Multi-resource Allocation Scheduling in Dynamic Environments

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1 MANUFACTURING & SERVICE OPERATIONS MANAGEMENT Vol. 00, No. 0, Xxxxx 0000, pp in ein INFORMS doi /xxxx c 0000 INFORMS Muli-reource Allocaion Scheduling in Dynamic Environmen Woonghee Tim Huh Sauder School of Buine, Univeriy of Briih Columbia, Vancouver, BC, Canada, Nan Liu Deparmen of Healh Policy and Managemen, Mailman School of Public Healh, Columbia Univeriy, New York, NY, USA, Van-Anh Truong Deparmen of Indurial Engineering and Operaion Reearch, Columbia Univeriy, New York, NY, USA, Moivaed by ervice capaciy-managemen problem in healh care conex, we conider a muli-reource allocaion problem wih wo clae of job (elecive and emergency) in a dynamic and non-aionary environmen. Emergency job need o be erved immediaely, while elecive job can wai. Diribuional informaion abou demand and reource availabiliy i coninually updaed, and we allow job o renege. We prove ha our formulaion i convex, and he opimal amoun of capaciy reerved for emergency job in each period decreae wih he number of elecive job waiing for ervice. However, he opimal policy i difficul o compue exacly. We develop he idea of a limi policy aring a a paricular ime, and ue hi policy o obain upper and lower bound on he deciion of an opimal policy in each period, and alo o develop everal compuaionally-efficien policie. We how in compuaional experimen ha our be policy perform wihin 1.8% of an opimal policy on average. Key word : Muli-reource Allocaion, Markov Deciion Proce, Healhcare Operaion Managemen Hiory : 1. Inroducion We conider a muli-reource allocaion problem wih wo clae of job (elecive and emergency) in a dynamic and non-aionary environmen. Emergency job need o be performed immediaely, while elecive job can wai. Thi paper i primarily moivaed by ervice capaciy-managemen problem in healh care conex, where a limied amoun of capaciy mu be allocaed among diinc paien demand ream. Example include walk-in and cheduled paien in a primarycare faciliy, and emergency and non-emergency paien for eing (uch a magneic reonance imaging) or a urgical procedure. In managing uch yem, he manager can chooe how many elecive job (paien) o allocae o each day, and hu how much capaciy remaining in he day can be reerved for emergency job (paien). We refer o hee deciion a allocaion cheduling deciion. (We ue he erm paien and job inerchangeably.) The main goal of allocaion cheduling i o fulfill demand for elecive paien in a imely manner, and o leave ufficien lack capaciy o mee emergency demand. In making he above radeoff in allocaion cheduling, he deciion maker mu anicipae he demand for emergency and elecive job, a well a he paern of reource availabiliy over ime. Allocaion cheduling i furher complicaed by he fac ha any job may require muliple reource, e.g., urgeon, nure, operaing room and equipmen, and a lack of any neceary reource could reul in cancellaion or poponemen. Compounding he complexiy i he fac ha cheduling deciion are ofen made in environmen where informaion abou demand and reource availabiliy i highly dynamic, non-aionary, and correlaed. For example, in urgical cheduling, everal facor accoun for non-aionariy and correlaion: 1

2 Auhor: Muli-reource Allocaion Scheduling in Dynamic Environmen 2 Manufacuring & Service Operaion Managemen 00(0), pp , c 0000 INFORMS 1. Saffing paern: Salaried aff accoun for mo of he urgical-uie co (Dexer e al. 1999), and affing cheduling i ubjec o ime-of-week and ime-of-year flucuaion. 2. Medical equipmen: The availabiliy of uch device ha can reduce urgical ime (for example, ee Kuenkuler (2004)) affec he conumpion rae of oher reource uch a operaing room. 3. Paien cheduling paern and demand growh: Surgical demand i non-aionary and ubjec o periodiciy and rend, a evidenced by Moore e al. (2008). 4. Cyclic reamen: For cerain urgical ubpecialiie (for example, chemoherapy and colorecal liver meaae), demand i correlaed over ime ince a reque for a procedure ypically reul in ubequen reque. In hi paper, we conider an allocaion cheduling problem in uch a dynamic environmen, where demand and capaciy conrain may be random, non-aionary, and ime-correlaed. Reque for elecive paien arrive in each period, and a deciion mu be made o fulfill a number of hee reque in he period and waili he re. Thi deciion mu aify capaciy conrain for he period wih repec o muliple ype of reource. There i a per-paien per-period co for wailiing, and wailied paien may renege. Afer he cheduling deciion ha been made for he period, emergency demand arie. Emergency demand ha exceed available capaciy mu be aified uing urge capaciy a a co. The deciion maker mu deermine a cheduling policy o minimize he oal dicouned co over a finie horizon. While he andard ool of Markov Deciion Procee (MDP) can be ued o derive he rucure of he opimal policy, MDP canno be ued a a compuaional ool in hi eing becaue he compuaion explode in general wih he lengh of he horizon. We analyze he opimal policy and derive efficien approximaion a well a upper and lower bound on he opimal deciion, baed on which we propoe an efficien cheduling policy. Our work i cloely relaed o hoe dealing wih he allocaion of medical ervice capaciy among diinc demand ream. Thi opic ha araced growing aenion in he operaion managemen lieraure (Gupa 2007). In general, hree ype of deciion problem have been conidered: (1) who o erve nex, (2) when o chedule he arriving paien and (3) how much capaciy o reerve for a paricular cla of paien. For he fir problem (who o erve nex), Green e al. (2006) analyze he problem of cheduling paien for a diagnoic faciliy hared by oupaien, inpaien and emergency paien. They aume only one paien will arrive or will be erved in a ingle period. In he econd problem (when o chedule), referred o a advanced cheduling, paien are cheduled ino fuure dae upon heir arrival. Parick e al. (2008) preen a mehod for dynamically cheduling muli-prioriy paien o a diagnoic faciliy, and Liu e al. (2010) develop dynamic policie for a primary care clinic aking ino accoun paien cancellaion and no-how behavior. Advanced cheduling i ued in conex where i i imporan o fix appoinmen dae oon afer hey are requeed. The hird problem (how much capaciy o reerve) i he ubjec of our paper. Gerchak e al. (1996) conider he problem of reerving urgical capaciy for emergency cae when he ame operaing room are alo ued for elecive cae, and characerize he rucure of he opimal cheduling policy. Ayvaz and Huh (2010) exend he work of Gerchak e al. (1996) by conidering independen bu non-aionary arrival and capaciy realizaion in each period. They alo conider he poibiliy of allowing ame-day ervice for elecive cae, he opion of rejecing elecive cae, and muliple clae of elecive cae. Boh e of auhor ue MDP ool for analyi and compuaion, and heir mehodology canno be readily adaped o evolving informaion abou demand and capaciie. Our conribuion in hi work can be ummarized a follow. We formulae he allocaion cheduling problem in fully dynamic environmen; our model i he fir o exploi evolving and poibly correlaed informaion abou he diribuion of demand and capaciy, o he be our knowledge. Our model explicily capure reource uncerainy (Cardoen e al. 2010) involving

