The Optimal Number of Hospital Beds Under Uncertainty: A Costs Management Approach

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1 Joural of Optimizatio i Idustrial Egieerig, Vol.11, Issue 2, Summer ad Autum 2018, DOI: /joie The Optimal Number of Hospital Beds Uder Ucertaity: A Costs Maagemet Approach Saeed Khalili a, Mohammad Ghodoosi b,*, Javad Hassapour c a Master's Graduate, Departmet of Idustrial Egieerig, Yazd Uiversity, Yazd, Ira. b Faculty Member, Departmet of Idustrial Egieerig, Torbat Heydarieh Uiversity, Torbat Heydarieh, Ira c Faculty Member, Departmet of Idustrial Egieerig, Qucha uiversity Advaced Techology Uiversity, Qucha, Ira Received 06 February 2016; Revised 08 March 2017; Accepted 20 November 2017 Abstract Equippig hospital beds uses a great deal of a hospital's resources. Therefore, it is essetial to cosider the hospital beds' efficiecy. To icrease its efficiecy, a fuzzy urestricted model for maagig hospital expeses is preseted i this paper. The lack of beds i hospitals leads to patiets admissio loss ad cosecutively profit loss. O the other had, icreasig the bed cout leads to a icrease i equipmet expeses. Therefore, i order to determie optimal bed capacity, it is of utmost importace to cosider these two costs simultaeously. I our paper, hospital admissio system is modeled with a multiserver queuig system (M/M/K). Therefore, to calculate the total cost fuctio, limitig probabilities of multiserver queueig model is used. Furthermore, due to ucertai ature of parameters, such as iterest rate ad hospitalizatio profit i various future time periods, these ucertaities are covered by fuzzy logic. Fially, to determie the optimal bed cout, Lee ad Li's fuzzy rakig method is used. This model is implemeted o a case study. Its goal is to determie the optimal bed cout for emergecy uit of Razi hospital i Torbat Heydarieh. Cosiderig the high capability of Markovia chais i modelig differet circumstaces ad the various queueig models, the proposed model ca be exteded for various hospital uits. Keywords : Optimal umber of hospital beds, Costs maagemet, queuig theory, Fuzzy rakig techiques. 1. Itroductio Hospital beds require large amout of ivestmet of a hospital s resources. Therefore, maagig hospital capacity ad appropriate productivity are vitally importat. Due to some disagreemet o the effectiveess of hospital beds i Ira, the study ad research is defied by the Health Miistry. Although it bega i 1995, i the absece of a appropriate method of evaluatio ad measuremet of the effectiveess of hospital beds, this project is eeded to develop a ew idea to determie the optimal umber of hospital beds. Icorrect ad iefficiet maagemet of hospitals has led to a lot of ecoomic pressure. Furthermore, it may cause a waste of resources, a icrease i patiet lists, or caceled surgeries, etc. I recet decades, the growth of the populatio has pushed up the demad for hospital services (Coile & Associatio, 2002). May hospitals do ot have eough resources ad space. This causes a icrease i the costs of the developmet of hospitals i the future (Coile & Associatio, 2002). Icreased receptio of patiets ad lack of hospitals lead to icreased capacity of hospitals. By icreasig the umber of patiets ad the legth of time, a patiet is hospitalized ad the eed for more capacity icreases, ad vice versa. This issue causes challege i determiig optimal capacity (Bazzoli et al., 2003). *Correspodig author address: m.ghodoosi@torbath.ac.ir 2. Literature Review Before studyig recet studies o determiig the optimal capacity of a hospital, it is ecessary to ote that some researchers have studied the capacity maagemet issue uder the ame of demad maagemet ad some have used demad ad capacity maagemet. However, the issue of demad maagemet ad capacity maagemet must be cosidered to be two differet cocepts. Demad maagemet tries to brig the patiets' or customers demad timig ad patiets or customers demad volume uder cotrol of a orgaizatio maagemet team, with the implemetatio of marketig strategies (Taylor, 1980). Capacity maagemet, however, guaratees that sufficiet capacity to meet ad supply the market demad exists (Klasse & Rohleder, 2002). Capacityrelated decisio makig is oe of the most importat strategic cocers for idustry s maagers, ad it affects the way a idustry respods to curret ad future demads (Getz, 1983). I the capacity evaluatio field, most of the previous studies have had a particular approach of their ow to capacity maagemet i differet idustries. For example, most of the researchers i tourism idustry have focused o the issue of capacity maagemet usig a reveue maagemet approach (Cradall & Marklad, 1996). However, the literature review o this field shows a shortage of comprehesive aalysis i the capacity 129

