An Efficient Outpatient Scheduling Approach

Transcription

1 > T-ASE R< An Efficient Outpatient Scheduling Approach Haibin Zhu, Senior Meber, IEEE, Ming Hou, Senior Meber, IEEE, Chun Wang, Meber, IEEE, and MengChu Zhou, Fellow, IEEE Abstract Outpatient scheduling is considered as a coplex proble. Efficient solutions to this proble are required by any health care facilities. Our previous work in Role-ased Collaboration (RC) has shown that the group role assignent probles can be solved efficiently. Making connections between these two kinds of probles is eaningful. This paper proposes an efficient approach to outpatient scheduling by specifying a bidding ethod and converting it to a group role assignent proble. The proposed approach is validated by conducting siulations and experients with randoly generated patient requests for available tie slots. The aor contribution of this paper is an efficient outpatient scheduling approach aking autoatic outpatient scheduling practical. The exciting result is due to the consideration of outpatient scheduling as a collaborative activity and the creation of a qualification atrix in order to apply the group role assignent algorith. Note to practitioners -As the Age Wave approaches, health care facilities are becoing relatively scarce worldwide copared with what are deanded. The varying availability, requireents, and preferences of both facilities and outpatients ake the proble of scheduling outpatient appointents on health care facilities extreely challenging. Traditional anually operated scheduling systes based on phone calls are out of date although they are still widely used due to lack of new effective scheduling systes. To solve such a proble requires an efficient Web-based syste to schedule the appointents instantly in order to ake full use of those expensive and critical facilities. It is able to optiize concerned perforance obectives in a clinical environent. The proposed approach provides a technical foundation for efficient Web-based scheduling systes, which can be applied directly to not only outpatient scheduling in the health care sector, but also in soe other real-world scheduling applications. Index Ters Outpatient Scheduling, Roles, Agents, and Role Assignent Manuscript received March 22, 22, revised June 7, 22. This work was supported in part by NSERC under Grant No , IM Eclipse Innovation Grant, and DRDC - Toronto Grant. H. Zhu is with the Departent of Coputer Science and Matheatics, Nipissing University, College Drive, North ay, Ontario, P 8L7, Canada (phone: ext. 4434; fax: ; e-ail: haibinz@nipissingu.ca). M. Hou is with Defence Research & Developent Canada - Toronto, 33 Sheppard Avenue West, Toronto, Ontario, M3M 39, Canada (phone: ; fax: ; e-ail: Ming.Hou@drdc-rddc.gc.ca). C. Wang is with Concordia Institute for Inforation Systes Engineering, Concordia University, 55 Ste Catherine West, EV 7.649, Montreal, H3G M8, Canada (phone: ext. 5628; fax: ; e- ail: cwang@ciise.concordia.ca). M. C. Zhou is with the MoE Key Laboratory of Ebedded Syste and Service Coputing, Tongi University, Shanghai 292, China (e-ail: engchu@gail.co). I I. INTRODUCTION N health care anageent, soe facilities, such as agnetic resonance iaging (MRI) scanning or coputed toography (CT) scanning, are expensive and critical for certain disease diagnoses [2, 9]. Norally expensive facilities ust be pre-scheduled for unanticipated requireents of inpatients. Soetie later, pre-scheduled tie slots ay becoe available for outpatients to use. At the sae tie, outpatients ay have different requests for the appointents and soe appointents ay not be available for soe tie slots, i.e., there are different constraints in assigning available tie slots of an expensive device to available patients. Therefore, instant re-scheduling is required at such a oent in ters of volue and urgency. ecause the proble of efficient scheduling of patient appointents on expensive resources is coplex and dynaic, it ust be solved with an efficient syste to re-schedule the appointents to avoid wastes and ake full use of the expensive and critical facilities, i.e., the obective of outpatient scheduling is to find an appointent syste for which a particular easure of perforance is optiized in a clinical environent - an application of resource scheduling under uncertainty [2]. In Role-ased Collaboration (RC) [23], Group Role Assignent (GRA) [27] is a coplex proble for which the exhaustive-search based algorith has exponential coplexity. An efficient algorith for GRA has been