Optimal Allocation of Cashiers and Pharmacists in Large Hospitals: A Point-wise, Fluid-based Dynamic Queueing Network Approach

Transcription

1 Otimal Alloation of Cashiers an Pharmaists in Large Hositals: A Point-wise, Flui-base Dynami Queueing Network Aroah Chung-Cheng Lu Deartment of Transortation an Logistis Management National Chiao Tung University, Hsinhu, Taiwan Tel: (+886) , jasonlu@gmail.om Hui-Ju Chen Deartment of Inustrial Engineering an Management National Taiei University of Tehnology (Taiei Teh), Taiei, Taiwan Tel: (+886) ext. 2340, vanillasoy@gmail.om Kuo-Ching Ying Deartment of Inustrial Engineering an Management National Taiei University of Tehnology (Taiei Teh), Taiei, Taiwan Tel: (+886) ext. 2308, kying@ntut.eu.tw Tsung-Chi Yu Grauate Institute of Information Management National Taiei University of Tehnology (Taiei Teh), Taiei, Taiwan Tel: (+886) ext. 2340, eavi400@hotmail.om Abstrat. This aer resents a bi-objetive otimization moel for etermining the otimal time-varying numbers of ashiers an harmaists in a large hosital. The two objetives of this moel are to minimize the waiting ost inurre by atients an the oerating osts inurre by the hosital. A oint-wise flui-base aroximation aroah is aote to onstrut a ynami queueing network that takes into aount time-varying (or non-stationary) arrivals of atients an esribes time-varying queue lengths. The ynami queueing network is then enasulate in the otimization moel that etermines otimal time-varying numbers of ashiers an harmaists. A test roblem instane is esigne base on a large hosital in the ity of Taiei, an the MINOS solver of GAMS is alie to solve the roblem instane. Numerial results show that the otimization moel an rovie an otimal alloation of manower that signifiantly reues both waiting an oerating osts. Keywors: Queueing theory, Mathematial rogramming, Healthare management, Manower alloation 1. INTRODUCTION In Taiwan, after the National Health Insurane system was imlemente in 1995, hositals sought to reue oerational osts an inrease their revenues. Serviing a larger number of atients is one wao inrease suh revenues. Moreover, inreasing satisfation among atients beomes an imortant onsieration for hosital managers. Most hositals aot various inies to evaluate their servie quality, one of them being atient waiting time, whih greatly affets atient satisfation. Aart from outatients waiting times for iagnoses from otors, the waiting times or queue lengths at ashier an harmay ounters have long been a major onern for hosital managers an their atients, eseially uring rush hour when extene wait times generate numerous omlaints. Although inreasing the number of ashiers an harmaists is a straightforwar way of reuing atients wait times, this aroah inevitably raises hositals oerating osts an may be ineffiient uring