2 Saeed Khalili et al. /The Optimal Number of Hospital Beds maagemet field for differet idustries (Pullma & Rodgers, 2010). It is otable that there is more research i the servicerelated fields i compariso to the other fields. This is because i most servicerelated fields, the service capacity has promiet effect o customer satisfactio ad level of service offered to them. Cosequetly, optimal capacity determiatio has bee proposed i areas such as hospitals, parkig areas, restaurats, hotels, ad residetial ceters. Furthermore, research ito the determiatio of optimal capacity has bee doe i the maufacturig sector. The models that are most widely kow correspod to warehousig. For example, (White & Fracis, 1971) studied the optimal warehouse capacity problem with both the determiistic ad probabilistic demad coditios ad proposed a solutio method based o the liear programmig, duality theory, ad etwork flow. They also cosidered the warehouse costructio costs, trasportatio costs, ad storage costs i a public warehouse. I the restaurat maagemet domai, the research of Hwag ad colleagues (Hwag et al., 2010) must be oted. Usig queuig models, they modeled a system for a local restaurat. The model was desiged i order to maximize profit ad icrease customer satisfactio. A study was also doe to determie the optimal capacity of hotel. By usig the ivetory model, (Gu, 2003) tried to aalyze ad optimize the capacity of a hotel ad casio i Las Vegas. He proved both of the properties of sigleperiod model (i.e., a perishable item ad the probabilistic demad used by the hotel idustry), ad used this for the optimizatio of the capacity of hotels. Accordig to his statemet, if the empty ad ready rooms are assumed as a product of hotel, the the rooms that do ot accept the customers are the perishable products. I the hospital issue, due to the high cost of preparig ad equippig the operatig rooms, may studies have bee performed to determie the optimum umber of operatig rooms. Differet approaches are preseted i this study to determie the optimal capacity that ca be simulated by discrete ad stochastic simulatio methods, as well as a combiatio of queuig theory or other methods (Hershey et al., 1981; Kao & Tug, 1981; Kokagul, 2008; McMaus et al., 2004; Mile & Whitty, 1995). For example, (Kokagul, 2008) proposed the combiatio of determiistic ad stochastic models for optimizig the capacity of sigle hospital beds, for which the umber of admissios ad hospitalizatio legth of each patiet is a radom ad upredictable process. The key parameters are preseted accordig to the level of service, the level of employmet, ad the rate of admissios. By usig data obtaied through simulatio, a oliear mathematical model is preseted to determie the optimal capacity. (Be Bachouch et al., 2012) used liear programmig for hospital bed plaig. I this research, two types of patiets were cosidered: elective ad acute cases. They took ito accout several costraits: icompatibility betwee pathologies, o mixedsex rooms, cotiuity of care, etc. (Lapierre et al., 1999) used a mathematical model to develop time series models by usig cesus data o a hourly basis (umber of patiets i a care uit at a give time) i order to calculate the size of each uit. This study used simulatio i order to miimize the amout of collected data. By represetig the time cesus as a time series, models produce a frequecy distributio to show the bed demad. (Walczak et al., 2003) used eural etworks to predict the legth of the stay of a patiet arrived at emergecy i order to optimize bed plaig. (Garg et al., 2010) developed a ohomogeeous discrete time Markov chai with timedepedet covariates i order to model the patiet flow i a cost or capacitycostraied healthcare system. I aother research, (Gog et al., 2010) used a multiobjective learig particle swarm optimizatio for the bed allocatio problem i hospitals i geeral. (D. K. Lee & Zeios, 2009) developed a semiclosed migratio etwork to illustrate patiet flow ito the cliic ad to illustrate the flow betwee the cliic ad the hospital. (L. Li & Beto, 2003) used structural equatio modelig, takig ito accout several static factors such as hospital size ad hospital locatio i order to address the hospital capacity maagemet issue. Several