developed based on the Hungarian algorith, also called Kuhn-Munkres (K-M) algorith [, 4]. It is of polynoial coplexity. This work builds a syste that transfers outpatient scheduling into a GRA proble. II. RELATED WORK It is well accepted that scheduling probles in health care services are iportant and coplex. Much research is conducted in the fields of operational research and industrial engineering [2, 5]. However, very few attepts have been ade to solve the through the developent of practical systes. Cayirli and Veral [2] present general proble forulation and odeling considerations for effective scheduling systes, and provide taxonoy of ethodologies used in previous literature. Godin and Wang [5] propose to allocate available diagnostic services tieslots to outpatients through an iterative bidding procedure which is a trigger to the idea of Call-For-Collaboration (CFC) in this paper. Gul et al. [6] copare several heuristics for scheduling an outpatient procedure center with respect to the copeting criteria of

2 > T-ASE R< 2 expected patient waiting tie and overtie. Guo et al. [7] present a siulation fraework for the evaluation and optiization of scheduling rules. Gupta and Denton [8] state that any factors affect the perforance of such systes including arrival and service tie variability, patient and provider preferences, available inforation technology and the experience level of the scheduling staff. Kaandorp and Koole [] odel the outpatient appointent scheduling proble, and present a ethod to obtain optial outpatient schedules in case of a finite nuber of possible arrivals. They do not ention the efficiency of their algoriths. Liu et al. [3] develop a fraework and propose heuristic dynaic policies for scheduling patient appointents by considering that patients ay cancel or not show up for their appointents. Their consideration is also another source of the proble presented in Section II, i.e., cancellation and no-show ay release tie slots for outpatients to reschedule. Patrick and Puteran [6] analyze the CT operations at Vancouver General Hospital, Canada and state that inpatient deands are fluctuating and the outpatient have priorities [2]. Our exaple is established based on this work [6]. Santibáñez et al. [9] develop a ixed integer prograing odel to schedule surgical blocks for each specialty into operating roos and applied it to the hospitals in ritish Colubia Health Authority, considering operating roo tie availability and post-surgical resource constraints. Vereulen et al. [2] present an adaptive approach to autoatic optiization of resource calendars. The allocation of capacity to different patient groups is flexible and adaptive to the current and expected future situation. III. A REAL-WORLD OUTPATIENT SCHEDULING PROLEM The MRI lab of a hospital has a future 2-week schedule fro Monday to Friday. ecause soe pre-occupied slots are released by in-patients, the newly available hours (slots) are presented as the shaded cells in Table I. Note that the preoccupied slots cannot be freed earlier than a period of tie, e.g., two weeks, because the slots ust be prepared for the inpatients. Hence, a two-week schedule is usually adopted. Now, only one work day is left before the new schedule. The adinistration hopes to use as any unoccupied slots as possible. The question is how to assign the available slots to the ost needed outpatients and instantly infors the outpatients the new scheduled tie slots. TALE I. AN EXAMPLE OF A 2-WEEK SCHEDULE. Conventionally, outpatient records in a database tell soe outpatients pre-filled requireents and availability in choosing adacent tie slots (also called requested bundles of tie slots, or siply, requested bundles). Those outpatients who can coe to fill newly available slots in Table I are shown in Table III. ased on such a table, an outpatient scheduling proble is odeled as an optiization proble and proved as an NP-hard proble [5]. To foralize the questions in this paper, we use the sybols in Table II. TALE II. THE SYMOLS USED IN THIS PAPER Sybol Meaning A A set of agents. A bundle of tie slots. F t (C, ) The set fored by the eleents of vector C[]. N The set of non-negative integers, i.e., {,, 2, }. P An n-diensional vector of vectors of the bidding blocks fro outpatients. P is a vector with k ( k ) bidding blocks requested by patient Π. P also expresses the preferences of patient (if l < k then patient prefers the l th block P [l] to the k th one P [k], l, k< k, l k). P An n-diensional vector of vectors of bidding