3 Auhor: Muli-reource Allocaion Scheduling in Dynamic Environmen Manufacuring & Service Operaion Managemen 00(0), pp , c 0000 INFORMS 3 no ju a ingle reource bu muliple reource. We prove ha, imilar o he problem wih impler independen and idenically diribued (i.i.d.) demand cae, he opimal amoun of capaciy reerved for emergency paien in each period decreae wih he number of paien waiing for elecive paien, bu he opimal policy i difficul o compue exacly due o ae-pace exploion. To circumven hi difficuly, we develop a mehodology baed on he idea of a limi policy aring a a paricular ime. Limi policie are ancillary policie ha we ue a a mean o define our propoed policie, where limi policie are ued o approximae he value funcion in he Bellman equaion. Compuaional reul how ha our propoed policie perform well. The remainder of he paper i organized a follow. In Secion 2, we inroduce our MDP formulaion for he muli-reource allocaion cheduling problem. We analyze hi dynamic model in Secion 3. In Secion 4, we inroduce mehod o calculae upper and lower bound on he opimal deciion and derive he approximae cheduling policie. In Secion 5, we prove ha he opimal deciion in each period i bounded by hoe of he approximae police. In Secion 6, we preen he reul of our numerical experimen. We provide our concluding remark in Secion Model In hi ecion, we provide he mahemaical decripion of he muli-reource allocaion cheduling problem, and inroduce ome of he noaion ued hroughou he paper. We conider a finie planning horizon of T period, numbered =1,..., T. Demand for elecive and emergency paien over he period are random variable denoed by d and e, repecively, =1,..., T. We ue d o denoe he vecor coniing of d and e. The number of elecive urgerie cheduled in period i q, =1,... T. Any remaining capaciy i ued o aify emergency urgerie. We give pecial noaion o wo imporan um. We ue Q o denoe he cumulaive number of elecive urgerie cheduled by ime, or q =1, and D o denoe he cumulaive number of reque from elecive paien by ime, or d =1. We aume ha each paien ue n reource. The available quaniie of hee reource are pecified by a non-negaive vecor u in each period. The number q of elecive urgerie cheduled require an amoun A 1 q of he reource, wherea he number e of emergency urgerie ha arie require an amoun A 2 e of he reource. The vecor A 1 and A 2 are column vecor in R n +. We call he n 2 marix A := [A 1 A 2 ] formed by hee vecor he uilizaion marix for period. We require ha he cheduled number q of elecive urgerie mu no exceed he available capaciy a, i.e., A 1 q u. The even in each period occur in he following equence. (i) A he beginning of each period, here are w 1 0 elecive paien reque on he waili. The number of elecive urgery reque for he period, namely d i oberved and added o he waili. The capaciy vecor u and he uilizaion marix A are hen oberved. (ii) The manager decide he number q of elecive paien reque o fulfill in he period, reerving enough pare capaciy for emergency reque ha may arrive laer in he period. There i a per-uni penaly b for each elecive paien reque on he waili ha i no fulfilled in he period. Afer he penaly ha been charged, he waili may be reduced by a random fracion ξ [0, 1] due o paien reneging. Each lo of a paien caue a lo of revenue c. We call he oal co due o waiing and reneging paien in each period he ime- waiing co. (iii) Afer he value of q ha been deermined, he number of emergency paien reque for he period, namely e, i oberved and fulfilled wih he remaining capaciy for he period and addiional urge capaciy a needed. The urge capaciy ued of reource j i charged a a uni penaly rae of p j, j =1, 2,..., n, and we le p be a column vecor coniing of p j. We call he oal penaly co due o ue of urge capaciy in period a he ime- overime co. Our model aume a ime-dependen (raher han wai ime-dependen) reneging rae. Though i i difficul o deal wih wai ime-dependen abandonmen rae in general, our model can handle

4 Auhor: Muli-reource Allocaion Scheduling in Dynamic Environmen 4 Manufacuring & Service Operaion Managemen 00(0), pp , c 0000 INFORMS he pecial cae where paien renege if heir wai ime exceed an exponenially diribued olerance hrehold. Indeed, he queuing lieraure ofen make uch an aumpion for echnical racabiliy (Ward and Glynn 2003). If elecive paien mu be noified of heir appoinmen a lea L period in advance, hen we can inroduce a cheduling lead ime of L period. In every period, a deciion i made o aign q paien who are among he w on he waili a, o receive elecive urgery in period + L. For impliciy, we aume ha L = 0 in he re of he paper. However, all of our reul exend naurally when L i a poiive ineger. One unique feaure of our model i ha all random quaniie inroduced above, uch a d, u and A, are allowed o be correlaed wih each oher and correlaed over ime. Due o he correlaion rucure, our model can evolve in a way ha i dependen on pa hiory. We alo noe ha our model canno be reduced o a ingle-reource cae by idenifying he boleneck reource. The reaon i ha he boleneck reource i policy-dependen. A rengh of our model i ha i can raegically mach capaciy wih demand. We aume ha, for each period, we have wha we call an informaion e ha i denoed by F. The informaion e F conain all of he informaion ha i available ju before he allocaion deciion i made in period (i.e., a he end of ep (i) in period ), including all pa demand and capaciie. In paricular, ince d, A and u are oberved a he beginning of period, hee quaniie are known deerminiically given F, wherea e and ξ are oberved afer he allocaion deciion in period, and o hey do no belong o F bu o F +1. The informaion e F i unaffeced by any deciion, and i herefore common o all policie. Noe ha F deermine he diribuion of demand, co, and capaciie for all curren and fuure period {, +1,..., T }. The random variable e and ξ are diribued o a join diribuion ha i condiional upon F, bu for eae of noaion, we may no repreen heir dependency on F when here i no ambiguiy. There i a dicoun facor of α. The goal of he problem i o find a feaible cheduling policy (i.e., one ha repec he capaciy conrain) ha minimize he oal expeced dicouned co over he planning horizon. We conider only policie ha are non-anicipaory, i.e., a ime, he informaion ha a feaible policy can ue coni only of F and he curren waili. We ue upercrip P and OP T o refer o a given policy P and an opimal policy repecively. Given a policy P, he ae of waili in he yem evolve following he equaion w P =(w P 1 + d q P )(1 ξ ). (1) Noe ha in our model, we aume ha all variable are coninuou variable. 3. Srucure of he Opimal Policy In hi ecion, we fir formulae he problem a a Markov Deciion proce (MDP), which provide a framework for finding an allocaion deciion ha provide he opimal rade-off beween overplanning and under-planning for emergency paien MDP Formulaion The MDP problem ha we formulae ha he objecive of minimizing he oal dicouned co over he finie horizon of lengh T. The deciion o make i he number of elecive urgerie cheduled in period, q, and i ake place a ep (ii) in Secion 2. Le B repreen he number of elecive urgerie waiing o be cheduled a period including hoe arriving a period, i.e., B = w 1 + d. (2) The ae a period (i.e., he informaion baed on which he cheduling deciion i made), hen, i denoed by (B,F ). Noice ha F conain all he informaion on pa demand and capaciie. The deciion q depend on he ae and hould belong o he e R(B,F ) {q : A 1 q u, 0 q B }, (3)

5 Auhor: Muli-reource Allocaion Scheduling in Dynamic Environmen Manufacuring & Service Operaion Managemen 00(0), pp , c 0000 INFORMS 5 which repreen a feaible e of capaciy allocaion in period. Le p τ repreen he ranpoe of p. The expeced co incurred in period, given he deciion q, can be wrien a L(q,B,F ) = (b + ce[ξ F ])(B q )+p τ E(A 2 e + A 1 q u ) +. (4) Conider a finie planning horizon of T period, where α [0, 1] i he dicouning facor. Le V (B,F ) denoe he opimal waiing and overime co incurred from period o T when he ae a he end of ep (i) in period i (B,F ). Noice ha, in he nex period + 1, he number of ouanding elecive paien i updaed by B +1 = (1 ξ )(B q )+d +1, which follow from (1) and (2). Thu, he Bellman equaion can be formulaed a follow: V (B,F ) = min G (q,b,f ), where (5) q R(B,F ) G (q,b,f )=L(q,B,F )+αe [ V +1 (B +1,F +1 ) ] F = L(q,B,F )+αe { V +1 ((1 ξ )(B q )+d +1,F +1 ) F }, (6) where he erminal funcion i given by V T +1 (B T +1,F T +1 )=vb T +1 for ome alvage value v 0. The MDP formulaion preened in hi ecion i no eay o olve in general becaue he informaion ae F can grow large a he period index increae and he deciion of q concern he availabiliy of muliple reource. We fir focu our aenion o he ingle-period co funcion in Secion 3.2, and hen we udy cerain rucural properie of hi MDP in Secion 3.3. We preen he proof in he Appendix Properie of he Single Period Co Funcion For a fixed B, he ingle period co funcion L a defined in (4) i convex in q. Therefore, he myopic problem of minimizing L a a funcion of q i no difficul. Following he definiion of ubmodulariy in Topki (1998), we call a funcion g ubmodular if g(x 1,y 1 )+g(x 2,y 2 ) g(x 1,y 2 )+ g(x 2,y 1 ) for all x 1 >x 2, y 1 >y 2. We can how he following reul. Lemma 1. For fixed F, he ingle period co funcion L in (4) i joinly convex and ubmodular in B and q. Furhermore, L i joinly convex and ubmodular in (B,z ), where z = B q. Below we conider a pecial cae when only a ingle reource conrain exi. In hi cae, he uilizaion marix A become a 1 by 2 marix and he reource availabiliy u reduce o a calar. The ingle period co funcion can be wrien a [ L(q,B,F ) = (b + ce[ξ F ])(B q )+pa 2 E[r ], where r = e u ] + A 1 q. (7) A 2 Noe ha r repreen he number of emergency paien ha could no be accommodaed in period by he remaining available capaciy afer aifying q number of elecive urgerie. The myopic problem of finding he opimal q for L become a varian of he newvendor problem, where he uncerain demand i given by e and he ocking quaniy i (u A 1 q )/A 2, a linear ranformaion of q. Denoe he opimal value for q and r in hi myopic problem by q m and r m, repecively. Lemma 2. Conider he cae of a ingle reource. Le r nv be he max {0, 1 (b + ce[ξ F ])/(pa 1 )} quanile of e F. Then, he value of q m minimizing (7) i given by (u A 2 r m )/A 1, where r m i he poin in he inerval [max{0, (u A 1 B )/A 2 }, u /A 2 ] ha i he cloe o r nv, i.e., max{0, (u A 1 B )/A 2 } if r r m nv < max{0, (u A 1 B )/A 2 }, = r nv if max{0, (u A 1 B )/A 2 } r nv u /A 2, u /A 2 if r nv > u /A 2.