maagerial isights emerged from the research: Hospital cost cotrol is affected by techology ad workforce; The hospital s quality relates to the quality of the equipmet ad workforce chose; Staff traiig, staff competece, ad career growth ifluece the decisios of maagemet; Locatio, size ad service mix have sigificat impact o hospital s capacity related decisios. Some studies used queuig models for solvig ad optimizig the idustry ad the healthcare problem. (Cochra & Roche, 2009) icreased the capacity of a emergecy departmet based o a queuig model. I this model, ohomogeeous arrival patters, oexpoetial service time distributios, ad multiple patiet types are assumed. I aother study, (X. Li et al., 2008) used a multiobjective decisio model based o queuig theory ad goal programmig was developed for the allocatio of beds i a hospital. I aother research, by itegratig queuig theory ad compartmetal models of flow, (Goruescu et al., 2002) tried to demostrate how chagig admissio rates, legth of stay ad bed allocatio ifluece bed occupacy, emptiess ad rejectio i departmets of geriatric medicie. A review of capacity maagemet studies idicated that i this area, the studies are very poor. The limited research available is used for specific coditios ad the slightest chages i assumptios lose the theories effectiveess. Therefore, i this study, a ucostraied model is 130

3 Joural of Optimizatio i Idustrial Egieerig, Vol.11, Issue 2, Summer ad Autum 2018, provided, usig queue theory to determie the optimal capacity of the hospital. I the proposed approach, there are several cotributios which make the study uique ad distict from other studies. These cotributios lead to superior results i hospital capacity maagemet domai. The highlights of our cotributios are: Urestricted fuzzy oliear model. This model is used i additio to fuzzy rakig. Together, this approach reduces executio time. Furthermore, compared to the correspodig restricted fuzzy oliear model, this approach is uderstood more easily by humas. The use of Markovia chais ad queueig theory. The high flexibility of Markov chais i modelig would lead to a suitable model for various hospitals ad coditios. Therefore, the resultig model will have a high level of flexibility. Itegratio of fuzzy logic ad queuig theory. This itegratio leads to the coverage of ucertai coditios which is a iheret coditio of servicig eviromets, especially hospitals. A ovel cost fuctio with the cosideratio of time value of moey. 3. The Proposed Model I this sectio, the proposed queuig model is preseted to determie optimal hospital capacity. Sice the proposed model is based o queuig model cocepts, it is ecessary to adapt the hospital admissio system to a queuig system. I this adaptatio: Hospital patiets are customers i the queuig system with a arrival rate of Hospital beds are servers i this queuig system, ad thus the umber of the hospital beds is the same as that of the servers of the queue system. The average duratio of patiets' hospitalizatio time i hospital equals the iverse of the expected service time of the queuig system ad is idicated by The average umber of occupied beds i a hospital equals the expected umber of customers i the queuig model which is represeted by Model assumptios ad symbols. The time betwee the patiets' arrivals ad the duratio of patiets' hospitalizatio time i the hospital follows a expoetial distributio. : The capacity of the hospital (i.e., the umber of the beds i hospital). : The probability of patiets hospitalized i the hospital i a log period (i.e., the percetage of the time i which the hospital has occupied beds). : The rate of patiets' arrivals to hospital. : The service rate to patiets i the hospital. : The et profit of patiets hospitalizatio i each bed of hospital for oe ight. b: The ecessary budget for equippig ad icreasig the capacity by a sigle bed for the hospital (i.e. the cost of allocatig ad buildig the ecessary space for a hospital bed ad the cost of hospital equipmet for the bed) i: Iterest rate N: The umber of periods i the plaig horizo. Accordig to the features of the hospital, the receptio system, ad the cosidered assumptio i the proposed model, the queuig model (M/M/m) is the best model for the hospital system. Sice the hospital beds are cosidered for the servers of the simulated queuig model of the hospital system, the queuig system used for the hospital is the model (M/M/k). The aim is to determie optimal umber of hospital beds ( ). For this result, two differet