Scheduling Home Hospice Care with Logic-Based Benders Decomposition

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1 Schedulng Home Hospce Care wth Logc-Based Benders Decomposton Alza Hechng 1 and J. N. Hooker 2 1 Compassonate Care Hospce Group, alza.hechng@cchnet.net 2 Carnege Mellon Unversty, Pttsburgh, USA, jh38@andrew.cmu.edu Abstract. We propose an exact optmzaton method for home hospce care staffng and schedulng, usng logc-based Benders decomposton (LBBD). The objectve s to match hospce care ades wth patents and schedule vsts to patent homes, so as to maxmze the number of patents servced by avalable staff, whle meetng requrements of the patent plan of care and schedulng constrants mposed by the patents and the staff. The Benders master problem assgns ades to patents and days of the week and s solved by mxed nteger programmng (MIP). The routng and schedulng subproblem decouples by ade and day of the week and s solved by constrant programmng. We report prelmnary computatonal results for problem nstances obtaned from a major hospce care provder. We fnd that LBBD s superor to state-of-the-art MIP and solves problems of realstc sze, f the am s to conduct staff plannng on a rollng bass whle mantanng contnuty of the care arrangement for patents currently recevng servce. Keywords: home health care problem, routng and schedulng, logcbased Benders decomposton, home hospce care 1 Introducton Home health care s one of the world s most rapdly growng ndustres, due prmarly to cost advantages as well as agng populatons. Home care allows patents to receve basc medcal or hospce care n comfortable and famlar surroundngs, rather than beng transported or admtted to facltes that are expensve to operate. It also reduces the rsk of acqurng drug-resstant nfectons that may spread n hosptals and nursng homes. The ncreasng avalablty of portable equpment and onlne consultaton makes home care feasble for an ever wder varety of condtons. The cost-effectveness of home health care depends crtcally on the effcent dspatch of health care ades, whom we call ades for short. Ths poses the home health care problem (HHCP), whch asks how home vsts can be scheduled and staffed so as to make the best use of ades whle meetng patent needs. Ades typcally start ther work shft at home or a central offce, travel drectly from one patent to the next, and return to home or offce at the end of the shft. The shft may be subject to a number of legal or contractual restrctons, such

2 as a maxmum work tme and the need for lunch/dnner breaks. Each medcal or hospce servce must be performed by an ade wth the proper qualfcatons, and servces may be restrcted to specfed days or tme wndows. It may be necessary for two or more ades to vst a patent at the same tme, to carry out more complcated treatments. We focus on hospce care, whch has a few dstnctve characterstcs. Ades frequently provde personal and household servces rather than medcal treatment, or they may smply offer companonshp. They tend to vst on a regular schedule over a perod of several weeks, such as three tmes a week n the mornng. It s often mportant for a gven servce to be provded by the same ade durng every vst, so far as s possble. Staff plannng s typcally over a longer tme horzon, perhaps several weeks. Because of the regularty of vsts and the need for staffng contnuty, the prmary challenge that arses n practce s to update the schedule and antcpate staffng needs as the patent populaton evolves. If patent turnover for the next few weeks can be forecast, then a schedule can be computed for the new populaton to determne what knd of work force wll be requred. We therefore address the problem of recomputng the staff assgnments and vstaton schedule when a specfed subset of the patents are replaced by new patents wth known requrements. Due to the mportance of contnuty, we requre that exstng patents be served by the same ade on the same days as before, but allow for adjustments n the tme of day. The models are easly modfed to mantan the tme of day as well, or to reschedule both the tme of day and days of the week. Due to the dffculty of the HHCP, nearly all exstng soluton methods are heurstc algorthms. Recent work can be found n [1 10]. The few exact methods nclude two branch-and-prce methods [11, 12] and a branch-and-bound method that reles on a travelng salesman algorthm [13]. We propose a very dfferent exact method that uses logc-based Benders decomposton (LBBD) [14 18] and s well suted to schedulng on a rollng bass. An exact method offers the advantage that one can know wth certanty whether a gven work force can cover antcpated patent needs, and therefore when hrng addtonal staff s really necessary. We fnd that LBBD makes exact soluton possble for applcatons of realstc sze when the problem s to reschedule on a rollng bass, rather than schedule all the patents from scratch. LBBD explots a natural decomposton of the HHCP nto an assgnment component (allocaton of patents to ades) and a routng and schedulng component(dspatchng and routng of ades). It combnes the complementary strengths of mxed nteger programmng (MIP) and constrant programmng (CP), wth MILP solvng the assgnment problem and CP solvng the routng and schedulng problem. LBBD s a generalzaton of classcal Benders decomposton [19] n whch the subproblem can be any combnatoral problem, not necessarly a lnear programmng problem. The Benders cuts are based on an nference dual of the subproblem, whose soluton s regarded as a proof of optmalty or nfeasblty,