6 Auhor: Muli-reource Allocaion Scheduling in Dynamic Environmen 6 Manufacuring & Service Operaion Managemen 00(0), pp , c 0000 INFORMS 3.3. Analyi of he Dynamic Model In hi ecion, we idenify ome rucural properie for he opimal policie. Recall he Bellman equaion defined in (5) and (6) and ha he erminal co funcion i V T +1 (B T +1,F T +1 )=vb T +1. The following lemma preen ome ueful properie of funcion V (B,F ) and G (q,b,f ). Lemma 3. For fixed F, G ( ) and V ( ) have he following properie: (a) G (q,b,f ) i joinly convex and ubmodular in q and B ; (b) G (q,b,f ) i increaing in B ; and (c) V (B,F ) i convex and increaing in B. (d) Furhermore, G i joinly convex and ubmodular in (B,z ), where z = B q. (Noe: In hi paper, we ue he erm increaing and decreaing o mean non-decreaing and non-increaing, repecively, unle oherwie pecified.) Le q max (B,F ) repreen he large opimal choice for q minimizing G (q,b,f ), given B and F. Le q min (B,F ) be he malle opimal choice for q. Since he feaible region for q, i.e., R(B,F ), i increaing in B, we have he following heorem a a direc reul of Lemma 3. Theorem 1. For fixed F, boh q max (B,F ) and q min (B,F ) are increaing in B. We can furher how ha, for fixed F, he incremen of q min (B,F ) aociaed wih ha of B i bounded above by he incremen of B ielf. Tha i, for 1 uni increae in B, he incremen of q min (B,F ) i a mo 1 uni. Similar reul alo hold for q max (B,F ). Theorem 2. For fixed F, q min (B +,F ) q min (B,F )+ and q max (B +,F ) (B,F )+ for any > 0. q max Thee rucural reul for he opimal cheduling policie hown above are ypically he be one ha can be obained in uch model; ee Gerchak e al. (1996) and Ayvaz and Huh (2010). Nex, we conider he impac of reource availabiliy on he opimal policy. Inuiively, if more capaciy i made available, hen he addiional capaciy will be diribued beween elecive cae and emergency cae. Tha i, he opimal value of q and r will increae in u, bu he amoun of increae in q and he amoun of increae in r will boh be bounded above by ome funcion of how much u increae. Such reul have been hown o be rue in Ayvaz and Huh (2010) under a eing where he capaciy realized in each period i independen wih each oher. However, in our model, uch inuiive monooniciy reul do no necearily hold due o correlaion beween demand and capaciy. Conider he following hypoheical cae. If a larger capaciy realizaion in hi period i rongly correlaed wih a maller emergency demand in he nex period, hen in he curren period he manager may wan o allocae le capaciy for elecive demand and reerve more capaciy for emergency demand, becaue he know ha in he nex period here i le need o reerve capaciy for emergency cae and hence more capaciy can be ued for elecive cae. However, if we can regulae he dependence rucure beween demand and capaciy in a way uch ha capaciy realizaion doe no influence he demand proce, we can ill how cerain monooniciy reul. Under he aumpion of Theorem 3, he informaion e F only need o conain he demand hiory, i.e., F = {d 1,e 1,d 2,e 2,..., d }, ince only he demand proce may be correlaed over ime. Le q max (u ) and q min (u ) repreen he maximum and minimum opimal value for q given B, A, u and F. We can hen how he following reul on how q max (u ) and q min (u ) change in u wih all oher argumen fixed. Le E j be an n-by-1 vecor where all of i enrie are 0 excep for ha he j h enry i 1. Le A 1j repreen he jh enry of A 1. Theorem 3. Suppoe (1) {d,=1, 2,..., T } i independen of {u,=1, 2,..., T } and {A,=1, 2,..., T }; (2) {u,=1, 2,..., T } i a equence of independen random vecor; and (3) {A,=1, 2,..., T } i a equence of independen marice.

7 Auhor: Muli-reource Allocaion Scheduling in Dynamic Environmen Manufacuring & Service Operaion Managemen 00(0), pp , c 0000 INFORMS 7 Then, q max E j ) q max furhermore, q max (u ) and q min (u ) increae wih u componenwie for fixed B, A and F, i.e., q max (u + (u ) and q min (u + E j ) q min (u ) for any calar > 0 and any fixed j =1, 2,..., n; (u + E j ) q max (u )+ /A 1j and q min (u + E j ) q min (u )+ /A 1j. Noe ha condiion (1) ay ha he demand proce i independen of he uilizaion and capaciy procee; condiion (2) and (3) imply ha he uilizaion marix i independen acro period and o i he capaciy. Even under hee condiion, he demand can ill be correlaed over ime, and he uilizaion marix and capaciy can alo be correlaed in any given period. 4. Developmen of Approximae Scheduling Algorihm In he previou ecion, we have derived ome rucural properie for he opimal cheduling policie. While hee reul provide ueful inigh, hey do no addre he cure of dimenionaliy in he compuaion of he opimal policy. The compuaion i epecially problemaic becaue he yem ae in our model i very large, conaining all hiorical informaion on demand and capaciie. To addre hi iue, we develop everal efficien policie. Our policie are baed on he idea of replacing he value funcion ha i commonly ued in he compuaion of opimal allocaion quaniie, wih approximaion. A we hall how, hee approximaion capure he long-erm impac of a deciion in erm of he ineviable and incremenal effec on fuure co Incremenal Co and Benefi of a Deciion In hi ecion, we will decribe a way o accoun for he long-erm impac of a deciion, eiher in erm of he incremenal co ha i inroduce, or in erm of he incremenal benefi ha i bring, compared o he deciion ha have been made before. Thi new co accouning cheme i crucial in he developmen of our approximae cheduling policie. Our approach in hi ecion i o decribe he ime- waiing co for each period a a um of conribuion from all deciion made in period =1,...,. (Recall ha he ime- waiing co coni of boh he waiing co for hoe in he waili and he penalie aociaed wih reneging in period.) Since we conider a capaciaed yem, he deciion in each period impac he e of poible ae ha he yem can reach in each fuure period. More pecifically, he waili in period i gradually deermined by he deciion in each period {1, 2,..., 1} a follow. Suppoe we fix a policy P. For any policy P, we can define wo e of affiliaed policie: Lower Limi Policie. For any period {1,..., T }, we denoe by P a policy ha mimic policy P in period {1,..., }, and hen accommodae a many a elecive cae a poible in period { +1,..., T }. Thu, for {1,..., T }, q P = { q P min{w P 1 + d,c } if if >, where c i he maximum number of elecive paien ha can be erved in period, i.e., c = max{q 0 A 1 q u }. (8) We call P he lower limi policy defined a ime. Upper Limi Policie. For any {1,..., T }, we define he upper limi policy defined a ime, denoed by P, o be a policy ha mimic P in period {1,..., } and hen chedule no more elecive paien in period { +1,..., T }. Inuiively, he lower limi policy lead o horer waili, while he upper limi policy reul in longer waili. Thi inuiion i formalized below in Propoiion 1. I i helpful o inroduce ome new noaion here. Le w P be he ize of he waili under policy P a he end of period and righ before reneging occur, i.e., w P = w P 1 + d q P.