costs have bee combied ito a objective fuctio: The first cost is caused by excess capacity ( ). It meas that if a hospital is costructed with excess capacity, this capacity would be left uused most of the time. Therefore, by costructig too big a hospital, excessive cost would be imposed o the ivestors ad a hospital with such a high capacity would ot be efficiet. This cost would be referred to as Cost of Excess Capacity ad is obtaied from equatio 2. K j K j. j. b (2) j The secod cost is caused by the costructio of a hospital with capacity less tha optimal capacity ( ). If a hospital with too small a capacity is costructed, the the hospital would be at full capacity most of the time ad the hospital would ot be able to meet the demads of patiets. This cost would be referred to as Cost of Capacity Shortages ad is obtaied i equatio 3... (1) 131

4 Saeed Khalili et al. /The Optimal Number of Hospital Beds K j 1 K j. j. Pj (3) By chagig hospital capacity, these two costs move i opposite directios to oe aother. Cost of Excess Capacity icreases ad Cost of Capacity Shortages decreases by icreasig capacity, ad vice versa (Figure 1). Fig. 1. The relatio betwee costs ad capacity The total Cost of costructig a hospital with capacity k is equal to the sum of the Cost of Excess Capacity ad the Cost of Capacity Shortages. The cost of costructig additioal capacity would be imposed o ivestors for a sigle time: that beig the costructio of the hospital. The cost of the capacity shortage, however, is icurred daily, over the life of the hospital. To be able to compare ad combie these two costs, the cost of excess capacity should be distributed alog time by cosiderig time value of moey. To get this result, the cost of excess capacity is multiplied by CapitalRecovery Factor (CRF). N N CRF ( A/ Pi,, N) [( i.(1 i) )/((1 i) 1)] π ( 4) Values of were obtaied usig queuig model of (M/M/ k ) (Gelebe et al., 1987). 0! k 0k k! ; ; k k (5) ( k 1) k ( 1)! k (6) k! k (0) [1 ] I the log term, i ad P are chageable ad ot predictable; therefore, a exact estimate of their values is ot possible. As a result, these values are etered ito the model with the help of fuzzy triagular estimates based o experts' opiios ad experiece at had. Subscripts L, M, ad R idicate the most pessimistic, the most probable, ad the most optimistic values of the correspodig fuzzy triagular parameter, respectively. 132 i i, i, i P P, P, P L M R j L M R Therefore, i ucertai coditios ad by cosiderig the fuzzy triagular umber calculatios, the mathematical model is exteded as i equatios (711). N i. 1 i K MiC T N. K.. b K.. P 1 i 1 0 K (7) (8) (9) (10) (11) This cost fuctio ca be used to determie the optimal umber of beds i differet wards of the hospital operatig uder differet coditios. To get this result, firstly, a appropriate queuig system must be defied with the cosideratio of the specific coditios of each departmet. Afterwards, should be calculated for the defied system with the use Markov chais ad equilibrium equatios. Fially, the value of the total cost fuctio ( ) for various hospital capacities (k) should be calculated ad the case that has miimum total cost would be the optimal hospital capacity Rakig fuzzy umbers..,, Due to fuzzy total cost fuctio i the previous sectio, a fuzzy rakig techique is required to determie the miimum value of cost fuctio ad the optimal hospital capacity. There are umerous fuzzy rakig techiques (Asady, 2010; Asady & Zedeham, 2007; Cheg, 1998; Chu & Tsao, 2002; Nejad & Mashichi, 2011; Wag & Lee, 2008). I this article, Lee ad Li s techique (E. S. Lee & Li, 1988) is used. I this approach, two criteria are used, amely mea ad stadard deviatio. Fuzzy

5 Joural of Optimizatio i Idustrial Egieerig, Vol.11, Issue 2, Summer ad Autum 2018, umber s mea ad stadard deviatio are obtaied from equatios 11 ad 12. (12) (13) Give triagular fuzzy umbers, their mea ad stadard deviatio are obtaied from equatios 13 ad Case Study (14) (15) Razi hospital is located ext to the passeger termial i the city of Torbat Heydarieh. The hospital bega its activities o Jauary, The hospital area is 14 thousad square meters. The hospital is operated by 250 medical, admiistrative, ad service staff. The hospital has ipatiet, outpatiet, ad paracliic departmets. Each hospital departmet icludes a variety of cliics ad hospital uits. Emergecy uit is a subset of the hospital outpatiet departmet. Table 1 Rakig fuzzy umbers Compariso of mea values ( ) ( ) The prioritizatio rule for rakig two