sets (also called bundles) of tie slots fro outpatients, where P is a vector with k ( k ) bidding sets (bundles) of tie slots requested by patient Π. Q: A R A qualification atrix. Q[i, ] expresses the [, ] qualification value of agent I for role. R A set of roles. S: Ω N S[i] expresses the size of block I (I Ω). S : Π N S [] expresses the requested block size by patient ( Π). T: A R {, } An assignent atrix. T[I, ] = eans that agent I is assigned to role, and T[I, ] = otherwise. T A vector of appointents for outpatients. V: R A V[] expresses the original agent assigned to role. W: R N A vector to express the priority values of outpatients (roles), where W be the priority value assigned to patient Π or R. g: Ω Π [, ] A preference scale to evaluate the relative preference aong different patients and requested blocks. The fitness of the requested size of the requested block fro versus the available size of block i. H: Ω Π [, ] I An eleent in Ω or A. J An eleent in Π or R. K ax{, n} The nuber of eleents in Ω, i.e., Ω. The nuber of eleents in Ω or A, i.e., Ω or A. n The nuber of eleents in Π or R, i.e., Π or R. w ax The axiu nuber in W. X An assignent indicator. X ()= if bundle is allocated to patient ( ', <n), x ()= otherwise. Δ The su of the priority values of the scheduled patients, i.e., x ( ) P' W. Δ The su of the priority values of on the scheduled tie x ( i) S'[ ] W. slots, i.e., i * Δ ax {Δ}. Π A set of the bidding outpatients. Ω A set of available tie slot blocks, or siply blocks. Ω, Ω A set of available tie slots. δ in{, n} ζ(p, i, ) To express if agent i belongs to F t (C, ). Ψ S '[ ] W. If each outpatient requires one bundle of tie slots and we do not consider the preferences of patients aong requested bundles, the proble can be foralized as [5]:

3 > T-ASE R< 3 ax Δ= x ( ) P ' W () subect to, x ( ) (2) ' i ', x ( ) (3) i x ( ) ', = ' x ( ) (4) P ', x () {,} (5) where constraint (2) ensures that each outpatient can be assigned at ost one requested bundle of tieslots; (3) ensures that each tie slot is assigned to only one patient; (4) ensures that the assigned bundle is requested by the patient; and (5) is a - constraint. Note that i that contains i. in (3) is read as su for all TALE III. THE AVAILAILITIES AND PREFERENCES OF OUTPATIENTS. ID Nae Priority Outpatients availabilities in a day value P To 3 Monday: {7, 8}, Wednesday: {2, 3}, Thursday: {3, 4} P2 Chris 2 Monday and Tuesday: {7,8,9} P3 Ana 3 Wednesday and Thursday: {3} P4 ob 2 Wednesday: {,2}, Thursday: {3, 4} P5 Don Any day: {},, {} P6 Jane 3 Thursday: {7, 8} TALE IV. AVAILALE TIME SLOTS (34). To solve the proble described in ()-(5), the available tie slots are nubered fro -33 (Table IV), outpatients are nubered fro -5, and their availability is transferred to Table V. TALE V. THE AVAILAILITIES AND PREFERENCES OF OUTPATIENTS. Patient Nae Priority Outpatients requested bundles ID value P To 3 {2, 3},{9, },{9, 2}, {26, 27} P2 Chris 2 {2, 3, 4},{6, 7, 8} P3 Ana 3 {}, {2}, {26} P4 ob 2 {9, },{9, 2}, {26,27} P5 Don {},, {33} P6 Jane 3 {3, 4}, {29, 3} TALE VI. ASSIGNED TIME SLOTS (NAMED SLOTS). y using IM ILOG CPLEX Optiization Studio V2.2 (ILOG), the proble is solved (Table VI) in 3s with the obective as 26 (Section V of the ultiedia docuent). If the nuber of bundles and patients increases, the consued tie increases exponentially. Experients also assert that such odeling could only work for relatively sall size outpatient scheduling probles [5]. IV. COLLAORATIVE OUTPATIENT SCHEDULING - OUR STRATEGY In fact, outpatient scheduling is dynaic. Available tie slots, the outpatients availability and preference are changing. The inforation in Table III ay not reflect the current states of all the outpatients. The odel in ()-(5) does not consider the preferences of outpatients. In our strategy, the outpatient scheduling proble is considered as a collaborative action, i.e., the patients are collaborating on this scheduling work. The operation scenario is that, upon a change in the available tie slots, the facility office or clinic sends out a Call for Collaboration (CFC) essage by eails or calls to all the already registered (scheduled or not yet scheduled) outpatients, and soe or all of the respond to the CFC essage by bidding for bundles of tie slots. The scheduling algorith then akes optial rescheduling based on their responses. We assue that the tie unit is in slot, and each outpatient has a priority value assigned by his/her doctor. The priority values of outpatients are taken as one indicator to express