8 Auhor: Muli-reource Allocaion Scheduling in Dynamic Environmen 8 Manufacuring & Service Operaion Managemen 00(0), pp , c 0000 INFORMS (Compare hi expreion wih he definiion of w P in (1).) In hor, we call w P he pre-reneging waili a. Thi variable will be ueful in co accouning. The following propoiion follow direcly from he definiion of he lower and upper limi policie, P and P, and inducion. Propoiion 1. For every policy P and any pair of and aifying 1 T, we have Furhermore, w P w P w P w P. (9) i increaing in, and w P i decreaing in. We remark ha he inequaliie in (9) are igh in he ene ha by defining P appropriaely, we can achieve eiher w P = w P or w P = w P. Thu, he inerval defined by w P and w P repreen he e of value ha a policy P could have obained for w P. Le R P ()={ w R + w P w w P }, (10) where R + repreen he e of all nonnegaive real number, and we call R P () he feaible region for w P a een a he end of period. The following reul i a corollary of Propoiion 1. Propoiion 2. For every policy P, { w P } = R P () R P 1() R P 1 (). We hink of each region R P () a conaining all poible value for w P, given ha deciion in he inerval [1,] have been finalized. A more deciion are made, and a hee e become more rericed, he achievable range of ime- waiing co under policy P poenially change. In paricular, we have R P ()={ w P } afer period- deciion have been made. We dicu below how o quanify he co and benefi brough by hee ucceive rericion. For any real number u and v, aifying u v, we define he ime- minimum waiing co of he inerval [u, v] a he value L([u, v],) = min α (b + cξ ) w = α (b + cξ )u. w [u,v] Tha i, i i he lea poible value for he waiing co a, given ha w mu be in [u, v]. I i eay o ee ha for each, he minimum waiing co i monoone in he e [u, v], i.e., if [u,v ] [u, v], hen L([u,v ],) L([u, v],). Thi monooniciy propery allow u o quanify reducion o he feaible e for w P in erm of increae in he reuling minimum waiing co a follow. Le L P () denoe he incremen in he ime- minimum waiing co due o addiional rericion impoed by he e R P () on R P 1() (i.e., impoed by he deciion made in period ). More preciely, a he beginning of period, w P i confined o he e R P 1(). A he end of period, he e of poibiliie for w P i reduced by R P (). Accordingly, we define L P () =L(R P (),) L(R P 1(),), (11) where we define R P 0 () =R 0 () ={ w R + 0 w d =1 }. Noice ha L P () i a random quaniy becaue he e R P () and R P 1() depend on fuure realizaion of demand and capaciy, a well a paien reneging. The co L P () i an incremenal co becaue i capure he co of a rericion induced by an addiional deciion. By he monooniciy of he minimum waiing co and Propoiion 2, L P () i alway non-negaive. Similarly, we define he ime- maximum waiing co of [u, v], where u v, a he value U([u, v],) = max α (b + cξ ) w = α (b + cξ )v. w [u,v] Le U P () denoe he decreae in he ime- maximum waiing co due o addiional rericion impoed by he e R P () on R P 1(), i.e., U P () =U(R P 1(),) U(R P (),). (12)

9 Auhor: Muli-reource Allocaion Scheduling in Dynamic Environmen Manufacuring & Service Operaion Managemen 00(0), pp , c 0000 INFORMS 9 We refer o U P () a an incremenal benefi ince i capure he benefi induced by an addiional deciion. A before, U P () i a random quaniy, and i can be hown ha U P () i alway nonnegaive. Uing he incremenal co defined in (11), we are able o how ha every conribuion o he waili in period can be aribued o ome deciion made in previou period {1,..., }. Analogouly, he lengh of he ime- waili would have been d =1 if no elecive paien had been cheduled in each period up o, and any reducion of he ime- waili (from d =1 ) can be aribued o ome deciion beween 1 and. Theorem 4. The waiing co incurred in period by policy P, (b + cξ ) w P, aifie α (b + cξ ) w P = L P () and α (b + cξ ) w P = α (b + cξ ) d U P (), =1 =1 =1 where α i he dicoun facor per period. Theorem 4 provide an alernae way of expreing he co. Define L P = L P (), (13) = and call i he aggregae incremenal co a. Thi capure he oal effec, in erm of co, of he ime- deciion q P a i poibly increae he lower bound on he ize of he pre-reneging waili for all fuure period. Similarly, define he aggregae incremenal benefi a a U P = U P (). (14) = Then, he waiing co during he horizon can be wrien a follow: α (b + cξ ) w P = L P () = L P () = L P, (15) =1 =1 =1 =1 = =1 or equivalenly, α (b + cξ ) w P = =1 α (b + cξ ) d U P () = =1 =1 =1 =1 α (b + cξ ) d U P. =1 =1 =1 Noe: So far in hi ecion, we have conidered he waiing co under policy P, and we remind he reader ha he oher co i he overime co, which we denoe by, for each period, O P = α p τ (A 2 e + A 1 q P u ) +. (16) A can be een from hi expreion, he overime co can be compued eaily from realized value uch a emergency demand e and he number of elecive urgerie q P, and here i no need for approximaing hi co.

10 Auhor: Muli-reource Allocaion Scheduling in Dynamic Environmen 10 Manufacuring & Service Operaion Managemen 00(0), pp , c 0000 INFORMS 4.2. Two Approximae Policie Baed on Incremenal Co and Benefi In Secion 4.1, we have analyzed he ime- waiing co in each period, and hown ha i can be expreed in erm of eiher he incremenal co or he incremenal benefi of deciion in period {1,... }. The incremenal co and benefi can be compued for each ample of demand and capaciie under a given policy. In hi ecion, we will how ha hee incremenal co and benefi can be calculaed forward in ime, a random quaniie ha depend on fuure demand, capaciie, and reneging behavior. Baed on hi, we will inroduce wo policie for our urgical cheduling problem, called he Lower Minimizaion Policy (LM) and he Upper Minimizaion Policy (UM), which are relaed o he Lower Limi policy and he Upper Limi policy, repecively. We fix he policy o be P and he curren period o be. We alo fix w P 1, he lengh of he waili carried over from period 1 o, and d, he number of elecive paien reque for period. Le q P be he number of elecive paien ha are cheduled in period. Once q P i decided, he e R P () defined in (10), for any, i no affeced by any fuure deciion; hu, he minimum and maximum waiing co L(R P (),) and U(R P (),), a well a incremenal quaniie L P () and U P (), are no affeced by fuure deciion eiher. All of hee quaniie depend only on exogenouly defined random elemen (demand, capaciie, uilizaion marice and reneging fracion) in he fuure. In he following propoiion, we eablih ome key properie (uch a monooniciy and convexiy) of he incremenal co and benefi a funcion of q P. Thee properie will become ueful in enuring ha he cheduling policie ha we will propoe can be eaily compued. Thee reul are hown for a ingle ample pah of informaion for exogenou random elemen, bu i hould be noed ha hee properie alo hold in he expeced ene. Propoiion 3. Fix he policy P, and he ize of he waili a he beginning of period, w 1, P for ome period. For any period, he following aemen hold for any ample pah of random variable beween period and, i.e., {d,..., d }, {A,..., A } and {u,..., u }: (a) L P () i non-negaive and decreaing in q P, and equal 0 when q P = min{c,w P + d }. Furhermore, L P () i convex and coninuou a a funcion of q P. (b) U P () i non-negaive and increaing in q P, and equal 0 when q P =0. In fac, U P () i a linear funcion of q P. If we inerpre he ime- incremenal co L P () a he addiional co impoed on period by he deciion in period, we can compue he aggregae impac of he deciion in period by umming hi quaniy over all poible value of recall how we defined he aggregae incremenal co in (13). Similarly we have defined he aggregae incremenal benefi in (14). I i raighforward o verify ha he aggregae incremenal co and benefi inheri all he properie of he coniuing erm. We have he following corollary o Propoiion 3. Corollary 1. The aemen of Propoiion 3 coninue o hold when L P () i replaced wih L P and U P () i replaced wih U P. A in Secion 4.1, he majoriy of hi ecion ha been devoed o he waiing co, and we remind he reader ha he overime co i given in (16). Now, we are ready o inroduce wo cheduling policie for our urgical cheduling problem. Lower Minimizaion Policy (LM). Under hi policy, we chooe he number of elecive urgerie, q LM, uch ha E[L P + O P F ] i minimized over he feaible e [0, min(c,w 1 P + d )], where P refer o LM. We can hink of L P + O P a a proxy for he co ha we can aociae wih he deciion in period ; i i approximae ince he impac on he waiing co i approximaed wih he aggregae incremenal co a, L P (which i baed on he lower limi policy). Condiioned on F, boh he ize of he waili a he end of period 1, w 1, P and he new elecive paien reque in period, d, are known deerminiically. In fac, hi earch can be performed efficienly ince L P i convex in q P (by Corollary 1) and O P i alo convex in q P (by he definiion of O P ).