fuzzy umbers M ad M is depicted i table 1. Compariso of sd values Prioritizatio result ( ) ( ) ( ) ( ) Sice Torbat Heydarieh is close to Mashhad, Torbat Mashhad road is the most importat route for Mashhad travelers. I recet years, icreases i passeger traffic have caused a icrease i accidets, ad cosequetly icrease patiet referrals to the emergecy uit of the Razi hospital. Therefore, i September 2013, capacity icrease of the emergecy uit of the hospital was approved. I late 2014, the hospital bega research projects i collaboratio with the Uiversity of Torbat Heydarieh i order to determie the optimal capacity of emergecy uit. A brief summary of the research project is reported i this paper Numerical results for emergecy uit of Razi hospital Suppose that patiets arrive at the emergecy room accordig to a Poisso process with a average of 22 patiets per day. The legths of stay of the patiets o hospital beds i the emergecy departmet have the expoetial radom variable with a mea of 0.5 days. Assume that the cost of creatio ad icrease of capacity by a sigle bed i the emergecy departmet is 1,500 currecy (currecy to be cosidered); that reveue from the hospitalizatio per patiet per day is approximately 50 currecy ad i the 120moth plaig horizo, a mothly iterest rate of approximately 2% is assumed. What is the optimal umber of hospital beds available for allocatio to the emergecy uit of the hospital? Accordig to the expert opiios, imprecise parameters are displayed as i the followig triagular fuzzy umbers. = (1.5%, 2%, 4%) = (40, 50, 55) 4.2. Solutio Usig the code writte i MATLAB, the calculatios of fuzzy values for the proposed cost fuctio for differet capacities ad their results are summarized i Table 2. It is worth otig that i the MATLAB code, the secod summatio s upper boud must be substituted by a large umber rather tha by ifiity. This large umber must be chose such that the additio of probabilities equals almost oe. I this example, we have chose a upper boud equal to 100. π 0/9998 Table 2. Fuzzy values of the cost fuctio K 0 ( , , ) 1 ( , , ) 2 ( , , ) 3 ( , , ).. 97 ( , , ) 98 ( , , ) 99 ( , , ) 100 ( , , ) By usig the code writte i MATLAB software for Lee ad Li method, fuzzy umbers for the cost fuctio ad capacities rage from the lowest to the highest umbers are reported i table 3. It is evidet from Table 3 that by costructig a hospital with a capacity of 18, the total cost would be at the lowest level possible, hece this capacity is chose as optimal. If, for ay reaso, the metioed capacity could ot be costructed, 19, 17, ad more would be the ext most preferable optios. 133

6 Saeed Khalili et al. /The Optimal Number of Hospital Beds It is worth otig that the amout of parameters i, μ, λ, b, p was ifluetial i determiig the best capacity. Therefore, a sesitivity aalysis of iput parameters is eeded to idetify sesitive ad isesitive parameters. It is obvious that more attetio should be paid to estimate the sesitive parameters Table 3. Prioritizatio of differet capacities based o fuzzy cost fuctio Priorities based o cost K 1 ( , , ) 2 ( , , ) 3 ( , , ).. 99 ( , , ) 100 ( , , ) 101 ( , , ) Table 4. Sesitivity aalysis of parameter i percetage chage i the parameter i Prioritize top 20 scearios The amout of chage 0% +10% +20% +30% +40% +50% low Table 5. Sesitivity aalysis of parameter percetage chage i the parameter Prioritize top 20 scearios The amout of chage 0% +10% Medium +20% Medium +30% +40% +50% 134

7 Joural of Optimizatio i Idustrial Egieerig, Vol.11, Issue 2, Summer ad Autum 2018, Table 6 Sesitivity aalysis of parameter µ percetage chage i the parameter µ Prioritize top 20 scearios The amout of chage 0% +10% +20% +30% +40% Low Medium +50% Table 7 Sesitivity aalysis of parameter b percetage chage i the parameter b Prioritize top 20 scearios The amout of chage 0% +10% +20% +30% +40% low +50% low Table 8. Sesitivity aalysis of parameter p percetage chage i the parameter p Prioritize top 20 scearios The amout of chage 0% +10% +20% +30% +40% +50% Low Medium Medium 135