the ranking and copetence on a tie slot aong outpatients. In this way, if a bundle of 2 tie slots is assigned to a patient who has a priority value of 3, we collect 3 2 into the su of the priority values on the assigned tie slots. Then, the outpatient scheduling syste is designed to find an assignent schee for all patients such that the su of priority values on the assigned slots is axiized and their preferences are satisfied. We assue that the bidding patients ay have their original appointents when CFC is initiated, and they will contribute available slots when they are rescheduled. CFC is a process that is initiated anually or autoatically according to a schedule or newly available inforation, such as, tie slots becoe available and it is needed to reschedule. It ends when the iteration of rescheduling is done. The syste process is described as follows, where Ω" ( Ω' ) is a new set of available tie slots after one iteration of rescheduling, Ω'=Ω" expresses that no new slot is allocated in the rescheduling process. CallForCollaboration Process: Input: Ω' Output: T' Repeat Step : Receive: Π, W and P'; Until (tie is due). Repeat Step 2: Rescheduling (Ω', Π, W, P', Ω", T'); Until (Π= Φ Ω'=Ω"); Send out or post new schedules T'; Rescheduling Process: Input: Ω', Π, W, P'

4 > T-ASE R< 4 Output: T' and Ω". Step : Maxiize Δ while their preferences are satisfied; Step 2: For and return T' and Ω". To decrease the search space in the re-scheduling process, we propose to introduce soe restrictions in a bidding process: ) Each round of CFC considers a group of continuous tie slots as a whole, called tie slot block or siply block with a size attached (i.e., the nuber of continuous tie slots). Now, we obtain a set of available blocks Ω. A size vector S: Ω N ={,, 2, } is introduced, where S[i] expresses the size of block i (i Ω); 2) Patients ay choose soe fro these blocks (See Table VII, the available blocks have different nuber of slots) and specify how any slots they require. A size vector, S : Π N is introduced, where S [] expresses the requested size of the requested block by patient ( Π); 3) The sequence of the outpatients choices shows their preferences; and 4) A patient is allocated at ost one block with the requested size. TALE VII. THE AVAILALE TIME LOCKS () FOR TALE I. As for the proble in Table I, we ay redraw it shown as in Table VII. Now, the proble can be re-specified. We introduce a function g: Ω * Π [, ], where Ω * is the power set of Ω, as a preference scale to evaluate the relative preference aong different patients and requested bundles, i.e., g(i, ) = ( k - l) /k l =,,, k-,i P, i P [l] ; otherwise. (6) h(i, ) is introduced to evaluate the atching scales of the request, i.e., h(i, ) = S' [ ] /S[ i] ( S' [ ] S[ i]) ( i<, <n) (7) ( S' [ ] S[ i]) For exaple, suppose that other conditions are the sae. If patient x bids for the 3 rd (i=2) choice in 5 choices (k x =5); and patient y bids for the 2 nd (i=) in 3 choices (k y =3), we prefer the latter, i.e., y, because g(2, x)=3/5 < g(2, y)=2/3. If patient x requests 3 slots in a 5-slot block z, i.e., S[z]=5, S [x]=3, and patient y requests 4 slots in the sae block z, S [y]=4, we prefer y, because h(z, x)=s [x]/s[z]=3/5 < h(z, y) = S [y]/s[z]=4/5. With this adustent, the outpatient scheduling proble becoes to find an assignent schee for all patients such that Δ is axiized; all patient preferences are best satisfied; and all patient requests are best atched. It can be re-foralized as a three-obective optiization proble: subect to,, i, ax Δ = ax ax x ( i) S'[ ] W (8) i x (i ) (9) i g(i, ) x (i ) () i h(i, ) x (i () i ) i (i ) x ip (i ) x i) S'[ ] x (2) ( S[i] (3) i,, x (i) {,} (4) where () ensures that any outpatient can only obtain one fro the available tie blocks; (2) ensures that if a tieslot block is assigned to an outpatient, it ust belong to the block set the outpatient has requested; constraint (3) tells that each block can be allocated to ore than one patient, however, the total requested sizes of the assigned patients should not be larger than the size of the original block; and (4) is a - constraint. It is evident that the bundle requireent is reoved and therefore the proble is siplified. Note that this is a typical ulti-obective optiization proble to which a siple solution is weighted su [6]. However, it is still tie-consuing based on the odel (6)- (4). For exaple, a rando case (, n = 4) cost 54 inutes to be solved by a weighted su ethod by using ILOG (The Model and Data I in Section VI of the