11 Auhor: Muli-reource Allocaion Scheduling in Dynamic Environmen Manufacuring & Service Operaion Managemen 00(0), pp , c 0000 INFORMS 11 Upper Minimizaion Policy (UM). Thi policy pecifie he choice of q UM [0, min(c,d + w P 1)] uch ha E[O P U P F ] i minimized, where P denoe UM. Thi expreion, E[O P U P F ], i anoher proxy for he co we aociae for period, where we approximae he aving on waiing co wih he aggregae incremenal benefi, U P. A before, we can how ha hi expreion i convex in q P. In each period, our policie minimize a ingle-dimenional convex funcion, which can be performed efficienly if we are able o evaluae he funcion a a given poin efficienly. To evaluae he funcion a any given poin, we need o evaluae he expeced co of running a limi policy, which can be compued uing a Mone Carlo imulaion. The effor required o compue he co of a limi policy on a ample pah of imulaion i a mo proporional o he lengh of he horizon, and hu he compuaional effor cale very nicely (i.e., polynomially) wih he ize of he problem. Compared o a myopic policy which deermine he cheduling deciion in each period by olving he newvendor problem (EC.1) (ee Lemma 2), he wo policie inroduced above ake ino accoun he impac of curren deciion on fuure (waiing) co. Hence, hey end o be le myopic and i i hoped ha hey would perform beer han he newvendor-baed myopic policy Comparion o Exiing Approache The problem of how o ue evolving demand foreca o devie effecive upply-chain-managemen policie in hee eing ha been he ubjec of a coniderable body of reearch. We refer he reader o Iida and Zipkin (2001) and Dong and Lee (2003) for more comprehenive dicuion. Many work focu on characerizing he rucure of he opimal policy for he ingle-iem periodic-review, ochaic invenory problem, and in many model, including model wih Markov-modulaed demand, correlaed demand and foreca evoluion (Iida and Zipkin (2001), Özer and Gallego (2001), and Zipkin (2000)), he opimal policy can be hown o be ae-dependen. In general, he ae pace of hee problem grow exponenially wih he number of period, and a a reul, many auhor have developed compuaionally efficien heuriic o compue policie, e.g., Dong and Lee (2003), Lu e al. (2006), and Iida and Zipkin (2001). One cla of heuriic, which we call Look-Ahead Opimizaion (LA), characerize he porion ha can be compued immediaely (wihou conidering fuure deciion) of he long-erm co of a deciion and chooe a policy ha minimize hi co in each period. Chan (1999) and Chan and Muckad (1999) are he fir o conider hi approach, and hey udy uncapaciaed and capaciaed muli-iem invenory model wih linear co. They define a penaly funcion, which we will call marginal holding co, which accoun for he holding co incurred over he re of he horizon due o he invenory ordered in hi period. Their policy order in each period o minimize he expeced period backorder co plu he marginal holding co. Levi, Pál, Roundy and Shmoy (2005) how ha in ingle-iem invenory problem, he deciion of LA are lower bound on he deciion of an opimal policy. Truong (2011) prove ha LA i an approximaion algorihm wih a wor-cae performance raio of 2. Levi, Roundy, Shmoy and Truong (2008) define anoher lookahead co called marginal backlogging co for he capaciaed ingle-iem invenory problem. They prove ha he policy ha minimize he marginal backlogging co plu he period holding co in each period, ha invenory poiion ha are upper bound on he invenory poiion of an opimal policy. The algorihm we develop in hi paper are inpired by he Look-Ahead Opimizaion approach above. We alo aemp o capure he long-erm impac of a cheduling deciion, and make deciion o opimize hi impac. However, here alo imporan difference beween our approach and hoe previouly underaken. Fir, previou approache characerize he minimum impac of a deciion on wo eparae caegorie of fuure co, namely holding and backorder co, o derive he marginal holding and marginal backorder co, repecively. The minimizaion of each of hee marginal co plu he

12 Auhor: Muli-reource Allocaion Scheduling in Dynamic Environmen 12 Manufacuring & Service Operaion Managemen 00(0), pp , c 0000 INFORMS myopic (period ) co lead o a eparae (upper or lower) bound. In our problem eing, we characerize he long-erm impac of a deciion on a ingle caegory of co, namely he co having elecive paien wai. Thi co i analogou o he holding co in invenory problem. We view he impac of hi co from diameric perpecive. We analyze boh he minimum impac and he maximum impac of a deciion on fuure waiing co. A a reul, we quanify boh an incremenal co and an incremenal benefi o a deciion. We minimize he myopic co plu he incremenal co o obain new lower bound and policie, and we minimize he myopic co minu he incremenal benefi o obain new upper bound and a differen cla of policie. A we will how in our compuaional experimen, he policie obained by maximizing he incremenal benefi are uperior o hoe obained by minimizing he incremenal co. Second, in invenory problem, he problem dynamic are impler. The marginal holding and backorder co can be aed in erm of he curren ae and exogenou fuure random variable, uing cloed-form expreion. In our problem, he incremenal co and benefi depend more inricaely on he equence of exogenou random variable ha are realized in he fuure, including reource availabiliy and reource conrain, reneging, and emergency arrival. Thu, here i no eay way o expre hee co and benefi. We mu define hem implicily, in erm of he co incurred by cerain limi policie. The deciion of a limi policy are predefined, o ha he expeced co of he enire policy can be calculaed efficienly on every ample pah. Becaue of hi implici definiion, he analyi of hee policie and he proof of he bound are coniderably more echnical. The advanage of working wih limi policie, however, i ha hey are a very rich cla of policie and hey provide a general mehod o generae poenial bound. We believe ha hi mehod of generaing bound will find applicaion in many oher eing. 5. Bound on he Opimal Deciion The wo policie inroduced in Secion 4, LM and UM, are baed on he incremenal co and benefi, which are eimaed by conidering he limi policie. Thee eimaed co provide lower and upper bound on he waiing co. Now, we will how ha he deciion of UM and LM provide lower and upper bound on he opimal deciion in each period. Thee bound provide addiional moivaion and uppor for anoher policy, which chedule he number of elecive urgerie in each period o be he average of hoe uggeed by he LM and UM algorihm. We will compare he performance of hee policie o he performance of an opimal policy in he nex ecion. Under LM, he aggregae incremenal co funcion ha we ue o deermine he number of elecive urgerie q LM in each period i baed on he logic ha all fuure capaciy would be allocaed for elecive cae. Inuiively, hi would overeimae he fuure capaciy employed in elecive urgerie, and hu undereimae he penaly co aociaed wih keeping an elecive paien reque in he waili. Therefore, we expec he reuling waili under LM o be larger han ha under an opimal cheduling policy. Similarly, we expec he waili under UM o be maller compared o ha of an opimal policy. The following reul formalize our inuiion above. Theorem 5. For each {1,..., T }, boh w LM Theorem 6. For period {1,..., T }, boh w UM w OP T w OP T and w LM and w UM w OP T w OP T hold almo urely. hold almo urely. Finally, we ummarize below he main reul of hi ecion, ha LM and UM yield lower and upper bound on he opimal policy. Thi reul follow from he proof of Theorem 5 and 6. Corollary 2. For each period {1,..., T }, uppoe we fix he waili ize w 1 and informaion e F. Then, he deciion produced by LM and UM aify q LM q OP T q UM almo urely, where q OP T i he deciion of an opimal policy.