2 off-eak hours. Instea, alloating aroriate numbers of ashiers an harmaists at ifferent time erios uring a given ao strike a balane between atient wait times an hositals oerating osts is essential for hosital managers to maximize their revenues, while maintaining the level of servie esire. Most hosital managers alloate manower at ashier ounters an isensaries uring eak an off-eak hours aoring to their ast exerienes. However, this manual aroah may not be aequate for etermining the otimal number of ashers an harmaists in resonse to time-varying emans. Very few revious stuies in this area have eveloe otimization moels for manower alloation in hositals, although some have aresse the otimization roblems in alloating bes to inatients from ifferent ivisions, an sheuling aointments with otors, nurses, an harmaists. In a stuy onute by Creemers et al. (2012), an otimization moel for assigning a set of reefine outatient time slots to otors from ifferent ivisions was resente in orer to minimize the total exete weighte waiting time of atients. Xu an Liu (2011) eveloe an otimization moel to solve the be arrangement roblem in one hosital, where a atient s riority is etermine bhe severity of his/her illness in orer to reue the amount of time bes sen unouie (in other wors, to inrease the utilization of bes). Some ast stuies have aime to reue the waiting times of atients, for instane, for ashiers (Chao, 2010), harmaies (Afolabi an Erhun, 2003; Nukwe et al., 2011), onsulting rooms (Jerbi an Kamoun, 2009; Palvannan an Teow, 2010), an bes (Joustra et al., 2009; Hu et al., 2011). To the authors knowlege, none of the revious researh has eveloe otimization moels for otimal manower alloation for ashiers an harmaists. The lak of a systemati aroah for alloating otimal time-varying numbers of ashiers an harmaists motivates us to evelo the otimization moel we resent here. This aer resents a bi-objetive otimization moel to etermine otimal manower alloation at the ashier ounter, isensary, an harmay at ifferent time erios of a given ay in a large hosital. The main eision variables are the time-varying numbers of ashiers an harmaists an the two objetives are to minimize the total atient wait time an the total oerating ost of the hosital. To effetivelake into aount time-varying or non-stationary arrival rates of atients an esribe queue lengths in suh a ynami system, the oint-wise flui-base aroximation metho is aote to onstrut a ynami queueing network. Then, the bi-objetive otimization moel is formulate base on the ynami queueing network. We alhe weighte sum metho (Zaeh, 1963; Hwang an Masu, 1979) to aress the bi-objetive otimization moel, where the two objetives are transforme into a single objetive by assigning weight arameters. In this transformation, the single objetive is to minimize the weighte waiting osts of atients an oerational osts of the hosital. The otimal solution obtaine bhe transforme metho is a Pareto otimal solution of the original bi-objetive otimization roblem when a set of weight arameters is given. Reaers an refer to Miettinen (1999) an Steuer (1989) for information on multi-objetive rogramming. For further isussion about the hoie of weights, lease see Marler an Arora (2004).

3 Disensary : resrition flow : atient flow v t Presritions are generate Cashier S t Pharmay Figure 1: The ynami queueing network onstrute for the manower alloation roblem in this stuy 2. PROBLEM DESCRIPTION Tyially, in a large hosital in Taiwan, otors will generate an transfer resritions to the isensary after examining atients, where harmaists fill resritions that will be elivere to the harmay for atients to ik u. In the meantime, atients leave onsulting rooms an wait for ashiers to aheir bill before iking u their resritions aoring to their assigne serial number at the harmay ounter. This system roess an be esribe as a ynami queueing network, as shown in Figure 1. The queueing system onsists of three waiting oints (Cashier, Pharmay, an Disensary) an two flow subsystems (atient flow an resrition flow). The uer half of this system reresente by otte-line ars is the resrition-flow subsystem, while the lower half of this system reresente by soli-line ars is the atient-flow subsystem. In the atient-flow subsystem, the ar oming from the left to the Cashier enotes the arrival of atients at the Cashier at time t (λ t ). The ar going from the Cashier enotes the servie amount at Cashier at time t (ν t ), whih equals the arrival amount of atients at the Pharmay at time t (λ t ). Patients will enter the queue at the Pharmay after they have been servie (i.e., aheir bill) at the Cashier. The ar leaving the Pharmay enotes the servie amount at the Pharmay at time t (ν t ) (i.e., the atients who have reeive their fille resritions an leave the hosital). In the resrition-flow subsystem, the ar oming from the left to the Disensary enotes the arrival of resritions at the Disensary at time t (λ t ). The ar going from the Disensary enotes the servie amount