ultiedia docuent). TALE VIII. AN EXAMPLE FOR OUTPATIENT SCHEDULING ID Nae Priority Patients lock Original appointent value biddings for blocks size P To 3, 3, 6, (8-9 of Thursday in the 3 rd week) P2 Chris 2, (2-4 of Monday in the 4 th week) P3 Ana 3 3, 6, 8 ( of Thursday in the 3 rd week) P4 ob 2 3, 6, 8 2 3(- of Tuesday in the 4 th week) P5 Don,, 9 (7 of Friday in the 3 rd week) P6 Jane 3 4, ( 3-4 of Friday in the 4 th week) Table VIII is the assued inforation collected by one round of CFC siilar to the requests fro Table V. It is assued that the responses to CFC are for the available blocks fro to 9. The blocks originally assigned to the 6 responded patients are - 5. Please note that it is not hard to collect the inforation shown in Table VIII. For exaple, a group e- ail can be sent out to all the outpatients. The interested ones can click on a link provided by the e-ail to provide their preferences aong the available blocks. The priority values are found based on the electronic docuents of outpatients in the health care office. So are the original appointents. To solve the proble expressed by forula (6-4) efficiently, the GRA algorith ust be iteratively called because it can only assign one available block to each patient in each iteration, i.e., it can only solve the proble by replacing (3) with (5):

5 > T-ASE R< 5 i, x ( ) (5) i After each GRA assignent, soe available blocks ay be still available for those outpatients who have not yet scheduled. We need ore GRA processes until no available tie slot blocks cover the requests of outpatients. This iteration ay affect the global optiization as described by forula 6-4, but leads to an efficient solution. That the three optiization goals are synthesized to for one goal is another factor to affect the optial solution. Note that, the aor idea is to transfer soe constraints to nubers in order to apply the optiization algorith. We adit that not all the constraints can be transferred to proper nubers, but there are indeed soe constraints that can be processed this way. V. GROUP ROLE ASSIGNMENT y E-CARGO [23-27], we ean Environents, Classes, Agents, Roles, Groups and Obects. To deal with the role assignent probles, we ephasize a role set denoted by A and an agent set denoted by R. Agents in A are nubered as,,, and - ( = A ); and roles in R are nubered as,,, and n- (n = R ). Definition : A role range vector is a vector of the lower ranges of roles denoted as L[] N ( < n). Definition 2: A qualification atrix is defined as Q: A R [, ], where, Q[i, ] expresses the qualification value of agent i for role. The iproveent of the efficiency of the algorith for the outpatient scheduling proble ainly coes fro the foration of the Q atrix. Definition 3: A role assignent atrix is defined as T: A R {, }. If T[i, ] =, agent i is assigned to role and T[i, ] = otherwise ( i < ; < n). Note that T also expresses a group. Definition 4: A group qualification is defined as the su of the assigned agents qualifications, i.e., n Q [ i, ] T[ i, ]. i Definition 5: A role r is workable if it is assigned enough current agents to play it, i.e., T [ i, ] L[] [23-27]. Definition 6: A group expressed by T is workable if all its roles are workable. Definition 7: The group role assignent (GRA) proble. is to find an assignent atrix T that akes the group qualification is the largest, i.e., ax{ n Q [ i,] T[ i,] } i subect to ( T [ i,] =L[])( < n). i i VI. FROM THE OUTPATIENT PROLEM TO THE GROUP ROLE ASSIGNMENT PROLEM In ters of GRA, we consider patients as roles and tie slot blocks as agents. This consideration is explained next. Definition 8: An original agent vector V is an n-vector of the original agent assigned for a role, i.e., V: R A, where V[] eans the agent originally assigned to role. We use F t (P, i) to express the set fored by the eleents of vector P[i]. If F t (P, ) F t (P, ') Φ, we say that roles and ' are copeting on agents (blocks) expressed by F t (P, ) F t (P, '). We keep other sybols as described in Sections III and IV. In solving the outpatient scheduling proble, the ost iportant step is to build an appropriate Q to reflect the values required in the assignent process. The vectors W and P and functions g and h can be used to for Q with the following forula ( i<, <n): Q[i, ] = P, i, ) g( i, ) h( i, ) ( i) ( W[ ]/ w ) ; where, ζ(p, i, )= ( ax i F t i F ( P, ); t ( P, );...