13 Auhor: Muli-reource Allocaion Scheduling in Dynamic Environmen Manufacuring & Service Operaion Managemen 00(0), pp , c 0000 INFORMS Compuaional Experience While we have noed ha he opimal policy for our cheduling problem can be difficul o compue, a number of policie emerge from our dicuion o far. We have inroduced wo policie, he Lower Minimizing (LM) and Upper Minimizing (UM) policie, in Secion 4. We now udy how hey perform. Our experimenal eing i no mean o imulae real-life inance of he allocaion cheduling problem. Raher, we aim o generae a ufficienly rich e of mall problem ample, where he opimal policy can be evaluaed wih reaonable compuaional effor, o ha we can compare he performance of our algorihm wih ha of an opimal policy or a naive myopic policy. We will ue he following variaion of our baic model in order o make our experimen more comparable o hoe of Gerchak e al. (1996). In he experimen, we conider he cae where only a ingle reource conrain exi. Inead of having cae a he uni for emergency demand e, =1,..., T, we ue uni of urge capaciy a he uni o pecify e. Similarly, we ue p o capure he co of uing each uni of urge capaciy o aify emergency demand. We redefine he column A 2, =1,..., T, o ha A 2 e capure he amoun of he normal capaciy a conumed by a quaniy of emergency demand ha i equivalen o e uni of urge capaciy. Thi variaion i lighly more flexible in ha i allow he model o capure variabiliy in he capaciy uage of differen emergency cae. I can be hown ha all of our reul hold under hi variaion. We conider horizon lengh T = 5, 14 period. The waiing penaly for poponemen of an elecive paien by one day i b =1, 2, 10. The penaly for uing one uni of urge capaciy i p = 24. For impliciy, we aume ha he dicoun facor i 1 and here i no reneging. We e average demand and capaciy value o be imilar o hoe choen by Gerchak e al. (1996). The capaciy i 960 minue per day, correponding o wo eigh-hour hif. The demand are randomly generaed for each problem inance a we hall nex decribe. We pecify he demand rucure in each experimenal inance a a binary probabiliy ree. Le U[a, b] denoe a random variable ha i uniformly diribued on [a, b]. For each problem inance, we generae he branch probabiliie of he binary ree in each period from U[0, 1]. We generae he newly arriving demand for elecive paien in each period a each node in he ree uing U[0, 12]. Elecive urgerie alway ake 60 min per eion. We generae he mean number of minue demanded by emergency paien a each node from U[320, 480]. We aume ha he acual demand for emergency paien i uniformly diribued beween 0 and wice he mean. Once a problem inance (i.e., a binary ree) i generaed, he manager i aware of he demand rucure (including he average a each node a well a branch raniion probabiliie) and he curren ae, bu doe no know how he ae would evolve and how demand would be realized. We alo conider a cla of problem inance, which we call igh inance. In hee inance, he mean demand for emergency paien a each node i generaed from a much higher range, namely from U[660, 800], much cloer o he capaciy of 960. The re of he problem daa i generaed for hee inance in he ame way a before. For each combinaion of he parameer, we generae a number of inance, i.e., binary probabiliy ree, for he demand rucure baed on he aforemenioned diribuional aumpion (ee Table 1). For each inance, we compue he opimal policy OP T uing an exhauive earch. To faciliae compuing he opimal policy we conider only ineger-valued demand and allocaion quaniie. In addiion o UM and LM, we have implemened wo oher policie. The fir policy, a hybrid one denoed by H, i moivaed by he reul of Secion 5. Since he oucome of LM and UM form bound on OP T, our propoed policy H enure ha he waili in each period i he average of wha i would have been under LM and UM. The econd policy we conider i a myopic one, which minimize he ingle-period co given in (4), and we denoe hi policy by M. The oucome of our experimen how ha ha UM i he be-performing policy. In 6600 problem inance, he average performance of UM i wihin 1.8% of OP T compared o 35% for he naive myopic policy M. Under a range of problem configuraion, U M performance i never

14 Auhor: Muli-reource Allocaion Scheduling in Dynamic Environmen 14 Manufacuring & Service Operaion Managemen 00(0), pp , c 0000 INFORMS Table 1 Performance of Approximae Policie Se up Mean of performance raio Variance of performance raio T p b Tigh? No. inance LM UM H M LM UM H M No No No Ye Ye Ye No No No Ye Ye Ye more han 3.6% away han OP T, wherea M performance can be a much a 234% wore han ha of OP T. The myopic policy doe no perform well ince i doe no anicipae he long-erm impac of a waili. Since he co of poponing an elecive paien by one period i mall, a wai li i bad only if i i no reolved over many period. Since he myopic policy ee only one period ahead, i everely over-allocae capaciy o emergency urgerie. The performance aiic are ummarized in Table 1, where he performance raio i defined a he raio of he oal dicouned co under he cheduling policy in queion o ha under an opimal policy. We ee ha UM conienly ouperform all oher policie acro a range of experimenal eing. I exhibi boh maller average performance raio wih repec o OP T, a well a maller variance for he performance raio. By comparion, LM and H are wihin 4.8% and 2.4% of OP T on average, repecively. Though hey are lighly wore han UM, hey eem o be much beer han M. Recall ha o derive UM and LM, we ue he fuure co of a limi policy a a ubiue for he fuure co of an opimal policy. The limi policy ued o derive UM allocae all available capaciy in each period o emergency urgerie, wherea he limi policy ued o derive LM allocae all available capaciy o elecive cae. Since he co of over-planning for emergency urgerie and making paien wai i ypically much maller han ha of under-planning, he co of he former limi policy i cloer o opimal han ha of he laer. Hence, a an approximaion, UM i expeced o perform beer han LM. Since UM i conienly beer han LM, H i alway wore han UM becaue i performance lie beween hoe of UM and LM. We alo find ha LM, UM, and H all perform much beer han he myopic policy M, and hu he myopic policy may no be a good policy o ue. All of our policie deeriorae in performance wih longer horizon and higher waiing co, alhough he degradaion in performance for UM i relaively mall. All of he policie do beer under capaciy-igh inance becaue he deciion pace in hee inance i more highly conrained. Hence he deciion underaken by differen policie are cloer ogeher and cloer o hoe of OP T. 7. Concluion In hi paper we develop a capaciy allocaion model for allocaion cheduling which explicily conider muliple reource ha are needed o erve wo clae of paien wih differen wai-ime eniiviie. Our model allow he demand, reource uilizaion and capaciy availabiliy o be random, non-aionary and ime-correlaed. We prove imilar rucural reul for he opimal oluion a in he eing wih i.i.d. demand. Our primary heoreical conribuion i a mehod o obain upper and lower bound on he deciion of an opimal policy in each period. We alo

15 Auhor: Muli-reource Allocaion Scheduling in Dynamic Environmen Manufacuring & Service Operaion Managemen 00(0), pp , c 0000 INFORMS 15 develop everal compuaionally-efficien policie which are hown o perform very well in our numerical experimen. Our work i moivaed by problem in healh care ervice capaciy managemen, in paricular problem in urgical cheduling. Many imporan feaure in our model, uch a dynamic and non-aionary environmen and muli-reource conrain, are well recognized in he urgicalcheduling conex (Cardoen e al. 2010, Moore e al. 2008, Dexer e al. 2005). Previou lieraure in hi area uually aume a implified eup, uch a i.i.d. demand rucure and a ingle generic reource conrain. Our work i able o bring heory cloer o pracice by conidering a much more general eup. In our model, an elecive paien doe no receive an appoinmen a he ime of joining he waili. Thi i no an uncommon pracice in public healh yem (uch a Canada, UK and Auralia). In he UK and Auralia, waili are kep for elecive paien, and hey are conidered ueful in managing urgery chedule (Edward 1997). There are everal way o improve our model for pracical applicaion. Fir, i would be ueful o conider paien heerogeneiy in reource conumpion wihin each paien ype, i.e., emergency and elecive. Second, we have aumed ha paien only conume reource for one period, and i would be inereing o conider a model ha allow paien o occupy ome reource (e.g., bed) for muliple period. Third, our model pecifie how many elecive paien o admi for he curren period bu no he iming or equence of ervice. I would be inereing o develop a equenial deciion model ha joinly make hee deciion. All exenion above require ubanially revied model and analyi, and we leave hem for fuure reearch. Acknowledgmen We hank he Edior, he Aociae Edior and referee for conrucive and poiive feedback. Reference Ayvaz, N., W.T. Huh Allocaion of hopial capaciy o muliple ype of paien. Journal of Revenue & Pricing Managemen 9(5) Cardoen, B., E. Demeulemeeer, J. Beliėn Operaing room planning and cheduling: A lieraure review. European Journal of Operaional Reearch 201(3) Chan, E., J. Muckad Markov chain model for muli-echelon upply chain. Unpublihed manucrip. Chan, E.W.M Markov Chain Model for muli-echelon upply chain. Cornell Univeriy, Augu. Dexer, F., A. Macario, R.D. Traub, M. Hopwood, D.A. Lubarky An operaing room cheduling raegy o maximize he ue of operaing room block ime: compuer imulaion of paien cheduling and urvey of paien preference for urgical waiing ime. Aneheia & Analgeia 89(1) Dexer, F., E. Marcon, R.H. Epein, J. Ledoler Validaion of aiical mehod o compare cancellaion rae on he day of urgery. Aneheia & Analgeia 101(2) Dong, L., H.L. Lee Opimal policie and approximaion for a erial muliechelon invenory yem wih ime-correlaed demand. Operaion Reearch Edward, R.T NHS waiing li: oward he eluive oluion. London: Office hof Healh Economic. Gerchak, Y., D. Gupa, M. Henig Reervaion planning for elecive urgery under uncerain demand for emergency urgery. Managemen Science 42(3) Green, L.V., S. Savin, B. Wang Managing paien ervice in a diagnoic medical faciliy. Operaion Reearch 54(1) Gupa, D Surgical uie operaion managemen. Producion and Operaion Managemen 16(6) Iida, T., P. Zipkin Approximae oluion of a dynamic foreca-invenory model. Working paper. Kuenkuler, J VCU neurourgeon develop new device for performing deep brain urgery. Available a: hp:// neurourgeon develop new device for performing deep brain. Levi, R., R. O. Roundy, D. B. Shmoy, V. A. Truong Approximaion algorihm for capaciaed ochaic invenory conrol model. Operaion Reearch 56(5)

