The Impact of Flexibility And Capacity Allocation On The Performance of Primary Care Practices

Transcription

1 University of Massachusetts Amherst Amherst Masters Theses February The Imact of Flexibility And Caacity Allocation On The Performance of Primary Care Practices Liang Wang University of Massachusetts Amherst Follow this and additional works at: htts://scholarworks.umass.edu/theses Part of the Management Sciences and Quantitative Methods Commons, and the Oerations Research, Systems Engineering and Industrial Engineering Commons Wang, Liang, "The Imact of Flexibility And Caacity Allocation On The Performance of Primary Care Practices" (2010). Masters Theses February Retrieved from htts://scholarworks.umass.edu/theses/486 This thesis is brought to you for free and oen access by ScholarWorks@UMass Amherst. It has been acceted for inclusion in Masters Theses February 2014 by an authorized administrator of ScholarWorks@UMass Amherst. For more information, lease contact scholarworks@library.umass.edu.

2 THE IMPACT OF FLEXIBILITY AND CAPACITY ALLOCATION ON THE PERFORMANCE OF PRIMARY CARE PRACTICES A Master's Thesis Presented by LIANG WANG Submitted to the Graduate School of the University of Massachusetts Amherst in artial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN INDUSTRIAL ENGINEERING AND OPERATION RESEARCH Setember 2010 Deartment of Mechanical and Industrial Engineering

3 THE IMPACT OF FLEXIBILITY AND CAPACITY ALLOCATION ON THE PERFORMANCE OF PRIMARY CARE PRACTICES A Master's Thesis Presented by LIANG WANG Aroved as to style and content by: Hari Balasubramanian, Co-chair Ana Muriel, Co-chair Senay Solak, Member Donald L. Fisher, Deartment Head Deartment of Mechanical and Industrial Engineering

4 ACKNOWLEDGEMENTS I sincerely thank my advisors, Dr. Hari J. Balasubramanian and Dr. Ana Muriel, for their countless suort throughout my study and numerously insightful discussions during my work on this thesis. I also thank Dr. Senay Solak for the comments and time he devoted to imrove the quality of my work. I want to thank my arents, without their continuous suort and unlimited love, this work would not have been ossible. I also want to give my thanks to my girlfriend, Linli Zhang, who is graduated from University of Massachusetts Amherst and now is attending the MBA rogram in University of Texas Austin. For many years, she devotes her love to encourage me to overcome the challenges imeding in the way and strive for the excellence, I am deely grateful for all she did for me. iii

5 ABSTRACT THE IMPACT OF FLEXIBILITY AND CAPACITY ALLOCATION ON THE PERFORMANCE OF PRIMARY CARE PRACTICES SEPTEMBER 2010 LIANG WANG B.S., HUAZHONG UNIVERSITY OF SCIENCE AND TECHNOLOGY M.S., UNIVERSITY OF MASSACHUSETTS AMHERST Directed by: Professor Dr. Hari Balasubramanian and Professor Dr. Ana Muriel The two imortant metrics for any rimary care ractice are: (1) Timely Access and (2) Patient-hysician Continuity. Timely access focuses on the ability of a atient to get access to a hysician as soon as ossible. Patient-hysician continuity refers to building a strong or ermanent relationshi between a atient and a secific hysician by maximizing atient visits to that hysician. In the ast decade, a new aradigm called advanced access or oen access has been adoted by ractices nationwide to encourage hysician to do today s work today. However, most clinics still reserve re-scheduled aointments for long lead-time aointments due to atient reference and clinical necessities. Therefore, an imortant roblem for clinics is how to otimally manage and allocate limited hysician caacities as much as ossible to meet the two tyes of demand re-scheduled (non-urgent) and oen access (urgent) while simultaneously maximizing timely access and atient-hysician continuity. In this study we use a quantitative aroach to aly the ideas of manufacturing rocess flexibility to caacity management in a rimary care ractice. We develo a closed form exression for iv

6 caacity allocation for an individual hysician and a two hysician ractice. In the case of multile hysicians, we use a two-stage stochastic integer rogramming aroach to investigate the value of flexibility under different levels of flexibility and rovide the otimal caacity allocation solution for each hysician. We find that flexibility has the greatest benefit when system utilization is balanced and when the individual hysicians have unequal utilizations. The benefits of flexibility also increase as the ractice gets larger. v

7 TABLE OF CONTENTS Page ACKNOWLEDGEMENTS... iii ABSTRACT... iv LIST OF TABLES... viii LIST OF FIGURES... x CHAPTER 1 INTRODUCTION Background on rimary care Current rimary care ractices Team care and hysician flexibility Caacity allocation between re-scheduling and oen access LITERATURE REVIEW Quantitative models for rimary care ractice Research related to flexibility MODELING APPROACH Assumtions Model formulation Formulation for dedicated flexibility Formulation for two hysicians with full flexibility Formulation for general configuration VALUE OF FLEXIBILITY Practice without any flexibility vi

8 4.2 Two hysicians with oen access flexibility Value of flexibility in a ractice with three hysicians Results for three hysicians with symmetric demand distributions * Ni of three hysicians with symmetric demand distributions Results for three hysicians with asymmetric demand distributions * Ni of three hysicians with asymmetric demand distributions Trends in the total * Ni values for all three hysicians Value of flexibility in a ractice with six hysicians Results for six hysicians with symmetric demand distributions Results for six hysicians with asymmetric demand distributions Conclusion IMPLICATIONS FOR PRACTICE CONCLUSIONS FUTURE WORK APPENDICES A. THEOREMS PROOF B. PROGRAMS FOR THE STUDY OF FLEXIBILITY BIBLIOGRAPHY vii