4 at the Disensary at time t (ν t ). Sine the maximum servie aaity at the Pharmay at time t is etermine bhe servie amount at the Disensary at time t, this ar also enotes the maximum servie aaity of the Pharmay at time t (S t ) (i.e., the fille resritions whih will be elivere to the Pharmay). We an make the following assumtions as we evelo the otimization moel: The time-varying arrival rates at eah servie faility, system servie rates, emloyees salaries er time unit, an atients waiting osts er time unit are given. The time sent by atients on walking from onsulting rooms to the Cashier an from the Cashier to the Pharmay are ignore for simliity an without loss of generality. The time sent on elivering resritions from the Disensaro the Pharmay is ignore for simliity an without loss of generality. Presritions onsiere in this stuy involve ommon or hroni iseases rugs. Drugs for rare iseases rugs are not inlue. 3. POINT-WISE FLUID-BASED APPROXIMATION APPROACH AND OPTIMIZATION MODELS A oint-wise flui-base aroximation aroah is aote to onstrut the ynami queueing network, whih is enasulate in the roose otimization moel for etermining otimal time-varying numbers of ashiers an harmaists in a large hosital. The notations use to resent the otimization moel are as follows. Parameters: λ t λ t The atient arrival rate at the Cashier at time t The resrition arrival rate at the Disensary at time t λ t ρ t ρ t C wait W wait The atient arrival rate at the Pharmay at time t The aaity utilization ratio at the Cashier at time t The aaity utilization ratio at the Disensary at time t The waiting ost of atients(ollars/time) Weight for the waiting ost of atients T servie The servie rate at the Cashier for eah time T servie The servie rate er unit time at the T servie Disensary The servie rate er unit time at the Pharmay C servie The unit-time servie ost of a ounter at the Cashier (ollars/time) C servie The unit-time servie ost of a harmaist C servie W servie I T at the Disensary (ollars/time) The unit-time servie ost of a ounter at the Pharmay (ollars/time) Weight for the servie ost of the hosital The length of eah time unit (minutes) The number of time units y max The uer boun of the number of ounters at the Cashier y max The uer boun of the number of y max harmaists at the Disensary The uer boun of the number of the ounters at the Pharmay Enogenous Variables: x t x t x t ν t ν t The queue length of atients at the Cashier at time t The queue length of resritions at the Disensary at time t The queue length of atients at the Pharmay at time t The servie amount at the Cashier at time t The servie amount at the Disensary at

5 ν t S t S t time t The servie amount at the Pharmay at time t Deision Variables: The servie aaity at Cashier at time t The servie aaity at Disensary at time t The number of Cashier ounters oen at time t (manower alloation) The number of harmaists available in Disensary at time t (manower alloation) The number of Pharmay ounters oen at time t (manower alloation) The multi-objetive otimization moel for the M/M/1 manower alloation roblem: T Minimize I C wait ( t=1 x t + T t=1 x t ), I Subjet to T t=1 + C servie T (C servie C servie t=1 + T t=1 ) (1) x t+1 = x t + λ t ν t, t, (2) x t+1 = x t + λ t ν t, t, (3) x t+1 = x t + λ t ν t, t, (4) ν t S t ρ t = 0, t, (5) ν t S t ρ t = 0, t, (6) ν t = ν t = T servie t, (7) S t = T servie y t, t, (8) S t = T servie y t, t, (9) ρ t = x t (x t + 1), t, (10) ρ t = x t (x t + 1), t, (11) 1 y t y max, t = 1, 2,, T, (12) x t 0, t = 1, 2,, T, (13) 1 y t y max, t = 1, 2,, T, (14) x t 0, t = 1, 2,, T. (15) 1 y t y max, t = 1, 2,, T, (16) x t 0, t = 1, 2,, T, (17) The objetive funtion (1) is to minimize the waiting ost for atients at the Cashier an Pharmay (the first term) an the oerational ost of the hosital at the Cashier, Disensary, an Pharmay (the seon term). Note that the queue length at the Disensary is not taken into aount sine atients o not wait for resritions at the Disensary. Constraints (2)-(11) are establishe bhe oint-wise flui-base aroximation aroah. Seifially, onstraints (2)-(4) are the flow onservation equations at the Cashier, Disensary, an Pharmay, resetively. Constraints (5)-(7) etermine the servie amounts at the Cashier, Disensary, an Pharmay, resetively. Constraints (8) an (9) etermine the maximum servie aaities at the Cashier an Disensary, resetively. Constraints (10) an (11) are the aaity utilization ratios at the Cashier an Disensary, resetively, for the M/M/1 queueing system. Constraints (12), (14) an (16) resent the minimum an maximum manower requirements at the Cashier, Disensary, an Pharmay, resetively, set bhe hosital, where the lower bouns all orreson to 1 an the uer bouns orreson to y max, y max an y max, resetively. Constraints (13), (15) an (17) require nonnegative queue lengths. By alying the weighte-sum metho to solve the bi-objetive otimization roblem, we assign two weights (W wait ) an (W servie ) to the waiting ost an oerational ost in the objetive funtion, resetively. The objetive funtion (1) an be transforme into a single-objetive with weight arameters as follows: T I [W wait C wait ( t=1 x t + T t=1 x t ) + W servie (C servie T t=1 + C servie T C servie t=1 + T t=1 )], (18) where the transforme objetive (18) is the total weighte ost of this system. In aition, we an establish the otimization moel for M/G/1 queueing system. In this moel,

