback are the resultng run tmes. he complexty of the underlyng model ncreases dramatcally and the model becomes computatonally ntractable as the sze of the problem nstance ncreases. Wthn one hour of run tme our large nstances under consderaton could not be solved to optmalty. he resultng gap was stll at 95.6%. he proposed model can be solved wthn a reasonable amount of tme for small problem nstances. For larger nstances ncludng more than 40 transportaton requests however the use and development of a (meta-)heurstc becomes unavodable. We are plannng to further extend ths model n a two-fold way: Frst we would lke to relax the assumpton that every patent may only be escorted to/from one sngle appontment, by consderng several appontments that need to be consdered sequentally. Next we would lke to combne the optmzaton of the underlyng routng problem (gven startng and endng tmes of appontments) wth the schedulng problem at hand. By smultaneously optmzng both the underlyng schedulng problem (for the ndvdual appontments, by makng sure that at most one appontment may be scheduled per room) and the resultng routng problem, we expect to acheve even better results. REFERENCES [1] M. Gronalt, R.F. artl, M. Remann, New savngs based algorthms for tme constraned pckup and delvery of full truckloads. European Journal of Operatonal Research (2003), vol. 151, no. 3, pp [2] J. Belën, E. Demeulemeester, and B. Cardoen. A decson support system for cyclc master surgery schedulng wth multple obectves. Journal of Schedulng (2009), vol. 12, no. 2, pp [3] B. Cardoen, E. Demeulemeester, and J. Belën. Operatng room plannng and schedulng: a lterature revew. European Journal of Operatonal Research (2010), vol. 201, no. 3, pp [4] A. Beaudry, G. Laporte,. Melo, S. Nckel, Dynamc transportaton of patents n hosptals. OR Spectrum (2010), vol. 32, no. 1, pp [5]. N. Psarafts, A dynamc programmng soluton to the sngle-vehcle, many-to-many mmedate request dal-a-rde problem. ransportaton Scence (1980) vol. 14, no. 2, pp [6] J. C. Cordeau, G. Laporte, he Dal-a-Rde Problem: models and algorthms, Annals of Operatons Research (2007) vol. 153, no.1, pp [7] M.W.P. Savelsbergh, M. Sol, he general pckup and delvery problem, ransportaton Scence (1998) vol. 29, pp

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