9 LIST OF TABLES Table Page 4.1 Assumtions for 3 hysicians with symmetric demand distributions in Symmetric Case 1 (10/14) Statistics of objective value for different flexibilities with 100% utilization in Symmetric Case Statistics of total demands for 3 hysicians with 100% utilization in Symmetric Case Measurement for different flexibilities in term of system revenue in Symmetric Case 1 (10/14) Measurement for different flexibilities in term of timely access rate in Symmetric Case 1 (10/14) Measurement for different flexibilities in term of continuity rate in Symmetric Case 1 (10/14) Measurement for different flexibilities in term of system revenue in Symmetric Case 2 (14/10) Measurement for different flexibilities in term of timely access rate in Symmetric Case 2 (14/10) Measurement for different flexibilities in term of continuity rate in Symmetric Case 2 (14/10) Measurement for different flexibilities in term of system revenue in Symmetric Case 3 (6/18) Measurement for different flexibilities in term of timely access rate in Symmetric Case 3 (6/18) Measurement for different flexibilities in term of continuity rate in Symmetric Case 3 (6/18) Measurement for different flexibilities in term of system revenue in Symmetric Case 4 (18/6) Measurement for different flexibilities in term of timely access rate in Symmetric Case 4 (18/6) viii

10 4.15 Measurement for different flexibilities in term of continuity rate in Symmetric Case 4 (18/6) Assumtions for 3 hysicians with asymmetric demand distributions in Asymmetric Case Measurements of system revenue with asymmetric demands in Asymmetric Case Measurements of timely access rate with asymmetric demands in Asymmetric Case Measurements of continuity rate with asymmetric demands in Asymmetric Case Assumtions for 3 hysicians with asymmetric demand distributions in Asymmetric Case Measurements of system revenue with asymmetric demands in Asymmetric Case Measurements of timely access rate with asymmetric demands in Asymmetric Case Measurements of continuity rate with asymmetric demands in Asymmetric Case Assumtions for 6 hysicians with symmetric demand distributions Measurement of system revenue for 6 hysicians (symmetric) Measurement of timely access rate for 6 hysicians (symmetric) Measurement of continuity rate for 6 hysicians (symmetric) Assumtions for 6 hysicians with asymmetric demand distributions Measurement of system revenue for 6 hysicians (asymmetric) Measurement of timely access rate for 6 hysicians (asymmetric) Measurement of continuity rate for 6 hysicians (asymmetric) ix

11 LIST OF FIGURES Figure Page 3.1 System configuration for dedicated flexibility System configuration for two hysicians sharing oen access demands System configuration for artial and full flexibility Dedicated case with demand rates 10 and 14 for re-scheduling and oen access resectively. And a closer view of the value near the otimal oint Dedicated case with demand rates 16 and 8 for re-scheduling and oen access resectively. And a closer view of the value near the otimal oint Two hysicians have flexibility in oen access ractice where N1 = 19 and N2 = Two hysicians have flexibility in oen access ractice where N1 = 16, N2 = Box-Whisker Plot comarison of objective values for different flexibilities with 100% utilization in Symmetric Case Comarisons of different flexibilities in term of system revenue in Symmetric Case 1 (10/14) Comarisons of different flexibilities in term of timely access rate in Symmetric Case 1 (10/14) Comarisons of different flexibilities in term of continuity rate in Symmetric Case 1 (10/14) An examle of diversion rocess in 2-chain and full flexibility chain flexibility imrovement under different demand ratios for all symmetric cases Full flexibility imrovement under different demand ratios for all symmetric cases Distributions of the differences of Ns between flexible configurations and dedicated case when the system is 40% utilized in Symmetric Case 3 (6/18) x

12 4.13 Distributions of the differences of Ns between flexible configurations and dedicated case when the system is 80% utilized in Symmetric Case 3 (6/18) Distributions of the differences of Ns between flexible configurations and dedicated case when the system is 100% utilized in Symmetric Case 3 (6/18) Distributions of the differences of Ns between flexible configurations and dedicated case when the system is 120% utilized in Symmetric Case 3 (6/18) Distributions of the differences of Ns between flexible configurations and dedicated case when the system is 160% utilized and in Symmetric Case 3 (6/18) Distributions of the differences of Ns between flexible configurations and dedicated case when the system is 40% utilized in Symmetric Case 2 (14/10) Distributions of the differences of Ns between flexible configurations and dedicated case when the system is 80% utilized in Symmetric Case 2 (14/10) Distributions of the differences of Ns between flexible configurations and dedicated case when the system is 100% utilized in Symmetric Case 2 (14/10) Distributions of the differences of Ns between flexible configurations and dedicated case when the system is 120% utilized in Symmetric Case 2 (14/10) Distributions of the differences of Ns between flexible configurations and dedicated case when the system is 160% utilized in Symmetric Case 2 (14/10) System revenue comarison between Asymmetric Case 1 and 2 for flexible configurations Timely access comarison between Asymmetric Case 1 and 2 for flexible configurations Continuity comarison between Asymmetric Case 1 and 2 for flexible configurations Distributions of the differences of Ns between flexible configurations and dedicated case when the system is 40% utilized in Asymmetric Case Distributions of the differences of Ns between flexible configurations and dedicated case when the system is 80% utilized in Asymmetric Case Distributions of the differences of Ns between flexible configurations and dedicated case when the system is 100% utilized in Asymmetric Case xi

13 4.28 Distributions of the differences of Ns between flexible configurations and dedicated case when the system is 120% utilized in Asymmetric Case Distributions of the differences of Ns between flexible configurations and dedicated case when the system is 160% utilized in Asymmetric Case Average Ns values for three hysicians with asymmetric demand distributions Comarison of system revenue imrovement between 3 and 6 hysicians Comarison of timely access imrovement between 3 and 6 hysicians Comarison of continuity imrovement between 3 and 6 hysicians xii

14 CHAPTER 1 INTRODUCTION The US healthcare system, by all accounts, is in a state of crisis and cannot be alleviated without fundamental change and reform. With exenditures of about $2.2 trillion, or 16.2% of the GDP [1], the US healthcare system ranks the second among the members of the World Health Organization (WHO) and ranks at the to among industrialized countries [2]. This exenditure is exected to increase continuously to around 20% of the GDP in less than a decade [1, 3]. One might think that, given this immense sending, health outcomes would imrove corresondingly. However, the current situation is that about million Americans lack health insurance. Most of them believe the insurance is too exensive to afford. The WHO ranks the US as 37 th in overall system erformance and 72 nd among the 192 member states in terms of overall level of health [4]. A solution to the current crisis in healthcare requires a multi-ronged effort involving multile asects of the healthcare system. Healthcare olicy makers agree that one of the key areas that needs to be addressed is rimary care. The World Health Reort 2008 [5] is aroriately titled Primary Health Care Now More Than Ever. 1.1 Background on rimary care Primary care roviders (PCP) form the backbone of most modern health care systems and are tyically the first oint of contact between atients and systems. They manage a atient s general health issues and rovide reventive medicine, atient education and routine hysical exams, In addition, they review a atient s medical history and take care 1