6 instea of (10) an (11), aaity utilization ratio funtion onstraints with a general servie time istribution (M/G/1 queue) are (19) an (20) as: (Cashier) ρ t = x t +1 (x t ) 2 +2 (Cs ) 2 x t +1 1 (C s ) 2, t, (19) (Disensary)ρ t = x t +1 (x t ) 2 +2 (Cs ) 2 x t +1 1 (C s ) 2, t. (20) The objetive funtion an the other onstraints are the same as those for the M/M/1 queueing system. 4. NUMERICAL EXPERIMENTS 4.1 Data rearation To examine the roose otimization moel, we generate a roblem instane base on real ata rovie by a large hosital (M) in Taiei. The atient arrival ata were ollete bhe hosital between Deember 2013 an January 2014, inluing 43 weekays (i.e., every Monao Friay). Arrival ata were ollete every 10 minutes from 7AM to 11PM on eah of those ays. A time interval (I) orresons to 60 minutes. The servie rate T servie is 60 (ersons/time). The uer boun of the number of Cashier ounters (y max ) is 10. Base on an emloyee s monthly salary, the servie ost at the Cashier ounter an be omute as 4 ollars/minute er erson. Then C servie is 240 ollars/time unit er erson. The servie rate T servie is The uer boun of the number of harmaists y max is 17. The servie ost at the Disensary is 7 ollars/minute er erson. This imlies that C servie is 420 ollars/time er erson. Thus, the servie rate T servie is 120 (ersons/time). The uer boun of the number of Pharmay ounters y max is 8. The servie ost at the Pharmay is 7 ollars/minute er erson, or C servie is 420 ollars/time er erson. Consier that the average salary er month is 45,888 ollars in the year 2014 rovie by Diretorate-General of Buget, Aounting an Statistis, Exeutive Yuan R.O.C. (Taiwan); the atients waiting ost ( C wait ) is 5 ollars/minute er erson. Here, we assign the same weight for the atients waiting ost an the hosital s oerational ost. In other wors, W wait an W servie are both set to be 0.5. The MINOS solver of GAMS was alie to solve the otimization moel in Setion 3. The numerial exeriments were imlemente on a omuter with Core i GHz CPU, 4.00GB Ram, equie with Mirosoft Winows 7. With the above arameters, the CPU time for GAMS to solve the roblem instane was seons. 4.2 Numerial Results We omare the otimal solution of the moel with hosital M s urrent manower alloation. The otimal numbers of ashiers o not signifiantly iffer from those of hosital M s urrent alloation. At the Disensary, the otimal numbers of harmaists are less than those of the hosital s urrent alloation in the morning an afternoon; while the otimal number is more than that of the hosital s urrent alloation in the evening. In aition, the otimal number of Pharmay ounters is less than those of the hosital s urrent alloation in the morning an afternoon; while in the evening, they are almost the same. The otimal manower obtaine bhe moel an reue the queue length at the Cashier from (ersons) to 6.05 (ersons). The numerial results show that at the Pharmay, the otimal solution an reue the queue length from (ersons) to 0 (ersons), eseially in the evening. Furthermore, the servie ost orresoning to the Cashier ereases by 2, ollars with the otimal solution, omare to the hosital s urrent alloation. The waiting ost ereases by 26, ollars. At the Pharmay, the servie ost ereases by 10,824 ollars while the waiting ost ereases by 290,631.9 ollars. The total servie ost ereases from 105, ollars to 90, ollars if the otimal solution is use as