3 rather than a lnear programmng dual. LBBD has reduced soluton tmes by orders of magntude relatve to conventonal methods n a varety of problems [14,15,17,18,20 22,16,23 34].In oursolutonofthe HHCP, the Bendersmaster problem assgnsadesto patents and to daysofthe week onwhch these patents are servced. The Benders subproblem s the routng and schedulng problem that results from the assgnment obtaned by solvng the master problem. The subproblem decouples nto routng and schedulng mcro-problems that correspond to each ade and each day of the week. Infeasble mcro-problems gve rse to Benders cuts that are added to the master problem. The process repeats untl all the mcro-problems are feasble. Our prmary methodologcal contrbuton s to dentfy a relaxaton of the schedulng subproblem that, when ncluded n the master problem, results n sgnfcantly faster soluton. The only prevous applcaton of LBBD to the HHCP of whch we are aware s a heurstc method n an unpublshed manuscrpt [4]. It solves the master problem wth greedy heurstc and the subproblem wth CP, whle creatng a schedule for only one day. 2 The Problem The problem can be stated as follows. For each patent j there s a tme wndow [r j,d j ] durng whch a vst to that patent must take place, as well as the vst duraton p j. It s assumed that each patent requres one type of vst. If a patent requres two or more types of vsts, the patent s regarded as two or more dstnct patents(wth nonoverlappng tme wndows f the vsts should not overlap). Ades must be qualfed to serve assgned patents, but ths requrement actually makes the problem easer to solve and s therefore not consdered here. Each ade departs from home base b and returns to home base b (whch could be the same as b ). The allowable shft hours of ade are specfed by a tme wndow [r b,d b ] for departure from the orgn base and a wndow [r b,d b ] for arrval at the destnaton base. Travel tme between patent (or home base) j and patent j s t jj. We formulate the problem for a cyclc 7-day schedule wth no vsts on weekends.eachpatentj requresv j vstsperweek,wth v j {1,2,3,5}.Twcea-week vsts must be separated by at least 2 days, and thrce-a-week vsts by 1 day. The varables are desgned to facltate a decomposton scheme n whch the schedulng subproblem s solved by constrant programmng. Bnary varable δ j = 1 when patent j s servced, and bnary varable x j = 1 when ade s assgned to patent j. Bnary varable y jk = 1 when ade vsts patent j on day k, so that y jk x j for all, j, k. There are sequencng varables π kν that represent the νth patent vsted by ade on day k. Varable s jk ndcates the tme at whch ade s vst to patent j starts on day k. We maxmze the number of patents that can be covered by a gven work force. Ths not only determnes whether the work force s adequate, but t tends to mnmze dle tme and drvng tme n an ade s schedule. The problem can

4 be stated as follows: δ j max j x j = δ j, y jk = v j δ j, all j,k y jk x j, all,j,k y bk = y b k = 1, all,k y j,k+τ 1 y jk, τ = 1,4 v j, all,j,k wth v j {2,3}, 1 k v j +1 δ j,x j,y jk {0,1}, all,j,k, n k = j y jk, alldff { π kν ν = 1,...,nk }, all,k (a) (b) (c) (d) (e) (f) (g) (1) π kν {j y jk = 1}, all,k, and ν = 1,...,n k (h) π 1k = b, π nk k = b, all,k () r j s jk d j p j, all,j,k (j) s πkν +p πkν +t πkν π k,ν+1 s πk,ν+1, all,k, and ν = 1,...,n k 1 (k) Constrant (b) defnes δ j and ensures that every patent s vsted by the same ade on the requred number of days. Constrant (d) says that an ade s start and end home base must be vsted every day. Constrant (e) controls the spacng of assgned days. Constrant (g) defnes varable n k to be the number of patents assgned to ade on day k and requres that the correspondng sequence varables take dstnct values. Constrant (h) says that an ade s vsts that are sequenced on a gven day are n fact those assgned to the ade on that day. Constrant () ensures that the start and end home base are vsted frst and last, respectvely. Constrant (j) enforces tme wndows. Constrant (k) ensures that a vst does not start before the ade can arrve from the prevous vst. When updatng an exstng schedule, we need only fx y jk = 1 when patent j remans n the populaton and s assgned to ade on day k. To requre that exstng patents be served at the same tme of day as before, ther tme wndows can be set equal to the vst perod. To allow exstng patents to be served on dfferent days of the week than before, we can fx the varables x j rather than y jk. 3 Benders Subproblem The subproblem decouples nto a separate mcro-problem for each ade and each day. Each s a feasblty problem that checks whether there s a schedule that observes the tme wndows whle takng account of the vst duratons and travel tmes. If not, a Benders cut s generated as descrbed below. The subproblem formulaton conssts of the schedulng constrants n(1) after the daly assgnment varables y jk are fxed to the values ȳ jk they receve n the