15 of referrals to medical secialists for secondary and tertiary care. 94% of atients value their PCP as a source of first contact care and aroximately 90% are satisfied with their coordinated referrals [6]. The imortant benefits of an effective rimary care system are well documented in the clinical literature. For instance, Starfield, Shi and Macinko (2005), among others [7, 8, 9], show that imroving rimary care generates several romising results: Imroves access to health services for relatively derived oulation grous. Assist in the revention and early management of health roblems due to education and early detection. Builds stronger relationshis between atients and their PCP and reduces the amount of wasteful exenditures by minimizing inaroriate referrals to secondary and tertiary care roviders. Desite its ivotal role in the overall system, rimary care is at grave risk due to a dysfunctional financing and delivery system [7]. A study by the American College of Physicians (2006) oints out the current dilemma faced by the rimary care: the demand for healthcare grows steadily and dramatically with an estimated growth rate of 38% from 2000 to 2020, yet the number of students secializing in rimary care kees declining due to lower salaries combined with higher workloads [7, 10]. This imbalanced situation involving increasing demand and shortage of suly leads to worse quality of care, longer waiting times, and increased dissatisfactions, all of which aggravate the crisis in the healthcare system. To imrove rimary care ractices and overcome the roblems that are imeding the healthcare system from erforming otimally, two imortant metrics are introduced: (1) 2

16 Timely Access and (2) Patient-hysician Continuity. These are two of the six recommended aims by the Institute of Medicine (2001) [11]. Timely Access focuses on the ability of a atient to get access to care as soon as ossible. Not getting timely aointments lowers atient satisfaction and increases the likelihood of sending the atients to the Emergency Room (ER) more frequently [12, 13]. The inability to get a timely aointment esecially hinders the aroriate management of chronic diseases that could have been effectively treated in a rimary care ractice. Patient-hysician continuity refers to building a strong or ermanent relationshi between a atient and a secific hysician so that the atient can see his/her own PCP as much as ossible. Continuity is considered as one of the hallmarks of rimary care. Gill and Mainous (1998) oint to several studies which show that atients who regularly see their own PCP are (1) more satisfied with their care; (2) more likely to take medications correctly; (3) more likely to have roblems correctly detected; and (4) less likely to be hositalized [14]. Continuity is more imortant for atients with a comlex medical history and chronic roblems since they can be treated more aroriately by their own hysicians who are familiar with their conditions. From the hysician s ersective, continuity is also beneficial since workloads are more focused. 1.2 Current rimary care ractices Various tyes of rimary care ractices currently exist in the U.S., for examle, those consisting of family hysicians, general internists and ediatricians. Though many of them are conducted by one single hysician, more than 65% of rimary care ractices are grou ractices consisting of more than one hysician [15]. To establish the connection 3

17 between atient and hysician, each hysician has a anel, which is the set of atients he/she is resonsible for. The hysician takes aointments from his/her resective anel and only treat atients from other anels in excetional cases. Physician aointments are usually scheduled into 15- or 20-minute slots. Reimbursement to hysicians in rimary care is based largely on 20-minute visits, and a full-time hysician tyically has 24 aointments in a working day based on eight hours. Broadly seaking, aointments for rimary care can be classified into two tyes: (1) Non-urgent or re-scheduled aointments and (2) Urgent or acute aointments. Nonurgent aointments come from atients with chronic conditions who need regular treatments, and atients requiring annual exams or a first time assessment. Urgent aointments are demands that come in on a daily basis from atients requiring immediate attention from their PCPs. If their own hysicians are unavailable at the walkin time, atients have to get their care at an emergency room. In traditional ractices of aointment scheduling, urgent aointments received higher riority and were scheduled as soon as ossible, while non-urgent requests were usually ostoned u to several weeks or even months. To address the issue of long backlogs and intolerant waiting times, a new aradigm called advanced access or oen access has been adoted by ractices nationwide [16]. Under oen access, all atients, regardless of urgent or non-urgent status, are given same-day aointments with their own hysician who are encouraged to do today s work today. The key of a successful imlementation of oen access is to balance the demand and suly aroriately, which means anels should be sized roerly and hysicians might work overtime occasionally 4

18 [17]. In common ractice, oen access schemes are imlemented simultaneously with traditional re-scheduling methods in most clinics. 1.3 Team care and hysician flexibility Another aroach to overcome deficiencies in rimary care ractices is to allow the concet of team care to lay a central role to imrove quality of care, something which is recommended by the Institute of Medicine (2001) in its reort Crossing the Quality Chasm: New Health System for the Twenty First Century [18]. Team care brings with it the idea of hysician flexibility, which imlies that atients will not only be seen by their dedicated hysician, but also by suort staff or other hysicians in the team. This actually haens routinely in ractice without any installation or secial configuration. While, the flexibility of allowing a hysician to see atients from any of other hysician anels might imrove timely access, hysician flexibility can be detrimental to continuity and increase the chances of misdiagnosis. One question that arises naturally is: what is the maximum level of flexibility that will still rovide an accetable level of continuity given two different demand streams? The levels of flexibility that will be comared and investigated in this thesis are shown in Figure 1.1. Figure 1.1 Different flexibility configurations that tradeoff continuity with timeliness. 5