16 Auhor: Muli-reource Allocaion Scheduling in Dynamic Environmen 16 Manufacuring & Service Operaion Managemen 00(0), pp , c 0000 INFORMS Levi, Reef, Marin Pál, Robin Roundy, David B. Shmoy Approximaion algorihm for ochaic invenory conrol model. Michael Jnger, Volker Kaibel, ed., Ineger Programming and Combinaorial Opimizaion, Lecure Noe in Compuer Science, vol Springer Berlin / Heidelberg, Liu, N., S. Ziya, V.G. Kulkarni Dynamic cheduling of oupaien appoinmen under paien no-how and cancellaion. Manufacuring & Service Operaion Managemen 12(2) Lu, X., J.S. Song, A. Regan Invenory planning wih foreca updae: approximae oluion and co error bound. Operaion reearch 54(6) Moore, I.C., D.P. Srum, L.G. Varga, D.J. Thomon Obervaion on urgical demand ime erie: deecion and reoluion of holiday variance. Aneheiology 109(3) Özer, Ozalp, G. Gallego Inegraing replenihmen deciion wih advance demand informaion. Managmen Science Parick, J., M.L. Puerman, M. Queyranne Dynamic muli-prioriy paien cheduling for a diagnoic reource. Operaion reearch 56(6) Topki, D.M Supermodulariy and complemenariy. Princeon Univeriy Pre, Princeon, NJ. Truong, V. A The pediaric vaccine ockpiling problem. Under review. Ward, A.R., P.W. Glynn A diffuion approximaion for a markovian queue wih reneging. Queueing Syem 43(1) Zipkin, P.H Foundaion of invenory managemen, vol. 2. McGraw-Hill Boon, MA.

17 e-companion o Auhor: Muli-reource Allocaion Scheduling in Dynamic Environmen ec1 Online Appendix Lemma EC.1. If V (z) i a convex funcion in z and a i a poiive conan, hen V (x ay) i ubmodular in x and y. Proof: Take arbirary x 1 >x 2. Since V ( ) i convex, i marginal incremen i increaing and herefore V (x 1 ay) V (x 2 ay) decreae in y, which complee he proof.. Proof of Lemma 1: The join convexiy of L(q,B,F ) in B and q i eviden from (4). I i eay o ee he ubmodulariy of he fir erm in L(q,B,F ). Since he econd erm of hi funcion only involve q, i i alo ubmodular in q and B. Thee prove he deired reul wih repec o B and q. Now define L(z,B,F )=L(B z,b,f ), where z repreen B q. Then, from (4), L(z,B,F ) = (b + ce[ξ F ])z + p τ E(A 2 e + A 1 (B z ) u ) +, which can alo be hown o be convex and ubmodular in (z,b ) by Lemma EC.1.. Proof of Lemma 2: Subiuing r =(u A 1 q )/A 2, he problem of minimizing (7) wih repec o q i equivalen o minimizing he following expreion wih repec o r : (b + ce[ξ F ]) A 2 A 1 r + pa 2 E e [e r F ] +, (EC.1) where max{0, (u A 1 B )/A 2 } r u /A 2. Wihou he conrain on r, hi i he newvendor problem wih he uncerain demand e, he purchae co (b+ce[ξ F ]) A 2 A 1, and he backlog penaly pa 2. Thu he opimal value of r for hi unconrained newvendor problem i r nv a noed above. Conidering he bound on r and he convexiy of (EC.1) wih repec o r, we obain he deired reul.. Proof of Lemma 3: We prove hee reul by inducion. I i eay o ee ha he erminal co V T +1 (B T +1,F T +1 )=vb T +1 i a linear and increaing funcion of he final backlog B T +1. Now, uppoe ha he aemen are rue for + 1. We fir prove aemen (a) for. Since L(q,B,F ) i joinly convex in (q,b ) (by Lemma 1) and V +1 i joinly convex (inducion hypohei), i follow from (6) ha G i joinly convex in (q,b ). Alo, he ubmodulariy of G follow from he ubmodulariy of L (Lemma 1) and he join convexiy of V +1 in B and q (inducion hypohei and Lemma EC.1 in he Appendix). Thi complee he proof of (a) for. Saemen (b) for follow direcly ince L(q,B,F ) i increaing in B from (4), and he econd erm in he righ-ide of (6) i increaing in B from he inducion hypohei. Now we conider aemen (c) for. Since G(q,B,F ) i joinly convex in (q,b ) from aemen (a) and he feaible region of he minimizaion operaor in (5) i convex, we obain ha V (B,F ) i alo convex in B. Finally, o how V (B,F ) increae in B, conider he cae where B increae o B + δ for ome δ > 0. I i eay o ee ha G (q, B,F ) G (q, B + δ,f ) for any 0 q B (from aemen (b)). Furhermore, we can how ha G (B,B,F ) G (q, B + δ,f ) for any B <q B + δ ince we obain from (6) ha G (B,B,F )=L(B,B,F )+αe { V +1 ((1 ξ )(B B )+d +1,F +1 ) F } L(q, B + δ,f )+αe { V +1 ((1 ξ )(B + δ q)+d +1,F +1 ) F } = G (q, B + δ,f ), where he inequaliy follow from he definiion of L in (4) and he inducion hypohei on V +1. Therefore, i follow ha min 0 q B G (q,b,f ) min 0 q B +δ G (q,b + δ,f ), implying by he definiion of V in (5) ha V (B,F ) increae in B. Now we conider (d). From (a), G i joinly convex and ubmodular in (q,b ). Rewrie G in erm of B and z where z = B q a follow: G (z,b,f )=L(z,B,F )+

18 ec2 e-companion o Auhor: Muli-reource Allocaion Scheduling in Dynamic Environmen αe { V +1 ((1 ξ )z + d +1,F +1 ) } F. The join convexiy of G in (B,z ) follow from he convexiy of L (Lemma 1) and he convexiy of V (aemen(c)). The ubmodulariy of G in (B,z ) i a direc reul of he ubmodulariy of L in (B,z ) (Lemma 1).. Proof of Theorem 1: The reul follow from he ubmodulariy of G (q,b,f ) in (q,b ), and he definiion of q max (B,F ) and q min (B,F ).. Proof of Theorem 2: Since G i joinly convex and ubmodular in (B,z ) by Lemma 3, where z = B q, he large opimal choice of z given B, which we will call z max, i increaing in B, i.e., for any > 0, z max (B +,F ) z max (B,F ), where z max (B,F )=B q min (B,F ). Thu, q min (B +,F )=B + z max (B +,F ) B + z max (B,F )=q min (B,F )+, a required. The reul for q max can be hown by a imilar proof uing z min, he malle opimal choice of z given B.. Proof of Theorem 3: We how reul for q min and hoe for q max follow a imilar proof. Since F only conain demand hiory now, he G funcion defined in (6) become, uing (4), G (q, u ) = (b + ce(ξ F ))(B q )+p τ E e (A 2 e + A 1 q u F ) + + αe F+1,u +1,A +1 { V+1 [(1 ξ )(B q )+d +1,F +1 ] F }. Le u fix a j. To how ha q min (u + E j ) q min (u ) for fixed B, A, F and u k where k j, i uffice o how ha G defined above a a funcion of q and u j i ubmodular. Noice ha he fir and la erm above do no involve u, wherea he middle erm i clearly ubmodular in q and u j (by Lemma EC.1 in he Appendix). Alo noice ha he feaible region for q (ee (3)) i increaing in u j wih oher parameer fixed. Thee prove he monooniciy reul. Now, le v = u A 1 q. Noice ha v i an n by 1 vecor where n i he number of reource. Denoe i jh enry by v j. Thu, q =(u j v j )/A 1j. Define G (v j, u )=G (q, u ). Then, G (v j, u ) = (b + ce(ξ F ))(B (u j v j )/A 1j )+p τ E e (A 2 e + A 1 (u j v j )/A 1j u F ) + + αe F+1,u +1,A +1 { V+1 [(1 ξ )(B (u j v j )/A 1j )+d +1,F +1 ] F }. Uing an argumen imilar o he proof of Lemma 3, i can be hown ha G i ubmodular in (u j, v j ). Thu, he maximum opimal choice of v j a a funcion of u j, which we denoe by (u j ), i increaing in u j, i.e., v max (u j ) v max (u j + ). Thu, v max j j j q min (u ) = u j vj max (u j ) A 1j u j + v max j (u j + ) A 1j A 1j = q min (u + E j ) A 1j, which i he required reul.. Proof of Theorem 4: Since Propoiion 2 implie { w P } = R P (), i follow ha α (b + cξ ) w P boh L(R P (),) and U(R P (),). Then, ince R P 0 () = [0, d =1 ], i follow ha α (b + cξ ) w P = L(R P (),)=L(R P (),) L(R 0 (),)= implying he fir required equaliy. Similarly, we obain α (b + cξ ) =1 d α (b + cξ ) w P = U(R 0 (),) U(R P (),)= =1 which how he econd required inequaliy. ( L(R P (),) L(R P 1(),) ) = =1 i L P (), =1 ( U(R P 1(),) U(R P (),) ) = U P (), =1