5 prevous soluton of the master problem. The mcro-problem S k for each ade and day k s alldff { π ν ν = 1,..., nk } π 1 = b, π nk = b r j s j d j p j, all j P k s πν +p πν +t πνπ ν+1 s πν+1, ν = 1,..., n k 1 π ν P k, ν = 1,..., n k where P k = {j ȳ jk = 1} and n k = P k. If S k s nfeasble, we generate a smple nogood cut j P k (1 y jk ) 1 that prevents the same set of patents from beng assgned to ade on day k n subsequent assgnments. We can, n prncple, generate stronger cuts by determnng whether the same proof of nfeasblty remans vald when smaller sets of patents are assgned to ade on day k. Unfortunately, we do not have access to the mechansm by whch CP solver proves nfeasblty. We therefore tease out stronger cuts by resolvng S k for subsets of P k. S k can be rapdly re-solved because of ts small sze. We use the followng smple heurstc, whch has proved effectve n several studes [18,20,22,21,34]. We ntally set P k = P k, and for each j P k we do the followng: remove j from P k, re-solve S k, and restore j to P k f the modfed S k s feasble. Ths yelds a Benders cut that results n sgnfcantly better performance : j P k (1 y jk ) 1 (2) Whenever we derve a cut for a gven ade and day k of the week, we can generate a smlar cut for every other day of the week. However, the resultng prolferaton of cuts causes the soluton of master problem to bog down. We found that an effectve compromse s to sum the cuts for the remanng 4 weekdays. Thus for each cut (2), we also generate the cut (1 y jk ) 4 k k j P k 4 Benders Master Problem The basc master problem conssts of constrants (a) (f) of the orgnal problem (1) and the Benders cuts generated n all prevous teratons as descrbed above. It also contans a relaxaton of the subproblem, because computatonal experence n [22] and elsewhere ndcates that ncludng such a relaxaton s crucal to obtanng good performance of LBBD. We found the followng tme wndow relaxaton to be effectve. For each ade, defne a set {[r b,α l ] l L } of backward ntervals that begn wth the start of the ade s shft, and a set {[β l,d b ] l L } of forward ntervals that end wth the termnaton of the shft. For each backward nterval l L, let J l be the set of vsts whose tme wndow [r j,d j ] s a subset of the nterval, and defne J l

6 smlarly for forward ntervals. Let the backward augmented duraton p jk for a vst j, ade and day k be the duraton p j plus the mnmum transt tme from the prevous vst (whch may be the orgn base for the ade), and smlarly for the forward augmented duraton p jk. That s, p jk = p j +mn { t bj, mn j Q k {t j j} }, p jk = p j +mn { mn j Q k {t jj },t jb } where Q k s the set of vsts that are already assgned ade on day k, or that have not yet been assgned an ade. Thus the backward augmented duraton s a lower bound on the tme requred to reach and carry out a vst, and smlarly for the forward augmented duraton. We now observe that sum of the backward augmented duratons of vsts n J l must be at most the wdth ofbackwardntervall, and smlar forany forward nterval: j J l p jk y jk α l r b, l L ; j J l p jk y jk d b β l, l L (3) Ths because the vsts and travel to each vst must ft between the begnnng of the ade s shft and the end of the backward nterval, and smlarly for a forward nterval. Inequaltes (3), collected over all ades, comprse a tme wndow relaxaton. The backward and forward ntervals should be chosen so that the vsts that can take place wthn them have a large total duraton relatve to the wdth of the nterval, as ths results n tghter nequaltes (3). In the test nstances, the tme wndows of the vsts span ether most of the mornng or most of the afternoon. It was therefore natural to use one backward nterval endng at noon, and one forward nterval begnnng at noon, for each ade. Thus L = L = {1} and α 1 = β 1 = noon for each. Ths s a weak relaxaton when schedulng all patents from scratch, because the shortest travel tme from the last (or next) vst s a weak bound on the actual travel tme. However, t s more effectve n the rollng problem, because the shortest travel tme s computed only over patents who are already assgned ade on day k or are unassgned. 5 Computatonal Results We tested the LBBD algorthm on real-world data provded by a major hospce carefrm.toobtananntalschedule,weranagreedyheurstconan80-patent populaton usng 20 ades. Snce the heurstc could schedule only 48 patents, we ran the LBBD algorthm on 60 of these patents, ncludng 40 pre-scheduled by the greedy heurstc and 20 treated as new patents. LBBD scheduled all of the new patents usng 18 ades. The resultng 60-patent schedule was used as a startng pont for computatonal tests. It s better than a heurstc schedule but worse than an optmal one, as one mght expect when schedulng on a rollng bass.