19 In (a), atients may see any other hysician (full flexibility). This configuration leads to the highest level of timely access as resources are ooled, but may not ensure continuity. In (b), atients can only see their own dedicated hysician (no flexibility), which leads to the highest level of continuity, although timely access might not be guaranteed. Combing these two levels leads to configuration (c) artial flexibility, where atients and hysicians are chained such that each atient in addition to having his/her own hysician, also has one auxiliary hysician (AP). Having laid out the main issues, below we examine in more detail how the inherent flexibility of rimary care hysicians can be best managed, at different levels of the lanning hierarchy, to imrove timely access and continuity. 1.4 Caacity allocation between re-scheduling and oen access Though oen access has been successfully imlemented and adoted in rimary care ractices, most clinics still reserve re-scheduled aointments for long lead-time aointments due to atient reference and clinical necessities. Therefore, the most urgent roblem becomes finding how to otimally manage and allocate limited hysician caacities as much as ossible to meet the two tyes of demand re-scheduling and oen access. Qu and Shi (2009) roosed a two-level hysician caacities management scheme which combines the high level total caacity of the clinic and low level caacity of individual hysician care to find the otimal caacity allocation method for current oen access clinics with one hysician, or a hysician team that has caacities ooled [19]. We will use an alternative aroach to find the best allocation scheme for multile 6

20 hysicians and investigate the otimal allocation method for rimary care ractices with different levels of flexibility. 7

21 CHAPTER 2 LITERATURE REVIEW 2.1 Quantitative models for rimary care ractice The alication of otimization aroaches to rimary care is limited, yet growing. With the advent of advanced access roosed by Murray and Tantau [20, 21], research focusing on caacity lanning and allocating in rimary care is booming. For instance, Green et al. (2007) [17] develo a simle robability model to investigate the number of overtime aointments that a hysician could be exected to engage as a function of his/her anel size. To offset the effect of variability, they conclude that hysician caacity should be sufficiently higher than atient demand. Using a queuing model, Green and Savin (2008) determine the effect of no-shows on a hysician s anel size. This queuing model demonstrates an ability to estimate the relationshi between a hysician s backlog and his/her anel size, as well as atient no-show rates. Qu et al (2007) [22] develo an exression for the otimal number of slots that should be reserved for re-scheduled aointments in a day for a single hysician ractice. They find the otimal solution deends on the no-show rates of re-scheduled demand and oen access demand, as well as the distribution of oen access demand. In chater 3, we rovide a simler aroach for the same quantity, which in turn leads to more comlex and yet unexlored two hysician ractices. Koach et al (2007) [23] use discrete event simulation in an oen access scheduling environment to analyze the effects of clinical characteristics on continuity of care and clinic erformance. One rimary conclusion relevant to this research is that continuity of care is adversely affected as the fraction of 8

22 atients on oen access increases. They also roose that hysician team ractice would be the solution to the roblem. Guta and Wang (2008) [24] develo a model to establish aointment booking olicies that can maximize a clinic's revenue. They use a Markov decision rocess (MDP) that exlicitly accounts for atient references with resect to secific aointment times and multile hysicians, and also for different tyes of demand: re-scheduling and oen access. The main differences between their research and ours are: 1) In their aroach, the booking of re-scheduled aointments is driven by atient reference; by contrast, we try to balance re-scheduled demand and same-day demand. 2) The same-day demand in their model arrives before the beginning of the day and can be treated as deterministic information, while we focus on more dynamic behavior and rovide otimal bound for the atient flow management. 2.2 Research related to flexibility Lots of research investigating the benefits of flexibility has focused mainly on the manufacturing, but more recently has extended to include the service system and worker training and allocation. Jordan and Graves (1995) [25] have studied the imrovements arising from using a flexibility configuration in sales and caacity utilization in multiroduct and lant networks. They were the first to comare the benefits of artial flexibility to full flexibility in the field of assembly lines, and they concluded that artial flexibility (chaining), delivers almost the same benefits of a fully flexible system, yet needs only a small fraction of links and costs. Graves and Tomlin (2003) [26] extend this research to multi-stage suly chains and to a make-to-order environment where 9

23 flexibility is also used to hedge against variability (Muriel et al. (2006) [27]). Brusco and Johns (1998) [28] find that the benefits of artial flexibility decrease with additional cost. Similarly, Chou et al (2008) [29] distinguish between range (the different scenarios a system can adat to) and resonse (the cost of using additional flexibility links) and show that imroving resonse outerforms imroving range. This conclusion suggests that in rimary care ractice, the benefits of limiting the number of hysicians that can see a atient is likely to outweigh the higher range rovided by a fully flexible ractice where any hysician can see the atient. Flexible queuing systems have been studied by Sheikhzadeh et. al. (1998) [30] using a similar chaining configuration. They comare full flexibility, or "ooling", with a 2-chain configuration, i.e., one where two neighboring queues are linked to each server and two neighboring servers are connected to each queue. They find that the chained system works almost as well as the fully flexible system if the assumtion of homogenous demand and service rate holds. The analysis is generalized in Gurumurthi and Benjaafar (2004) [31] to flexible queuing systems with general customer and service flexibility under Poisson-distributed demand and service rates. They show that the otimal allocation deends on the characteristics of the demand and articular olicy imlemented. As in the case of cross-training in serial roduction lines (Ho et al. (2004) [32]), flexibility has been found to be beneficial when imlementing (1) caacity balancing, or balancing the exected workload among hysicians. In this case flexibility will allow the load to be shared among hysicians, which imroves overall timely access and hysician utilization; and (2) variability buffering, which refers to a flexibility configuration that 10

24 accommodates to variability in atient demand. They used a MDP to comare different strategies of cross-training and found that configurations arallel to chaining have the otential to be robust and efficient methods [32]. Though extensive studies have been conducted in manufacturing flexibility and its more recent alication to other areas, there are, however, key oerational differences that make the alication of flexibility to rimary care more comlex and worthy of further analysis, as we exlore in the next chater. 11