19 e-companion o Auhor: Muli-reource Allocaion Scheduling in Dynamic Environmen ec3 Proof of Propoiion 3: We have already oberved ha he incremenal co and benefi, L P () and U P (), are non-negaive. The monooniciy properie of L P () and U P () wih repec o q P, aed in (a) and (b), follow eaily from he definiion. If q P = min{c,w P + d }, hen he limi policie P and P 1 coincide; hence R P () and R P 1() hare he ame lower bound, implying L(R P (),)=L(R P 1(),), and L P () = 0. Thi prove he fir par of (a). Similarly, if q P = 0, hen he limi policie P and P 1 coincide; hence, R P () and R P 1() hare he ame upper bound. I follow ha U(R P (),)=U(R P 1(),), and U P () = 0, implying he fir par of (b). In he definiion of L P ()=L(R P (),) L(R P 1(),) given in (11), he econd erm doe no depend on q P, and hu i uffice o conider he convexiy of he fir erm, L(R P (),). Since R P ()={ w R + w P w w P } from (10), i follow ha L(R P (),) = min{α (b + cξ ) w w P Thu, i uffice o how he following claim: w P r prove hi, we check he bae cae ha w P w w P } = α (b + cξ ) w P. and wr P are boh convex in q P for all r. To =(w 1 P + d q P ) + and w P =(w 1 P + d q P ) + (1 ξ ) and wr P are convex in are boh convex in q P. Now, a an inducion hypohei, aume ha w P r q P, for r. Recalling from (8) ha c r+1 i he maximum number of elecive urgerie ha can be fulfilled in period r + 1, we can wrie w P r+1 =(w P r + d r+1 c r+1 ) + and w P r+1 =(w P r + d r+1 c r+1 ) + (1 ξ r+1 ), and hey are again convex in q P, compleing he inducion ep and proving he claim. Thu, L P () i convex in q P, and he convexiy implie coninuiy. Thi complee he proof of (a). Uing a imilar argumen, o how he lineariy of U P () in q P, i uffice o how he lineariy of U(R P (),)=α (b + cξ ) w P in q P. Since w P r+1 =(w P r + d r+1 ) and w P r+1 =(w P r + d r+1 )(1 ξ r+1 ), U(R P (),) i linear in q P. Thi complee he proof of (b). Proof of Theorem 5: Since w P = w P (1 ξ ) hold by he definiion of w P and w P, i uffice o prove ha w LM w OP T for any ample pah, i.e., we how he required reul for any realized value of random variable. Aume oherwie for a conracion. Le be he fir period in which w LM < w OP T. Then, by he choice of and from he definiion, w LM 1 w OP T 1 and w LM 1 q LM = w LM d < w OP T d = w OP T 1 q OP T. We conruc an alernae policy, denoed by P, a a policy ha follow OP T in period {1,..., 1}, aifie w P 1 q P = w 1 LM q LM in period, and chooe q P = min(q OP T,w P 1 + d ) in each period { +1,..., T }. The overime co a under policy P i given by p τ (A 2 e + A 1 q P u ) +. Noe ha P i well-defined in period ince q OP T q P = w 1 OP T w 1 LM +q LM q LM. We will analyze he co of P relaive o he co of OP T. For each period {1,..., 1}, he co we aign o period baed on he expreion given in (15) and (16), namely L P + O P, i idenical o he correponding co of OP T, namely, L OP T + O OP T. In period, he co ha P incur can be expreed a L P + O P = = α (b + cξ )( w P w P 1 )+α p τ (A 2 e + A 1 q P u ) + = f( w P )+g(q P ) h( w P 1), where we define f( w P )= T = α (b+cξ ) w P, g(q P )=α p τ (A 2 e +A 1 q P u ) +, and h( w P 1 )= T = α (b + cξ ) w P 1. We noe ha g i a convex funcion, and alo ha Corollary 1 implie he convexiy of f. Similarly, OPT incur L OP T + O OP T = f( w OP T )+g(q OP T ) h( w OP T 1 ).

20 ec4 e-companion o Auhor: Muli-reource Allocaion Scheduling in Dynamic Environmen Since P mimic OPT up o period 1, we know w P 1 = w 1 OP T, which implie h( w P 1 )=h( w OP T Le δ = q P q OP T. Since w P = w LM, we obain f( w P )+g(q LM ) = f( w LM )+g(q LM ) f( w LM + δ)+g(q LM 1 ). δ) = f( w OP T )+g(q LM δ), where he inequaliy follow from he definiion of LM ince he choice of q LM. Thu, O LM f( w OP T ) f( w P ) g(q LM minimize L LM + ) g(q LM δ) g(q OP T + δ) g(q OP T ) g(q P ) g(q OP T ), where he econd inequaliy follow from q OP T q LM δ and he convexiy and increaing propery of g. Thu, f( w P )+g(q P ) f( w OP T )+g(q OP T ). Therefore, we conclude L P + O P L OP T + O OP T. Now, for each period { +1,..., T }, from he definiion of q P, i i eay o how q P q OP T, w P w OP T and w P w OP T. Thu, we have O P O OP T. Suppoe we claim L P L OP T. Then clearly, L P + O P L OP T + O OP T, i.e., P ha oal co no more han ha of OP T. Thu, o prove L P + O P L OP T + O OP T, i uffice o prove he claim ha L P L OP T for each { +1,..., T }. From he definiion of aggregae incremenal co and aggregae incremenal benefi given in (11)-(14), i uffice o how w P w P 1 OP T w w OP T 1 for any {,..., T }. (EC.2) Conider he bae cae of =. From he definiion of he lower limi policie and he definiion of P, we have w P w P 1 = q P 1 q P = min(c,d + w P 1 ) min(w P 1 + d,q OP T ). If w P 1 + d q OP T hen w P w P 1 = min(c,d +w P 1 ) q OP T min(c,d +w 1 OP T ) q OP T OP T = w w OP T 1, where he inequaliy follow from w P 1 w 1 OP T which come from he definiion of policy P, and he la equaliy follow from he definiion of he lower limi policy. Oherwie, w P 1 + d <q OP T c implie q P 1 = q P = w P 1 + d, in which cae, w P w P 1 = 0. Since w P w P 1 0 hold for any policy P, we obain (EC.2) for =. We make an inducion hypohei ha (EC.2) hold for ome 1, where { +1,..., T }, and prove he reul for. From he definiion of he lower limi policie, w P w P 1 =(w P 1 + d c ) + (w P d c ) +. If w P by auming w P = 0, hen (EC.2) hold for ince w P w P 1 > 0. We conider he following hree cae. 0 hold for any policy P. Thu, we proceed Cae w P 1 > 0. From he definiion of P and he lower limi policie, i follow ha w P w P 1 OP T > 0, a well a boh w w P > 0 and w OP T 1 w P 1 > 0. All of hee quaniie are poiive in hi cae, and we obain w P w P 1 OP T w w OP T 1 =(w P 1 + d c ) + (w P d c ) + = (w P 1 + d c ) (w P d c ) and OP T =(w 1 + d c ) + (w OP T d c ) + = (w Then, he inducion hypohei for 1 implie (EC.2) for, i.e., w P Cae w P 1 Since (w P 1 1 = 0 and w OP T 1 w P w P 1 OP T w w OP T 1 + d c ) + (w P 1 1 > 0. Here, we have w OP T w P OP T 1 w P 1 > 0 and w OP T 1 + d c ) (w OP T d c ). OP T w =(w P 1 + d c ) (w P d c ) + and OP T =(w 1 + d c ) (w OP T d c ). w OP T 1 > 0, which imply. + d c ), he inducion hypohei implie he required reul.

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