8 Saeed Khalili et al. /The Optimal Number of Hospital Beds 4.3. Sesitivity aalysis Tables 48 show the MATLAB output results of the sesitivity aalysis for each iput parameters of the problem. I these calculatios, each parameter is icreased by 10% i each step separately. Afterwards, priority chages of the top 20 scearios were studied. Aalyzig the results i Table 4 idicates that by icreasig, the fial rakig of the scearios does ot chage substatially. Accordig to Table 4, this parameter is ot a sesitive parameter of the model because oly a chage greater tha 50 percet causes the fial rakig to chage. The results of the sesitivity aalysis for parameter λ i Table 5 idicate that this parameter is a sesitive parameter of the model, ad it has a sigificat effect o the fial rakig of the scearios. As could be expected, icreasig λ causes the model to select more capacity for the emergecy uit because by icreasig λ, the rate of patiets' arrivals to hospital is icreased so that more beds are eeded to deal with the situatio. Care should be take while estimatig λ, because this parameter is a sesitive parameter of the model. Table 6 idicates the results of the sesitivity aalysis for parameter μ. Icreasig μ meas duratio of hospitalizatio (1/μ) is reduced. Logically, by reducig the duratio of hospitalizatio, fewer beds are eeded. Accordig to Table 6, this parameter is a sesitive parameter of the model. Table 7 idicates the results of the sesitivity aalysis for parameter (b). This parameter affects the amouts of cost of excess capacity. Also, cost of excess capacity affects the fial cost cosiderig the formula (1). Therefore, parameter (b) affects the fial cost of the hospital idirectly. But, accordig to Table 7, this parameter is a isesitive parameter of the model. Sesitivity aalysis of parameter p i Table 8 idicates that although this parameter could ot be assumed as sesitive as λ, but accurate estimatio of this parameter is importat (i compariso with i, b), because it has caused drastic chages i the fial rakig i some cases. Also, icreasig this parameter resulted i higher priority for those scearios with more assiged umber of beds i the fial rakig. 5. Coclusio ad Further Studies I this paper, the wellkow problem of determiig the optimal capacity of hospital was addressed. For this purpose, the system of hospitalizatio has bee modeled followig a queuig system. With cosideratio give to the costs of capacity shortage ad the possibility of excess capacity, a urestricted model for the problem was developed. I order to adapt the models to real coditios, some of the parameters of the problem were cosidered as fuzzy umbers. Cosiderig the metioed matters ad the example, it ca be cocluded that to obtai the optimal hospital capacity usig queuig theory cocepts, oe must determie a queuig model that matches the hospital coditios ad calculate accordig to the queuig model. Various values are placed i the cost fuctio. The cost fuctio value is calculated for various hospital capacities ad the capacity that allocates the miimum value of cost fuctio to itself is chose as the optimal hospital capacity. I the metioed case study, m/m/k model is used to model hospital s emergecy uit. I accordace to results of table 3, 18 hospital beds are required i order to miimize the hospital s total costs. Furthermore, tables 4 through 8 show that λ, μ, ad p are sesitive parameters, ad it is importat to estimate them carefully. The proposed approach i this article is applicable to ay queuig model. Cosiderig the flexibility of Markovia chais ad the various queuig models, the approach makes it possible to model various hospital uits with differet coditios. For example, i some hospital uits such as operatig rooms, patiets' referece to beds or duratio of hospitalizatio may ot follow the fuctios of Poisso ad expoetial. I these cases, geeral fuctios ca be used, such as (G / M / 1), (G / G / m) or ( G / G /1). Furthermore, there are more complex hospital models whose model does ot match the queuig model completely. For such cases, it is suggested that a combiatio of these models ad simulatios be used. I the real world, arrival rate ad hospitalizatio rate are ot certai. By cosiderig them as fuzzy ad usig fuzzy probabilities, values ca be calculated ad etered ito the model. Ackowledgmet The authors are greatly grateful to the Uiversity of Torbat Heydaieh for their assistace i this study. Refereces Asady, B. (2010). The revised method of rakig LR fuzzy umber based o deviatio degree. 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10 Saeed Khalili et al. /The Optimal Number of Hospital Beds White, J. A, & Fracis, Richard L. (1971). Normative models for some warehouse sizig problems. AIIE Trasactios, 3(3), This article ca be cited: Khalili, S., Ghodoosi, M., & Hassapour, J. (2018). The Optimal Number of Hospital Beds Uder Ucertaity: A Costs Maagemet Approach. joural of Optimizatio i Idustrial Egieerig. 11(2), 2018, URL: DOI: /joie