25 CHAPTER 3 MODELING APPROACH 3.1 Assumtions As we model a ractice that imlements a re-scheduled aointment aradigm and an oen access scheme at the same time, we assume that the daily caacity for each hysician is the same and known in advance. In rimary care ractice, each aointment usually takes 20 minutes and ractitioners are aid by the number of 20 minute aointments. Since each hysician normally works eight hours a day, this leads to a caacity of 24 slots er hysician er day. We assume that the demands of re-scheduled and oen access aointments in ractice are indeendent of each other, and for each hysician, demands for rescheduled aointments and oen access aointments are also indeendent. Further, we assume that demand distributions of re-scheduled aointments and oen access aointments are known (can be estimated by historical records) and belong to the Poisson distribution. The oen access aradigm increases the timely access effect that leads to much lower atient no show rates [33]. To include the no-show effects in our model, we treat the actual show-u rate as a revenue associated with each accessing aradigm. Thus we consider the revenue associated with meeting one oen access demand to be higher than that of satisfying one re-scheduled aointment. To investigate the value of flexibility in rimary care ractice, we configure a system with three different flexibilities: full flexibility, artial flexibility ( 2-chain ) and no flexibility (dedicated). To encourage continuity, we assume that seeing a atient from 12

26 another hysician's anel will generate a slightly less revenue for a hysician comared to satisfying a demand from his/her own anel. 3.2 Model formulation We model the roblem as a stochastic integer rogramming roblem with stationary robability distribution and contribution (i.e. revenue). Below we show the notation for the dedicated cases (i.e. no flexibility) and for a scenario of 2 hysicians with full flexibility. The notation is as follows: N : Caacity of each hysician. M : Number of hysicians and therefore anel. We index hysicians with i [1.. M ]. C : Cost of missing one re-scheduled demand. C o : Cost of missing one oen access demand. N i : Number of slots allocated for re-scheduled demand of hysician i. d i : Demand for re-scheduled aointments of hysician i. o d i : Demand for oen access aointments of hysician i. i ( ) : Probability mass function of re-scheduled demand for hysician i. qi ( ) : Probability mass function of oen access demand for hysician i. Fi ( ) : Cumulative distribution function of re-scheduled demand for hysician i. Φi ( ) : Cumulative distribution function of oen access demand for hysician i. EC ( ) : Exected cost of missing re-scheduled demand for hysician i. i EC ( ) : Exected cost of missing oen access demand for hysician i. o i 13

27 ECi ( ) : Total exected cost of missing demands for hysician i. The notation for a general formulation (i.e. more than 2 hysicians with any configuration of flexibility) will be demonstrated in the subsequent sections Formulation for dedicated flexibility An individual hysician without any flexibility is defined as one who can only serve the atients from his/her own anel. The system configuration is shown below in Figure 3.1: Figure 3.1 System configuration for dedicated flexibility. Ni For each number of slots that are allocated for a given re-scheduled demand { 0,1,2,..., N} hysician is:, the exected cost of missing the re-scheduled demand for each i = i i i i di = Ni + 1 EC C ( d N ) ( d ) (3.2.1) and the exected cost of missing a given oen access demand for each hysician is: 14

28 N o o i i i o i i i i o di = 0 di = N di + 1 i ( ) ( ) ( ) EC = d C d N d q d + o o ( ) ( ) ( ) 1 F N C d N N q d i i o i i i i o di = N Ni + 1 (3.2.2) The total exected cost of missing demands for the anel of hysician i is equal to the sum of equation (3.2.1) and (3.2.2). Our objective is to find the otimal number of slots reserved for re-scheduled aointments * N i that minimizes the total exected cost of missing demands for hysician i. For the dedicated flexibility configuration, we can use theorem 1 to find * N i : Theorem 1. For the dedicated case, the otimal number of slots allocated for rescheduled aointments of each individual hysician does not deend on the distribution of the re-scheduled demand but relies on the total caacity N, the costs scale C / C and the inverse cumulative distribution function of his/her own oen access demand, secifically: The roof is shown in the aendix. C * 1 Ni = N Фi (1 ) (3.2.3) C o o Formulation for two hysicians with full flexibility In a fully flexible ractice, atients can be seen by any available hysician. We divide the case of two hysicians with full flexibility into two scenarios: (1) the hysicians also have full flexibility in re-scheduled aointments; (2) the hysicians only have full flexibility in oen access aointments. 15

29 For hysicians that have both full flexibility in re-scheduled and oen access ractices, the otimal value of * Ni can be determined by theorem 2: Theorem 2. For a system that has both full flexibility in re-scheduled and oen access ractices, the otimal value of each * N i should satisfy: M i= 1 ( ) N = M N Φ 1 C / C * 1 i o, where Φ 1 ( ) is the inverse cumulative distribution function where the mean rate equals to the sum of each individual oen access demand mean rate. With full flexibility in the re-scheduled and oen access ractice, both the demand and caacity of M hysicians can be aggregated roortionally. This means that we can use a single system, with aggregated caacity and demand, to substitute for the case of multile hysicians, and the otimal value of Further, considering each hysician individually, the number of that is no larger than N, but the sum of these indicated in theorem 2. * N i can be obtained from equation (3.2.3). * N i can be any value * N i should be always equal to the value Figure 3.2 System configuration for two hysicians sharing oen access demands. For the scenario where re-scheduled atients see their own hysician, but the timesensitive oen access atients can be seen by more than one hysician (the system 16

30 configuration is shown in figure 3.2), we use the following theorem to determine the otimal values of N and * 1 N : * 2 Theorem 3. The otimal number of aointment slots for each hysician i to make available to re-scheduled atients in a two-hysician artially flexible ractice, where the two hysicians share oen access demands, is the smallest integers N 1 and N 2 that satisfy: C C o [1 F ( N )] [1 Φ(2N N N 1)] + N2 d2 = ( d ) [1 Φ(2N N d 1)] (3.2.4) and C C o [1 F ( N )] [1 Φ(2N N N 1)] N1 1 ( d1 ) N N2 d1 d1 = 0 [1 Φ(2 1)] (3.2.5) where Φ( ) is the cumulative distribution function where the mean rate equals to the sum of each individual oen access demand mean rate. If both hysicians have the same distribution of re-scheduled demand (symmetric), then the otimal numbers of N and * 1 N are the same and equal to the smallest integer * 2 N such that: C C o [1 F ( N )] [1 Φ(2N 2N 1)] + N di = 0 i ( d ) [1 Φ(2N N d 1)] i i i (3.2.6) where i can be any one of the two hysicians. 17

