Stochastic Programming Models for Optimization of Surgery Delivery Systems

Transcription

1 Stochastc Programmng Models for Optmzaton of Surgery Delvery Systems Bran Denton Department of Industral and Operatons Engneerng Unversty of Mchgan

2 Collaborators Har Balasubramanan (Unversty of Massachusetts) Sakne Batun (Unversty of Pttsburgh) Born Berg (North Carolna State Unversty) Todd Huschka (Mayo Clnc) Andrew Mller (Unversty of Bordeaux) Hed Nelson (Mayo Clnc) Ahmed Rahman (Mayo Clnc) Andrew Schaefer (Unversty of Pttsburgh) Ths proect s funded n part by the Natonal Scence Foundaton through Grant Number: CMMI

3 Summary Introducton Surgery process and complcatng factors Examples: OR = Operatng Room Problem 1: Sngle OR schedulng Problem 2: Mult-OR surgery allocaton Problem 3: B-crtera schedulng of a surgery sute Other Related Research 3

4 Motvaton Health care expendtures n the Unted States exceeded $2 trllon n 2012 Surgery accounts for the sngle largest proporton of a U.S. hosptal s total expenses and revenues Effcent access to surgery s mportant for patent health and safety 4

5 Motvaton Surgery n the U.S. s performed n two types of facltes: Hosptals Open 24 hours a day Patent recover n the hosptal Complex surgeres Ambulatory Surgery Centers Normally open 7am to 5pm Patents admtted and dscharged same day Lower cost and lower nfecton rate than hosptals 5

6 Surgery Process Patent Intake: admnstratve actvtes, pre-surgery exam, gownng, ste prep, anesthetc Surgery: ncson, one or multple procedures, pathology, closng Recovery: post anesthesa care unt (PACU), ICU, hosptal bed Intake Surgery Recovery 6

7 Mayo Clnc, Rochester MN The work I wll dscuss s motvated by problems at Mayo Clnc n Rochester, MN, U.S.: 7

8 Ambulatory Surgery Center A typcal ambulatory surgery center: Recovery rooms Patent watng area Operatng rooms Blue lnes represent patent flow 8

9 Management Decsons Decsons that can be supported wth operatons research models: Number of ORs and staff to actvate each day Surgery-to-OR assgnment decsons Schedulng of staff and patents n ntake, surgery, and recovery How to desgn the sute (ntake rooms, recovery rooms, ORs) 9

10 Complcatng Factors Many types of resources to be scheduled: surgery team, equpment, materals Hgh cost of resources and fxed tme to complete actvtes Large number of actvtes to be coordnated n a hghly constraned envronment Uncertanty n duraton of actvtes Many competng crtera 10

11 Surgery Duraton Uncertanty Tme for surgery s random Emprcal dstrbuton for tonslectomy: 11

12 Surgery Duraton Uncertanty Emprcal dstrbuton for abdomnal surgery to repar a herna: 12

13 Problem 1: Sngle Operatng Room Schedulng 13

14 Sngle OR Schedulng Problem Descrpton: Gven a sngle operatng room wth multple surgeres to be completed, what s the optmal amount of tme to allocate for each surgery to mnmze the cost of : Watng of patents and surgery teams to start surgery Unutlzed (dle) tme of the operatng room Overtme wth respect to a fxed length of day 14

15 Sngle OR Schedulng Planned OR Tme (e.g. 8 hours) x 1 x 2 x 3 x 4 x 5 Example Scenaro: Idlng Watng Overtme Goal: Mn{ Idlng + Watng + Overtme}

16 Stochastc Optmzaton Model Problem: Fnd the planned tme for surgeres to mnmze the obectve mn{ x Random surgery tme Cost of Watng n 1 W S L C w E Z [ W ] Cost of Idlng n 1 C s E Z [ S ] C max( W 1 Z 1 x 1,0) max( W 1 Z 1 x 1,0) max( Wn Zn x d,0) Cost of Overtme L E Z [ L]} Planned tme for surgery Fndng the expectaton of watng, dlng, and overtme s dffcult. 16

17 Lterature Revew Sngle Server Queung Analyss: Mercer (1960, 1973) Jansson (1966) Brahm and Worthngton (1991) Assumes steady state s reached, and arrvals are random Heurstcs: Whte and Pke (1964) Sorano (1966) Ho and Lau (1992) Does not guarantee optmal soluton Optmzaton: Wess (1990) 2 surgery news vendor model Wang (1993) Exploted phase type dstrbuton property Denton and Gupta (2003) General stochastc prorammng formulaton 17

18 Stochastc Lnear Program ]} [ mn{ 2 2 n n L s w Z l c s c w c E x Z s w w 1 1 n n n n x d Z g l s w 0,, 1,..., 0, 0,, 0 g l n s w x x Z s w s.t. 18 The problem can be reformulated as a two stage stochastc lnear program:

19 Two Stage Recourse Problem Intal Decson (x) Uncertanty Resolved Recourse (y) mn{ Q ( x) E [ Q Z ( x, Z)]} Q(x) Solve usng outer lnearzaton k k k k k Q ( x, Z ) mn{ c y T x W y h, y 0} T T T W 1 W 2 W 3 x T W K 19

20 Example Comparson of surgery allocatons for n=3, 5, 7 wth..d. dstrbutons wth unform dstrbuton, U(1,2): 2 X 1.5 µ Surgery 20

21 General Insghts We can draw the followng conclusons: Smple heurstcs often perform poorly The value of the stochastc soluton (VSS) s hgh Large nstances of ths problem can be solved very quckly usng decomposton 1) Denton, B.T., Gupta, D., 2003, A Sequental Boundng Approach for Optmal Appontment Schedulng, IIE Transactons, 35, ) Denton, B.T., Vapano, J, Vogl, A., 2007, Optmzaton of Surgery Seqencng and Schedulng Decsons Under Uncertanty, Health Care Management Scence, 10(1),

22 Problem 2: Multple Operatng Room Surgery Allocaton 22

23 Mult-Operatng Room Schedulng c Problem Descrpton: Gven a set of surgeres to be scheduled on a certan day decde the followng: How many operatng rooms to make avalable to complete all surgeres Whch operatng room to perform each surgery block (set of surgeres completed by a gven surgeon) 23

24 Mult-Operatng Room Schedulng Operatng rooms OR 1 OR 2 OR 3 OR m Assgnment decsons Surgeres S 1 S 2 S 3 S n Decsons: How many operatng rooms (ORs) to open each day? Whch OR to schedule each surgery block n? Performance Measures: Cost of operatng rooms opened Overtme costs for operatng rooms

25 Extensble Bn-Packng 1 x 0 f OR actve otherwse 1 y 0 f surgery Otherwse assgned to OR Z s. t. mn{ y m 1 n 1 y m 1 y p, x c x f x c 1,..., m, 1,..., n 1 1,..., n y o dx 1,..., m v bnary, o } o 0 Cost of ORs + Overtme Surgeres only scheduled n ORs that are actve Every surgery goes n one OR Overtme f surgery goes past end of day, T 25

26 Symmetry There are m! optmal solutons: OR 1 OR 2 OR 3 OR m Addng the followng ant-symmetry constrants reduces computaton tme: x x 1 2 x x 2 3 x m x m 1 OR Orderng y y 11 m y m y Surgery Assgnment 26

27 Two-Stage Stochastc MIP 0, ) ( {0,1},, ),, ( ) ( ) ( ) ( 1 ), (.. )]} ( [ mn{ ) ( n m m v f o x y dx o y Z y x y s t o E c x c Q x The real problem s stochastc due to random surgery duratons.

28 Integer L-Shaped Method Branch and bound tree: IP0 Ths problem can be solved usng decomposton IP4 IP2 IP3 IP1 Master Problem: Z mn{ s. t. y m 1 x c f x } (, ) IP6 IP5 m 1 y, y x 1 ( ) {0,1}, 0 IP8 IP7 (optmalty cuts) E [ ( h Tx)] 28

29 Heurstc and Bounds Dell Ollmo (1998) provdes a 13/12 approxmaton algorthm for bn packng wth extensble bns Heurstc: n LB; repeat; LPT( n); f n ( o end( repeat); 0, ) n 1; Start wth lower bound (LB) Stop; n 1 LB d(1 z f c ) v c d Sort surgeres from longest to shortest Sequentally apply surgeres to emptest room 29

30 Robust Formulaton 0 {0,1}, ) ( 1 ), (.. )}, ( mn{ 1 1 m m f x y y x y t s y x F x c Z ), ( y x F 1 ) :, (, }, max{0,.. } max{ 1 ):, ( 1 : 1 m y y v m y z z y z z z dx y c s t 30 Robust formulaton seeks to mnmze the worst case cost. Worst case (adversary) problem Uncertanty budget controls how conservatve the soluton s

31 Results for sample test problems based on Mayo Clnc surgery center: 15 surgery nstances Varable Cost = Varable Cost = Robust IP Robust IP Instance MV_IP LPT_Heu Tau=2 Tau=4 Tau=6 MV_IP LPT_Heu Tau=2 Tau=4 Tau= average stdev max mn MV_IP = soluton of mean value problem, LPT_Heu = applcaton of longest processng tme frst heurstc. Results expressed as the rato of optmal soluton to soluton generated by MV_IP, LPT_Heu, Robust IP

32 General Insghts A fast LPT based heurstc works well on a large number of nstances LPT works well when overtme costs are low LPT s better (and easer) than solvng MV problem n most cases Robust IP s better than LPT when overtme costs are hgh Denton, B.T., Mller, A., Balasubramanan, H., Huschka, T., 2010, Optmal Surgery Block Allocaton Under Uncertanty, Operatons Research 58(4), , 2010