7 We compared the performance of LBBD and mxed nteger programmng (MIP) for dfferent rates of patent turnover n the 60-patent populaton. One nstance s generated for each number m = 6,...,23 of new patents, where the new patents are assumed to be the last m patents n the lst of 60. We desgnated 8 of the 18 ades as avalable to cover the new patents (along wth ther pre-assgned patents), because a mnmum of 9 ades were requred n nearly every nstance. Ths allowed us to test computatonal performance near the phase transton for the problem. We formulated an MIP model for the problem by modfyng the well-known multcommodty flow model for the vehcle routng problem wth tme wndows [35 37]. The model conssts of (a) (f) n (1) and the followng: w jb k + j j w jj k = w bjd + j jw j jk = y jk, all,j,k w bjk + w jj k = w jb k + jj k, all,j,k j j j jw s j k s jk +p j +t jj M jj (1 w jj k), all,j,j,k r b s bk d b, r j s jk d j p j, all,j,k plus smlar constrants n whch j and/or j s a home base. Here the bnary varable w jj k {0,1} represents flow and M jj = max{0, d j p j +t jj r j }. We mplemented LBBD usng the IBM ILOG CPLEX Optmzaton Studo verson The master problem was solved by CPLEX and the subproblem by the IBM ILOG CP Optmzer. The routng and schedulng mcro-problems were formulated wth a nooverlap constrant assocated wth sequencng and nterval varables. We solved the MIP model usng CPLEX. The CPLEX presolve routne removes varables n the MIP model and LBBD master problem that are fxed to 0 or 1 by preassgnments. The solver was run n Wndows 7 on a laptop wth an Intel Core 7 processor and 7.75 GB RAM. The results appear n Table 1. Snce ILOG Studo does not report soluton tme for LBBD, the tmes shown are total elapsed clock tmes as ndcated on the Studo console. They reflect overhead ncurred n settng up the problem and retrevng the soluton, whch can be a sgnfcant fracton of total tme for the smallest nstances. Both LBBD and MIP readly solve the smaller nstances, but MIP suffers a combnatoral blowup when there are more than 14 or 15 new patents. MIP s dsadvantaged by the fact that the number of varables grows quadratcally wth the number of new patents, whle n LBBD t grows only lnearly. LBBD therefore postpones the blowup sgnfcantly. Table 1 also shows that ncludng a subproblem relaxaton n the master problem s crucal to the performance of LBBD. Patent records suggest that a 5 8% turnover per week s typcal n practce. LBBD therefore allows staff plannng a month or so n advance for a patent populaton of 60. Ths s adequate for many real-world problem nstances, partcularly gven that mprovements n the LBBD model and subproblem relaxaton are lkely.

8 Table 1. Effect of patent turnover on computaton tmes n a populaton of 60 patents and 18 ades, 8 of whom are avalable for new patents. The new patents replace an equal number of exstng patents. Number of Benders teratons s shown, along wth computaton tme (mnutes : seconds). The last two columns show results for LBBD wthout a subproblem relaxaton n the master problem. New Patents LBBD MIP LBBD no relax Patents Scheduled Iters. Tme Tme Iters. Tme :10 0: : :15 0: : :34 0: : :34 0: : :31 0: : :32 0: : :47 0: : :15 1: : :00 20: : :20 11: : :45 142: : : : : : : : : : : : :56 6 Concluson We fnd that logc-based Benders decomposton solves the home hospce care problem on a rollng bass more rapdly than state-of-the-art mxed nteger programmng, and t scales up to problems of realstc sze. Unlke nearly all competng methods developed for ths problem, t computes an optmal schedule and therefore allows planners to determne wth certanty whether a gven work force can meet projected patent requrements. LBBD has the advantage that the routng and schedulng subproblems reman constant n sze as the patent populaton grows, whle the number of schedulng varables n MIP ncreases quadratcally. The performance of LBBD also benefts from an effectve tme-wndow relaxaton of the subproblem that we developed for ncluson n the master problem. LBBD s partcularly well suted for schedulng on a rollng bass because contnuty constrants strengthen ths relaxaton. Due to the senstvty of performance to the qualty of the subproblem relaxaton, future research wll focus on dentfyng tghter relaxatons, as well as ncorporatng constrants and objectves that more adequately reflect the complexty of the real-world problem.

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