31 The roof can be found in the aendix. Observe that distribution of re-scheduled demand for hysician i. * N i does not deend on the Formulation for general configuration We investigate a rimary care ractice involving more than two hysicians with full flexibility, artial (2-chain), and no flexibility using a stochastic integer rogramming aroach. The system configuration is demonstrated in figure 3.3. Figure 3.3 System configuration for artial and full flexibility. Let A be the set of all ossible links (i, j) such that atients in anel i can be served by hysician j, atients, and Ri is the revenue associated with hysician i seeing one of his re-scheduled o R ij is the revenue associated with hysician j seeing an oen-access atient of anel i. Let U be the uer bound of the realization of re-scheduled demand d is and 18

32 oen access demand o d is for scenario s, for instance, U = 50, which means d {0,1,2,.,50} and d o {0,1, 2,., 50}.We introduce the following variables: is is φ = 1 if iu is d is < N, otherwise φ = 0. where i iu is u is = d and u {0,1,2,., U}. is is φ iu is is introduced for ushing unused slots from re-scheduled aointments to oen access demands. The total number of binary variables φ iu is equals the number of hysicians times the value of the uer bound of the demand realization. But these binary variables don't deend on the number of scenarios, since they only deend on the realization of re-scheduled demand and have no relationshi with the oen access demand. x is : Number of atients re-scheduled with hysician i under demand scenario s. o x ijs : Number of oen access atients of anel i assigned to hysician j under demand scenario s. For all i = 1, 2,, M and ( i, j) A. We will consider demand scenarios s associated with a articular realization ( d, d,..., d, d ) of demand and with a robability q s. Our goal is to maximize the o o 1s 1s Ms Ms revenue of satisfying aointments, and following the notation reviously introduced, we can formulate the roblem as follows: Objective: Max [ ] (3.2.7) S M o o qs Ri xis + Rij xijs s= 1 i= 1 ( i, j) A Subject to: Ni N i = 1, 2,, M (3.2.8) N d + Nφ i = 1, 2,, M, s = 1,2,, S (3.2.9) i is iuis N d φ i = 1, 2,, M, s = 1,2,, S (3.2.10) i is iuis x is N i = 1, 2,, M, s = 1,2,, S (3.2.11) i 19

33 x is d i = 1, 2,, M, s = 1,2,, S (3.2.12) is o xijs N d jsφ juis 1,2,,, 1, 2,, i:( i, j) A j = M s = S (3.2.13) o xijs N N j + φ ju N i 1,2,,, 1, 2,, i:( i, j) A j = M s = S (3.2.14) o o xijs dis 1, 2,,, 1,2,, j:( i, j) A i = M s = S (3.2.15) φ {0,1} i = 1, 2,, M, u = 1, 2,, U (3.2.17) iu is o N, x, x 0 i, j = 1,2,, M, ( i, j) A, s = 1,2,, S (3.2.18) i is ijs is Equation (3.2.9) ensures that φ iu is = 1 if d is < N. Equation (3.2.10) ensures that i φ = 0 if iu is d is > N. Equation (3.2.11) limits the number of re-scheduled aointments i to the desired caacity. Equations (3.2.13) and (3.2.14) ensure that the total oen access aointments for hysician i do not exceed remaining caacity when φ iu is = 1 and φ iu is = 0 resectively. Equation (3.2.17) is the binary constraint. 20

34 CHAPTER 4 VALUE OF FLEXIBILITY 4.1 Practice without any flexibility We refer to the rimary care ractice without any flexibility as the dedicated case. Each hysician can only see the atients come from his/her own anel. If the caacity, i.e. the caacity for re-scheduled demand or the caacity for oen access demand, is used u, the remaining demand will have to be turned away and a cost will incurred. We can use equation (3.2.3) to directly decide the otimal number of slots that should be allocated for re-scheduled aointments of each hysician in the dedicated case. Notice that equation (3.2.3) has a newsvendor tye solution which does not deend on the distribution of rescheduled demand. C * 1 Ni = N Фi (1 ) (3.2.3) C Figure 4.1 and 4.2 show the total exected costs of missing demands in two instances for the dedicated case: the caacity of each hysician is 24 slots er day, and the cost of missing one re-scheduled aointment is set to 0.75 and the cost of missing an oen access demand is 0.9; these costs are equal to the tyical show rates of each tye of demand as indicated by Bennett and Baxley (2009) [33]. All demands belong to Poisson distribution. In Figure 4.1, the demand rates for re-scheduled and oen access aointments are 10 and 14 resectively. In Figure 4.2, we change them to 16 and 8. We can see that since the cost of missing one oen access demand is higher than missing a re-scheduled demand, the marginal gain of increasing the value of the beginning but trends to be flat when it aroaches the otimal oint. o N is significant at 21

35 Figure 4.1 Dedicated case with demand rates 10 and 14 for re-scheduling and oen access resectively. And a closer view of the value near the otimal oint. 22

36 Figure 4.2 Dedicated case with demand rates 16 and 8 for re-scheduling and oen access resectively. And a closer view of the value near the otimal oint. 23

37 4.2 Two hysicians with oen access flexibility When hysicians have full flexibility to share both re-scheduled and oen access atients, the ractice can be treated as a dedicated system with ooled demands and caacities. For the case that hysicians only have flexibility in oen access ractice shown in Figure 3.2, we can use conditions (3.2.4) and (3.2.5) to search the otimal value of N and * 1 N directly. The running comlexity is * 2 2 ( ) O N, where N is the caacity of each hysician. Particularly, if two hysicians have the same demand rate of reotimal value scheduled aointments, we can use the condition (3.2.6) to search the of * N ( N ) in O( N ) time. Figures 4.3 and 4.4 illustrates two examles: * Cost Distribution Figure 4.3. Two hysicians have flexibility in oen access ractice. where N1 = 19 and N2 =