(6) (7) γ()=s [] / ax{s [], S [],, S [n-]} (8) Note that, ζ(p, i, ) tells that tie slot blocks (agents) are only qualified for the requested roles (patients), i.e., it is to prevent the situation that an agent (tie slot block) is assigned a role (patient) who is not willing to accept; γ() considers the size of the assigned blocks to patient and note that we do not use S [] directly in order to keep Q[i, ] [, ]; g(i, ) in (6) ensures that if two available blocks are patient s choices, the block with a better preference is assigned, and that if two patients are copeting for one tie block, the patients preferences are serialized, e.g., if tie block z is patient x's 2 nd preference (its index is ) in 3 choices and is patient y's 2 nd preference (its index is ) in 5 choices then y is preferred (2/3 < 4/5); and h(i, ) in (7) expresses that a tie block (agent) has higher qualification if its size fits the tie block size better Fig.. The Q atrix for Table VIII. Fro Table VIII, W = [3, 2, 3, 2,, 3], w ax = 3, P = [[, 3, 6, 8],[, 5], [3, 6, 8], [3, 6, 8], [,, 2, 3, 4, 5, 6,7, 8, 9], [4, 9]], S = [2, 3,, 2,, 2], S= [2, 4, 3, 3, 3, 4, 4, 4, 2, 6], and V= [,, 2, 3, 4, 5]. For exaple, Q[3, 2] = (/3) (/3) [(3-)/3] (3/3)=.. y synthesizing all the above data, we get a qualification atrix Q shown in Fig.. Now, A is the set of blocks available at the tie of CFC; R is the set of n patients who have responded to CFC; and Q is an n atrix obtained by forula (6-8). Then, the outpatient scheduling proble is in fact becoing a GRA proble to find an n assignent atrix T to

6 > T-ASE R< 6 n ax{q [ i,] T[ i,] } (9) i subect to n T [ i, ] ( i<) (2) ( i T [ i,] =L[])( < n) (2) In E-CARGO, that one agent can only play one role in GRA directly follows the above requireent of outpatient scheduling that one block (agent) can be assigned to only one n outpatient (role), i.e., T [ i, ] ( i < ). In fact, we can even loosen the restriction in GRA, i.e., reove the requireent of, T [ i,] =L[]( < n), i because it is acceptable for soe bidding outpatients to obtain no block in one round of CFC. ased on the GRA algorith, we can obtain atrix T shown as in Fig. 2 for the Q atrix in Fig.. Fig. 2. The Assignent Matrix T. Let x (i) in (8) = T[i, ] ( i < ; < n). We obtain Δ * as 26. We can translate the atrix T in Fig. 2 to a list of assignent tuples as the scheduling result of the st round CFC (naed slots in Table VIII). TALE IX. THE ASSIGNED TIME SLOTS. TALE X. THE NEW AVAILALE TIME SLOTS. Cobined with the original agent (tie slot block) vector V, the new list of available tie slot blocks becoes (shaded blocks Table X). ecause all the patients are scheduled, no ore iteration of GRA is required. The above list can be used for the 2 nd round of CFC. Note that block is fored by cobining two original appointents, i.e., (slots 8-9 and slot of Thursday in the 3 rd week). VII. THE ALGORITHM AND COMPLEXITY The coplexity of the efficient GRA algorith is polynoial [27]. If we transfer the outpatient scheduling proble into the GRA proble, the outpatient scheduling proble is solved in polynoial tie. The following algorith OutpatientRescheduling ainly describes the pre-process, the use of the GRA algorith, and the post-process. Note: the following algorith is described in a Java-like language; a=b eans to check if a is equivalent to b and a:=b is to assign the value of b to a. Input: Π: A set of outpatients bidding <x, y, Z, v>, where x is the identification of the outpatient; y: the priority value of the outpatient; Z is the list of the bids of tie slot blocks of the outpatient, where the position of a block in the list expresses the patient s preference; u is the size of the requested block; and v is the original tie slot block of the outpatient. Ω: A set of the available tie slot blocks expressed as <c, d, e, f >, where c is the starting slot nuber; d is size of the block; e is the day nuber of the week; and f is the nuber of the week. Output: χ: A list of tuples <a, b> where a eans a patient and b eans a tie slot block. Ω: a new set of available tie slot blocks. OutpatientRescheduling(Π, Ω, χ) { := := Ω ; n := n := Π ; while ( > and n < n); { Step : Transfer Ω into a list Ώ, where Ώ[i] Ω; ( i -); Step 2: Transfer Π into 4 lists P, W, C, and V, i.e., P[]:=c.x, W[]:=c.y, C[]:=c.Z, V[]:=c.v (cπ, n-); Step 3: Note that h, g, Q, ζ, and γ are all n atrices corresponding to forula (6, 7, 6-8). w ax: := ax {c.y (cπ)}; for ( i -, n-) { if ( i C[ ] ) ζ[i, ] := ; else ζ[i, ] := ; if (S [] S[i]) h[i, ]:= S []/S[i]; else h[i, ]:=; α[]:= C[].length; β[i, ]:= index of i in C[]; g[i, ]:= (α(c, ) β(c, i, ))/α(c, ); Q[i, ] := ζ[i, ] h[i, ] g[i, ] γ [] W[]/w ax ; } Step 4: Note T is an n atrix. Initialize the assignent atrix T with {}; Call RatedAssignForOutpatients(Q, T,, n); Step 5: For the new list of appointents in <patient, tie slot block> and adust the available tie slot blocks. Initialize χ[] with NULL( n-); for ( i -, n-) if (T[i, ]=) {