33 Problem 3: Patent Arrval Schedulng 33

34 Patent Arrval Schedulng c Problem Descrpton: Fnd the effcent fronter of appontment tmes for patents havng a procedure n an ambulatory surgery center to trade off: Expected patent watng tme pror to surgery Expected length of day to complete all surgeres 34

35 Patent Arrvals Patent Check-n Watng Area Patent Dscharge Endoscopy Sute Endoscopy sute provdes mnmally nvasve procedures to screen for cancer: Patent Watng Tme Schedule Recovery Area Frst Patent Arrval Intake Area Length of Day Preoperatve Watng Area Operatng Rooms Last Patent Completon 35

36 Intake, Surgery, and Recovery Probablty dstrbutons for ntake (preparaton), surgery, and recovery from surgery, based on 1 year of data from Mayo Clnc: Mean tme for surgery about 20 mnutes Mean tme for recovery about 45 mnutes 36

37 Smulaton-optmzaton Decson varables: scheduled start tmes to be assgned to n patents each day Goal: Generate the effcent fronter of schedules to understand tradeoffs between patent watng and length of day Schedules generated usng a genetc algorthm (GA) Non-domnated sortng used to dentfy the Pareto set and feedback nto GA 37

38 Pareto Set The non-domnated sortng genetc algorthm (NSGA-II) of Deb et al.(2000) s used n the smulaton optmzaton. Rank 1 Solutons Rank 2 Solutons Rank 3 Solutons z 2 z 1 38

39 Selecton Procedure Sequental two stage ndfference zone rankng and selecton procedure of Rnott (1978) s used to compute the number of samples necessary to determne whether a soluton domnates Soluton domnates f: E[ W ] E[ W ] and E[ L ] E[ L ] 39

40 Genetc Algorthm Man features of the GA: Randomly generated ntal populaton of schedules Selecton based on 1) ranks and 2) crowdng dstance Sngle pont crossover: z 1 z 2 z 3.. z n z 1 z 2 - y 3.. y n y 1 y 2 y 3.. y n Parents y 1 y 2 - z 3.. z n Chldren Mutaton 40

41 Mean Average Sesson Length of Day Schedule Optmzaton Example of the progresson of the genetc algorthm GA Domnated Effcent Fronter Average Mean Watng Tme 41

42 General Insghts We drew the followng conclusons from our study: The smulaton optmzaton approach provdes sgnfcant mprovement to schedules used n practce Controllng the mx of dfferent types of surgery each day can sgnfcantly mprove both patent watng tme and overtme Gul, S., Denton, B.T., Fowler, J., 2011 B-Crtera Schedulng of Surgcal Servces for an Outpatent Procedure Center, Producton and Operatons Management, 20(3),

43 Other Research There are many opportuntes for future research. Followng are three bref examples of other related research: Schedulng when Operatng Rooms are treated as a shared resource among surgeons Dynamc (onlne) schedulng Schedulng of other parts of the hosptal 43

44 Poolng OR Capacty A more advanced model can be used to evaluate the beneft of surgeons sharng operatng rooms as a pooled resource Surgeon Surgeon OR 1 Surgeon OR 2 OR OR Overtme OR Turnover Tme Surgeon Idle Tme Surgeon Turnover Tme Batun, S., Denton, B.T., Huschka, T.R., Schaefer, A.J., The Beneft of Poolng Operatng Rooms Under Uncertanty, INFORMS Journal on Computng, 23(2),

45 Dynamc Schedulng In some healthcare envronments patent requests are made dynamcally and the number of patents s uncertan: Patents request appontments stochastcally Appontment decsons are made one at a tme p n L +1 p 1-p n U -1 p 1-p n U n U -1 1 n U Formulated as a multstage stochastc lnear program n L 1-p n L n L +1 Erdogan, S.A., Denton, B.T., Dynamc Appontment Schedulng wth Uncertan Demand, INFORMS Journal on Computng 25(1), ,

46 Cancer Center Schedulng Optmzaton models can be appled to other parts of the hosptal such as cancer centers: Lab Results sent to Clnc Phlebotomy Clnc Patent Arrves Regstraton Radology Bran Denton Infuson Center North Carolna State Unversty Drug Request sent to Infuson Emal: Pharmacy Center Webste: Pharmacy Lab Results sent to Infuson Center Mxed drug sent Woodall, J., Gosseln, T., Denton, B.T., A Smulaton Optmzaton Approach to Improve Patent Access n a Cancer Center, Workng Paper, Interfaces (n press) 46

47 Bran Denton Unversty of Mchgan Ann Arbor Mchgan, USA More detals about the papers dscussed can be found at my webste: 47

48 Thank You ありがとう Bran Denton North Carolna State Unversty Emal: Webste: 48