38 16 14 Cost Distribution Figure 4.4. Two hysicians have flexibility in oen access ractice. where N1 = 16, N2 = Value of flexibility in a ractice with three hysicians For a rimary care ractice with three hysicians or more, it is too comlicated to get any closed form or condition for the otimal value of flexibility in this circumstance, we will use the stochastic integer rogramming model introduced in section Three different levels of flexibility will be evaluated full flexibility, artial flexibility (2-chain) and no flexibility (dedicated) for a variety of * N i. To investigate the value of settings with symmetric system utilization (from and asymmetric demand distributions and different levels of 40% u to 160%). The system utilization refers to the scale between exected demands and available caacities. We will focus on three measures: system revenue, timely access rate and continuity rate. The system revenue stands for the 25

39 total revenue of meeting atient demands; timely access rate is the ercentage of atients can who get access to care; and continuity rate resents the ercentage of atients who see their own hysician. Our model rovides the otimal value of N, N,..., N, and * * * 1 2 M the otimal allocation of atients to hysicians (i.e., for each anel that how many atients should see their own hysician, and how many of them should be diverted to a different hysician). The comutational comlexity of our model heavily deends on the number of scenarios, which is the most influential factor, and the number of hysicians. We tested the model of the general formulation using IBM ILOG OPL 6.3 on a PC with Intel 2 Cores Dual 2 3G CPU and 8GB memory. For three hysicians with 100,000 scenarios, it takes 50 hours to get the results when the relative MIP (Mixed Integer Programming) ga tolerance is set to 1%. Although our stochastic integer rogramming model can theoretically investigate the value of flexibility for any flexibility configuration with any number of hysicians, the time-consuming nature of the otimization and evaluation makes it imractical. Fortunately, a comutationally effective samle average aroximation method was roosed by S. Solak [34] to rovide an efficient solution aroach for two-stage stochastic integer rogramming roblems. The basic idea of the samle average aroximation method used in our research is to create a manageable number of samles/scenarios to roduce an estimation of the otimal objective value and corresonding first stage solutions. We then further run a large number of scenarios to have a recise estimation of the objective value based on the fixed first stage solution. This rocess is reeated over a number of relications to rovide confidence intervals and 26

40 statistical guarantees on the quality of the estimation. To allow for a fair comarison, the 2-chain, full flexibility and dedicated case use the same set of scenarios. To investigate the value of flexibility for three hysicians under different levels of system utilization, we first focus on the symmetric demand distributions (i.e., all anels generate identically distributed demands) to gain insights on its effectiveness to hedge against demand uncertainty. We then analyze the imact of asymmetric demand distributions, where flexibility additionally hels to balance the average suly and demand across roviders. We also use several cases in which the demand ratio between rescheduled and oen access demand changes significantly Results for three hysicians with symmetric demand distributions Following the findings of Bennett and Baxley (2009) [33], we assume a tyical no show rate for re-scheduled demand of 25%, and a 10% no show rate for oen access demand. Thus, we assign the revenue of scheduling one re-scheduled demand as 0.75, and 0.9 for seeing one oen access atient. These values stand for the actual show rates. To encourage continuity in the system, we assume that there is a 0.05 cost of seeing atients from another hysician's anel. System utilization in our model is defined as the ratio of the exected total demand for the clinics and total available caacity. For instance, in a ractice with three hysicians, suose each hysician has a demand rate of 10 for rescheduled aointment and 14 for oen access demand. The total exected demand is =72, and the total caacity is 24 3 = 72, therefore, the system utilization is 100%. To make the system under-/over-utilized, a factor varying from 0.4 to 1.6 will be multilied to the mean demands rate to generate different levels of utilization. We use 27

41 four cases with demand ratios of 10/14, 14/10, 6/18 and 18/6 to investigate the value of flexibility for a ractice with three hysicians having symmetric demand distributions. Symmetric Case 1 (10/14). Table 4.1 summarizes the assumtions for the first case where the demand ratio between rescheduled and oen access demands is 10/14. Physician caacity 24 Number of hysicians in ractice 3 Scenarios for each relication 1000 Number of relications 50 Revenue of seeing one re-scheduled demand 0.75 Revenue of seeing one owned oen access demand 0.90 Revenue of seeing one diverted oen access demand 0.85 Mean demand rate for re-scheduled aointments [10, 10, 10] Mean demand rate for oen access aointments [14, 14, 14] Relative MIP tolerance ga 0.01% Table 4.1 Assumtions for 3 hysicians with symmetric demand distributions in Symmetric Case 1 (10/14). In our exeriments, one interesting and romising henomena is that the 95% confidence interval of the objective values (system revenue) resulting from 50 relications lies in a very narrow range, the variance over the mean is less than 1%. Therefore, we can use the mean objective value of 50 relications to achieve an accurate estimation of the real objective value over the whole oulation of scenarios. Comutational effort for the second ste of stochastic integer rogram can be saved due to this. Table 4.2 shows an instance of the objective value statistics for different 28

42 flexibilities when the system is balanced. Figure 4.5 resents the corresonding Box- Whisker lot. 2-chain Full Flex Dedicated Conf. Intervals (One-Samle) 100% utilization Obj 100% utilization Obj 100% utilization Obj Samle Size Samle Mean Samle Std Dev Confidence Level (Mean) 95.0% 95.0% 95.0% Degrees of Freedom Lower Limit Uer Limit Confidence Level (Std Dev) 95.0% 95.0% 95.0% Degrees of Freedom Lower Limit Uer Limit Table 4.2 Statistics of objective value for different flexibilities with 100% utilization in Symmetric Case 1. Figure 4.5 Box-Whisker Plot comarison of objective values for different flexibilities with 100% utilization in Symmetric Case 1. 29

43 A ossible exlanation for this concentrated distribution of objective values might be the low variation of the aggregate system demand distribution. Table 4.3 demonstrates the distribution of total arrival demand of 50 relications when the system is balanced (i.e., 100% utilization). Total demand Conf. Intervals (One-Samle) 100% utilization Demand Samle Size 50 Samle Mean Samle Std Dev Confidence Level (Mean) 95.0% Degrees of Freedom 49 Lower Limit Uer Limit Confidence Level (Std Dev) 95.0% Degrees of Freedom 49 Lower Limit Uer Limit Table 4.3 Statistics of total demands for 3 hysicians with 100% utilization in Symmetric Case 1. We can see that the value of total demand varies very little among the relications. Though the demands are samled from Poisson distribution and the realization varies dramatically in each scenario, for a sum of 1000 scenarios, the averaged total demand will closely aroximate the sum of mean demand rates. Since the objective value is equal to the revenue of demands which the system could satisfy, a "flat" total demand distribution among the relications will roduce a "concentrated" objective value estimation. As mentioned earlier, we will use the mean objective value estimated from 50 relications to aroximate the actual value over the whole scenario sace. 30