7 oose to the urrent alloation. That is, the hosital an save 15, ollars, or 15% of its urrent oerating ost. The total waiting ost an be reue from 34, ollars to 29,060.7 ollars (or 91.6%) using the otimal solution. Therefore, the otimal solution obtaine bhe moel an signifiantly reue the oerating ost of the hosital as well as inrease servie quality. 5. CONCLUSION AND FUTURE RESEARCH This aer aots the oint-wise flui-base aroximation aroah to onstrut the ynami queueing network, whih is then enasulate in the roose bi-objetive otimization moel for etermining the otimal number of ashiers an harmaists in a large hosital. The numerial results of the test instane generate base on ata for a large hosital in Taiei show that the roose otimization moel an rovie the otimal alloation of manower whih signifiantly reues waiting an oerating osts. In sum, the hosital an save about 15% of its oerating ost an erease 91.6% of the waiting ost er ay. In future stuies, the roose ynami queueing network moeling aroah an be alie to etermine the otimal manower alloation in several other areas, suh as hek-in ounters in airorts an restaurants as well as servie ounters in banks. REFERENCES Afolabi, M.O. an Erhun, W.O. (2003) Patients' Resonse to Waiting Time in An Out-atient. Troial Journal of Pharmaeutial Researh, 2, Asmussen, S.R. (2003) Ranom Walks. Alie Probability an Queues, in the series of Stohasti Moeling an Alie Probability, 51, Chao, Y. (2010) Outatient Queue Business Simulation Base on Aetable Waiting Time. International Conferene On Comuter Design An Aliations, 1, Chen, X., Zhou, X., an List, G.F. (2011) Using Time-varying Tolls to Otimize Truk Arrivals at Ports. Transortation Researh Part E: Logistis an Transortation Review, 47, Creemers, S., Beliën, J., an Lambreht, M. (2012) The Otimal Alloation of Server Time Slots over Different Classes of Patients. Euroean Journal of Oerational Researh, 219, Grassmann, W.K. an Tavakoli, J. (2009) Transient Solutions for Multi-server Queues with Finite Buffers. Queueing Systems, 62, Green, L. an Kolesar, P. (1991) The Pointwise Stationary Aroximation for Queues with Nonstationary Arrivals. Management Siene, 37, Hu, R. an Chang, C. (2011) Using Mathematial Moeling to Rationally Arrange Hosital Bes. International Conferene on Eletri Information an Control Engineering (ICEICE), Wuhan, China, Hu, R., Hu, J., an Wu, Y. (2012) A New Queueing Algorithm of Ohthalmi Hosital Bes Arrangement. Avane Materials Researh, , Hwang, C.L. an Masu, A. (1979) Multile Objetive Deision Making, Methos an Aliations: A State-of-the-art Survey. Sringer-Verlag. Jerbi, B. an Kamoun, H. (2009) Multiobjetive Stuy to Imlement Outatient Sointment System at Hei Chaker Hosital. Simulation Moelling Pratie an Theory, 19, Jomon, A.P. an Li, L. (2012) Moels for Imroving Patient Throughut an Waiting at Hosital Emergeny Deartments. The Journal of Emergeny Meiine, 43, Joustra, P., Sluis, E., an Dijk, N.M. (2010) To Pool or Not to Pool in Hositals: A Theoretial an Pratial Comarison for A Raiotheray Outatient Deartment. Annals of Oerations Researh, 178, Marler, R.T. an Arora, J.S. (2004) Survey of Multi-objetive Otimization Methos for Engineering.Strutural an Multiisilinary Otimization, 26, Miettinen, K. (1999) Nonlinear Multiobjetive Otimization. International Series in Oerations Researh & Management Siene, 12, Kluwer Aaemi Publishers. Nukwe, H.C., Omale, S., an Oanuga, O.O. (2011) Reuing Queues in A Nigerian Hosital Pharmayl. Afrian Journal of Pharmay an Pharmaology, 5, Palvannan, R.K. an Teow, K.L. (2010) Queueing for Healthare. Journal of Meial Systems, 36, Steuer, R.E. (1989) Multile Criteria Otimization: Theory, Comutation, an Aliation. Krieger, Malabar. Wang, W.P., Tier, D., an Banerjee, S. (1996) A Simle Aroximation for Moeling Nonstationary Queues. In Networking the Next Generation Proeeings of the Fifteenth Annual Joint

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