7 > T-ASE R< 7 χ[] := <P[], the first P[].u slots of Ώ[i]>; if (P[].u= Ώ[i].d ) Ώ[i]:=NULL; else { Ώ[i].c := Ώ[i].c + P[].u; Ώ[i].d := Ώ[i].d - P[].u; } } Step 6: Keep the unscheduled patients the original appointents. Π := Φ; for ( n-) if (χ[]=null) { χ[]:=<p[], V[]>; V[]:=NULL; Π := Π {P[]}; } Step 7: For the new set of available tie slot blocks. Ω := Φ; for ( i -) if (Ώ[i] NULL) Ω := Ω { Ώ [i]}; for ( n-) if (V[] NULL) Ω := Ω {V[]}; := Ω ; n := Π ; }; if (n =) every patient gets an assignent; else soe patients have no assignents; }//End of OutpatientRescheduling The algorith RatedAssignForOutpatients is described as follows. Input: Q: an n rated qualification atrix. Output: T: an n assignent atrix. RatedAssignForOutpatients(Q, T,, n) {Step : k = ax{, n}; Transfer the n atrix Q to a k k atrix M [29]; Step 2: K-M (M); //Call the K-M algorith; Step 3: For the assignent atrix T based on the result of K-M (M); Step 4: return T; } Note that, in the above algorith, the requireent of, T [ i,] =L[]( < n) in the RatedAssign algorith is i reoved, because it is acceptable for soe bidding outpatients to obtain no block in one round of CFC. Theore : Algorith RatedAssignForOutpatients has the coplexity of O(k 3 ) (k = ax{, n}). Proof: Step has the coplexity of O(k 2 ). Step 2 (K-M algorith) has O(k 3 ) [, 4] Step 3 has O(k 2 ). Step 4 has O(). Therefore, the total coplexity is O(k 3 ). Theore 2: Algorith OutpatientRescheduling has the coplexity between O(k 3 )(the best case) and O(δk 3 ) (the worst case), where k = ax{, n} and δ = in{, n}. Proof: ased on Theore, Step 4 has the coplexity of O(k 3 ). Step has O(). Step 2 has O(n). Step 3 has O( n). Step 5 has O( n). Step 6 has O(n). Step 7 has O(+n). Therefore, The total coplexity inside the do loop is O(k 3 ). As for the while loop, the coplexity is ainly deterined by three factors besides and n: () the original appointents brought in by the bidding outpatients; (2) the conflict bidding for the sae blocks; and (3) the requested blocks fro patients. The worst situation is that the nuber of the do loops is δ = in{, n}, i.e., all the patients are bidding for the sae block that is large enough for all the patients, and each do loop satisfies one patient within this block. The best is within one loop: () All the patients bid for different blocks and are satisfied in one loop; or (2) All the blocks are assigned to patients in one loop. In suary, the coplexity of algorith OutpatientRescheduling is between O(k 3 ) and O(δk 3 ) VIII. VERIFICATIONS AND COMPARISONS To verify the proposed approach, we conducted siulations, perforance experientations, and perforance coparisons. The siulations show that the optiality is satisfied. The ties used by the proposed algorith for soe rando probles are fro.44 illiseconds (s) to 2s copared with that of ILOG fro 28s to ore than 2 inutes. The perforance experients show that the tie used for large groups ( = 8, n= ) is practical, i.e., at ost 6.6 seconds and in average 4.6 seconds. All the results are included in the suppleental ultiedia docuent. IX. CONCLUSIONS AND FUTURE WORK This paper contributes an efficient approach to outpatient scheduling by a special treatent for collecting patients choices fro available tie slots. An exciting future task will be to generalize the proposed approach and to find a way to transfor as any constraints as possible into a qualification atrix of GRA. If all the constraints of a general scheduling proble can be transfored into a qualification atrix of GRA in polynoial tie, such a general scheduling proble will be solved within polynoial tie. Such idea ay be extended to other scheduling probles [, 3, 2, 5, 9]. More interest future tasks include: ) to ipleent an online service syste in real health care environents; 2) to find out an algorith if functions h and g are correlated; 3) to investigate if the abnorality in Figs. 3 and 4 of the suppleent docuent is a deterined phenoenon; 4) to introduce heuristics in solving such assignent probles; and 5) to conduct epirical studies on the proposed approach of Call for Collaboration. REFERENCES []. Alidaee, H. Wang, and F. Landra, On the Flexible Deand Assignent Probles: Case of Unanned Aerial Vehicles, IEEE Trans. on Autoation Sci. and Eng., 8(4), pp , Oct. 2.