44 Tables 4.4, 4.5, and 4.6 give the measurement and comarison of 2-chain flexibility, full flexibility and dedicated case under different levels of system utilization in the three dimensions of interest: system revenue, timely access rate and continuity rate. System Revenue Utilization 40% 80% 100% 120% 160% 2-chain Full Flex Dedicated chain vs Dedicated 0.00% 1.50% 3.66% 2.15% 1.89% Full vs Dedicated 0.00% 1.52% 3.73% 2.19% 1.93% Table 4.4 Measurement for different flexibilities in term of system revenue in Symmetric Case 1 (10/14). Timely Access Rate Utilization 40% 80% 100% 120% 160% 2-chain 100% 99.88% 95.29% 82.01% 62.66% Full Flex 100% 99.88% 95.29% 81.99% 62.65% Dedicated 100% 98.40% 91.78% 80.72% 62.24% 2-chain vs Dedicated 0.00% 1.50% 3.82% 1.59% 0.69% Full vs Dedicated 0.00% 1.50% 3.82% 1.58% 0.66% Table 4.5 Measurement for different flexibilities in term of timely access rate in Symmetric Case 1 (10/14). Continuity Rate Utilization 40% 80% 100% 120% 160% 2-chain 100% 98.24% 95.29% 97.03% 96.97% Full Flex 100% 98.52% 96.41% 97.68% 97.59% Dedicated 100% % % % % 2-chain vs Dedicated 0.00% -1.76% -4.71% -2.97% -3.03% Full vs Dedicated 0.00% -1.48% -3.59% -2.32% -2.42% Table 4.6 Measurement for different flexibilities in term of continuity rate in Symmetric Case 1 (10/14). 31

45 And Figures 4.6, 4.7 and 4.8 are the comarisons illustrated in lot form resectively. Figure 4.6 Comarisons of different flexibilities in term of system revenue in Symmetric Case 1 (10/14). Figure 4.7 Comarisons of different flexibilities in term of timely access rate in Symmetric Case 1 (10/14). 32

46 Figure 4.8 Comarisons of different flexibilities in term of continuity rate in Symmetric Case 1 (10/14). We can see that the highest benefit of both system revenue and timely access rate is achieved in the case where the system is balanced, i.e. when the exected demand equals the available caacity. When the system is under-utilized, most of the demands can be met and therefore result in lower benefits of flexibility. By contrast, when the system is over-utilized and more likely to miss the demand, flexibility still has the ability to shift demand to a less utilized hysician. Therefore, the grah of system erformance imrovement is not symmetric. The benefits of 2-chain flexibility are almost as high as those of full flexibility, with only a 0.07% detriment in terms of system revenue. One interesting result is that the timely access rates of 2-chain flexibility and full flexibility are nearly the same no matter what the level of utilization of the system is. This is consistent with the results reorted in the literature on flexibility in manufacturing settings. The difference in revenue is even 33

47 lower in our healthcare setting, since the rescheduled demand cannot be shared between hysicians; flexibility can only be used on the oen access demand. Intuition tells us that since full flexibility has more "outbound" links than 2-chain flexibility, it should have a better ability to absorb incoming demands and yield a higher timely access rate than 2-chain flexibility. This is indeed true for the dynamic setting of atient scheduling where allocation decisions are made as requests arrive, with limited knowledge of the overall demand that will need to be serviced (Hichen (2009) [35]). By contrast, in the aggregate demand setting catured by our two-stage stochastic integer rogramming aroach, the atient allocation is only erformed after the full system demand is known. Although, 2-chain flexibility achieves almost the same benefits as full flexibility, in our aggregate setting, there are instances where full flexibility will clearly dominate. For instance, consider a ractice with four hysicians, where each has 10 slots left for oen access, and the demands for oen access are 20, 20, 0 and 0 resectively. In this extreme case, the 2-chain flexibility can only meet 30 oen access demands the full flexibility can satisfy all of them. Since this tye of instance would occur with a low robability, from a statistical oint of view, the 2-chain flexibility has almost the same effectiveness to absorb the demand as full flexibility. Another henomena that deserves our attention is that the diversion rate, which equals one minus the continuity rate, of 2-chain flexibility is higher than that of full flexibility. Our initial intuition tells us that since full flexibility has more "outbound" links than 2- chain flexibility, it should have a higher robability that the demand will be diverted to other hysicians. In reality, however, a single atient redirection to an available hysician, which can be made directly under full flexibility, may require redirecting several atients 34

48 along the 2-chain if the initial atient s anel and available hysician involved are not connected. For examle, Figure 4.9 shows a case of three hysicians where each hysician has 10 slots left for oen access, and the demands are 16, 10 and 4 resectively. We can see that the total number of diversions under 2-chain flexibility is 12, but only 6 under the full flexibility. Since 2-chain flexibility requires more "jums" to shift the demands, the diversion rate of 2-chain is higher than that of full flexibility in our model. Figure 4.9 An examle of diversion rocess in 2-chain and full flexibility. While the number of redirections is greater in the 2-chain system, it is imortant to note that each atient will always see either one of two hysicians. We believe this results in stronger continuity and efficiency from the ersective of both the atient (who could quickly get to be familiar and comfortable with both hysicians) and the hysician (who would be able to follow the other s anel relatively well and share cases with only one other hysician). Symmetric Case 2 (14/10). To further study the imact of the demand ratio on system erformance, we reverse the ratio from 10/14 used in case 1 to 14/10. Tables 4.7, 4.8, and 4.9 give the measurement and comarison of 2-chain flexibility, full flexibility and 35