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Atkins, Surgical block scheduling in a syste of hospitals: an application to resource and wait list anageent in a C health authority, Health Care Manageent Science, vol., pp , 27. [2] I.. Vereulen, S. M. ohte, S. G. Elkhuizen, J.S. Laeris, P.J.M. akker, and J.A. La Poutre, Adaptive Resource Allocation for Efficient Patient Scheduling, Artificial Intelligence in Medicine, vol. 46, no., pp. 67-8, May 29. [2] N. Wu, and M. C. Zhou, "Schedulability Analysis and Optial Scheduling of Dual-Ar Cluster Tools with Residency Tie Constraint and Activity Tie Variation," IEEE Trans. on Autoation Science and Engineering, vol. 9, no., pp , Jan. 22. [22] M.. Wright, Speeding up the Hungarian algorith, Coputers & Operations Research, vol. 7, no., pp , 99. [23] H. Zhu and M.C. Zhou, Role-ased Collaboration and its Kernel Mechaniss, IEEE Trans. on SMC, Part C, vol. 36, no. 4, pp , July 26. [24] H. Zhu and M.C. Zhou, Role Transfer Probles and Algoriths, IEEE Trans. on SMC, Part A, vol. 36, no. 6, pp , Nov. 28. [25] H. Zhu and M.C. Zhou, M-M Role Transfer Probles and Solutions, IEEE Trans. on SMC, Part A, vol. 39, no. 2, pp , March 29. [26] H. Zhu, and M. Zhou, An Efficeint Solution to the Role Transfer Proble, IEEE Trans. on SMC, Part A, vol.42, no.2, pp , March 22. [27] H. Zhu, M. Zhou, and R. Alkins, Group Role Assignent via a Kuhn- Munkres Algorith-based Solution, IEEE Trans. on SMC, Part A, vol.42, no. 3, 22, pp Haibin Zhu (M 2-SM 4) is Full Professor of the Departent of Coputer Science and Matheatics, Director and Founder of Collaborative Systes Laboratory, Nipissing University, Canada. He has published 2+ research papers, four books and two book chapters. He is serving and served as cochair of the Technical Coittee (TC) of Distributed Intelligent Systes of IEEE SMC Society, guest (co-) editor for 3 special issues of prestigious ournals, and organization chairs for any IEEE conferences. He is a founding researcher of Role-ased Collaboration and Adaptive Collaboration. He is the receipt of any awards. His research interests include Collaboration Theory, Technologies, Systes, and Applications, Huan-Machine Systes, Multi-Agent Systes, and Distributed Intelligent Systes. For ore inforation please feel free to browse Dr. Zhu s Website at Ming Hou (M 5 SM 7) received the Ph.D. degree in huan factors engineering fro the University of Toronto, Toronto, ON, Canada, in 22. He is currently a Defence Scientist and the Head of the Advanced Interface Group, Defence Research and Developent Canada-Toronto, where he is responsible for providing infored decisions to the Canadian Forces on investent in and application of advanced technologies for operator achine interface requireents. His research interests include applied cognition, intelligent adaptive syste design, virtual/ixed reality, supervisory control of uninhabited vehicles, and e-learning. Dr. Hou is a eber of the Huan Factors and Ergonoics Society and the Association of Coputing Machinery. He was the Chair of the Syposiu on Huan Factors and Ergonoics at the 29 IEEE Toronto International Conference Science and Technology for Huanity. He has been the Co-Chair of the International Syposiu on Mixed and Virtual Reality since 24. Chun Wang (M 6) received the.eng. degree in 99 fro Huazhong University of Science and Technology, China, and the M.E.Sc. and Ph.D. degrees in coputer engineering in 24 and 28, respectively, fro the University of Western Ontario, Canada. He is currently an assistant professor at Concordia Institute for Inforation Systes Engineering, Concordia University, Montreal, Quebec, Canada. He worked as a software engineer and a proect anager for The China National Petroleu Co. fro 99 to 2. His research focuses on e-supply Chain, e-coerce, algorithic echanis design, and ulti-agent systes. MengChu Zhou (S 88-M 9-SM 93-F 3) received his.s. degree in Electrical Engineering fro Naning Univ. of Sci. and Tech., Naning, China in 983, M.S. degree in Autoatic Control fro eiing Inst. of Tech., eiing, China in 986, and Ph. D. degree in Coputer and Systes Engineering fro Rensselaer Polytechnic Inst., Troy, NY in 99. He oined New Jersey Inst. of Tech. (NJIT), Newark, NJ in 99, and is currently a Professor of Electrical and Coputer Engineering and the Director of Discrete-Event Systes Laboratory. He is presently a Professor at Tongi University, Shanghai, China. His research interests are in intelligent autoation, lifecycle engineering and sustainability evaluation, Petri nets, wireless ad hoc and sensor networks, seiconductor anufacturing, Web service, workflow, and energy systes. He has over 44 publications. He was invited to lecture in any countries and served as a plenary speaker for any int l conferences. He is a founding Editor of IEEE Press ook Series. He served as Associate Editor of IEEE Trans. on Robotics and Autoation (997-2), and IEEE Trans. on Autoation Sci. and Eng.(24-27), and is currently Editor of IEEE Trans. on Autoation Sci. and Eng., and Associate Editor of IEEE Trans. on SMC-Part A and IEEE Trans. on Industrial Inforatics. He served as Guest-Editor for any prestigious ournals including IEEE Trans. on Industrial Electronics. He was General Chairs for any int l conferences. Dr. Zhou has led or participated in proects with total budget over $M. He was the recipient of any prestigious awards. He was recently elevated to Fellow of Aerican Association for the Advanceent of Science (AAAS). For ore inforation please feel free to browse Dr. Zhou s Website at