Regulating patient care in walk-in clinics

Regulating Patient Care in Walk-in Clinics

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Regulating Patient Care in Walk-in Clinics Mostafa Pazoki, Hamed Samarghandi PII: DOI: Reference:

S0305-0483(19)30535-3 https://doi.org/10.1016/j.omega.2020.102200 OME 102200

To appear in:

Omega

Received date: Accepted date:

8 May 2019 14 January 2020

Please cite this article as: Mostafa Pazoki, Hamed Samarghandi, Regulating Patient Care in Walk-in Clinics, Omega (2020), doi: https://doi.org/10.1016/j.omega.2020.102200

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Highlights • This paper addresses regulating visit time for elevating overall patient satisfaction. • We concluded that if patients’ arrival rate is small compared to the clinic’s capacity, regulation is not required. • Otherwise, a regulation in the form of minimum visit time improves overall patient satisfaction.

1

Regulating Patient Care in Walk-in Clinics Mostafa Pazoki∗1,3,5 and Hamed Samarghandi2,4 1

2

HEC Montr´eal, Montr´eal, QC, Canada Edward School of Business, University of Saskatchewan, Saskatoon, SK, Canada, S7N 5A7 3 E-mail address: [email protected] 4 E-mail address: [email protected] 5 Corresponding Author

Declarations of interest: none.

Funding: This work was supported by NSERC Discovery grant [#2017-03743]. Abstract This paper studies the problem of government intervention in walk-in clinics regarding patient satisfaction. Since walk-in clinics benefit from the number of patients they serve (fee-forservice), it may be in their best interest to reduce the visit times; consequently, patient care and quality of service may be sacrificed to gain more revenue. For this matter, a walk-in clinic as a queuing system with stochastic arrival and visit times is studied. To identify the cases when the quality of service is compromised for maximizing clinic’s revenue and therefore government intervention may be required, we compare revenue maximization policies and patient satisfaction maximization policies under various scenarios defined based on the proportion of the arriving patients to the clinic’s capacity and also the existence of local competition. It is concluded that if patients’ arrival rate is relatively small compared to the clinic’s capacity, regulation is not required. Otherwise, a regulation in the form of minimum visit time can increase patient satisfaction.

Keywords: Patient Satisfaction, Queuing System, Governmental Intervention, Revenue Management. ∗

Present Address: Edward School of Business, University of Saskatchewan, Saskatoon, SK, Canada, S7N 5A7

1

Introduction

Walk-in clinics are considered as private businesses, and consequently, regulatory bodies in North America have rarely imposed regulations on them. Since private healthcare centers, including walkin clinics, are established mostly based on revenue making plans, patient care can be one of the most important sacrifices along the clinic’s profit maximization path. Specifically, this research is motivated to address the “one issue per visit” problem. The College of Physicians and Surgeons of Manitoba warned about this problem1 in 2012 after a woman died of heart attack, a week after being allegedly denied by a doctor to hear about her back pain. In another study in USA, it is estimated that the patients have 11 seconds to talk about their problems before getting interrupted by the doctors2 . This problem becomes more complicated when wait time enters the game; while some believe that limiting the visit time is necessary to reduce the wait time of other patients (generating revenue may be another hidden reason), others consider restricting access unethical3 . Therefore, this paper concentrates on visit time, as a specific aspect of patient care that is directly linked to the clinic’s revenue. Several studies discuss the main factors of patient satisfaction in the Emergency Department of hospitals. Walk-in clinics serve the patients by the same procedure, and the same findings could also apply to walk-in clinics. Among the surveys discussing patient satisfaction or willingness to return, researchers have come across multiple factors ranging from poor explanation of test results [1] to waiting time [2], and from general interpersonal care [3] to how caring the nurses are [4]. Among all these factors, we observe that they are mostly related to clinicians’ personal traits and training, except two of them: waiting time and visit time, which reiterate the main motivation of this research. Below, we discuss the importance of both factors in details. Visit time is recognized as one of the important factors in patient satisfaction. There are two groups of surveys that recognize this fact directly and indirectly. The first group considers face time or visit time directly in the surveys. Among these researches, we name [5] who specify visit time as one of the four factors pertaining to physician communication; [6] who gave a high score to the 1

Retrieved on April 5th, 2019, from URL: https://www.cbc.ca/news/canada/manitoba/manitoba-mds-warnedabout-limiting-patient-complaints-1.1260643?cmp=newsletter-Second+Opinion+-+March+16+2019 2 Retrieved on April 5th, 2019, from URL: https://www.cbc.ca/news/health/doctor-patient-visits-1.4755498 3 Retrieved on April 5th, 2019, from URL: https://www.cbc.ca/news/health/second-opinion-one-problem-visit1.5061506

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time physician and nurses spend listening to the patients; and, [7] that specifically conclude that visit time is an important patient satisfaction factor. The second group of researches recognize interpersonal interactions of clinicians, particularly the doctors, as one of the most important patient satisfaction factors. [8] cite interpersonal interactions with the doctor as the most critical factor in patient satisfaction. Interpersonal communication denotes doctor’s manner and the amount, quality and understandability of information conveyed by the doctor to the patient. In another study, [9] address a set of important factors for patient satisfaction and likelihood to recommend; the list consists of how well the doctor explains the plan of care, shows interest in patients concerns, conveys information, and guides the patient for home care. [10] name the patient-doctor communication, including the information conveyed, as one of the factors in post-visit patient satisfaction. Although this group of researches do not directly address visit time as a patient satisfaction factor, they consider time-consuming actions such as amount of information conveyed to the patients as important drivers of patient satisfaction. Therefore, it is safe to claim that visit time plays a role, although indirectly, in improving the factors related to clinicians’ interpersonal skills. Although visit time is a significant patient satisfaction factor, it is important to note that the waiting time of the patients who are present in the clinic to see a doctor will rise once the physicians increase the visit time of the patients. Since many of the referring patients require immediate care, total waiting time is also an important factor in calculating the quality of care, and consequently, patient satisfaction. [11] find a strong correlation between patient satisfaction in an emergency department and the wait time to see a doctor. In another study in the UK, [2] survey the patients of an emergency department and recommend that reducing the wait time will improve patient satisfaction. [4] studied an emergency department and discovered the five most important factors in patient satisfaction; total waiting time is one of them. Waiting time is also addressed as an important factor in patient satisfaction by [12]. [13] explore the literature from 1990 to 2002 and review the main factors that impact patient satisfaction in emergency departments. Among the factors that are reiterated to be important is the waiting time to see a physician. [14] claim that waiting time and doctor’s manner are the two most important factors in patient satisfaction. The latter has also shown up in some surveys done for ED and walk-in clinics. Finally, Press Ganey (PG) scores are addressed to be related to perceived waiting time, patient communication and interpersonal interactions (6, 15, 16, 17, 18). To this end, there is enough empirical evidence to 4

support the importance of visit time and waiting time in patient satisfaction. Thus, a balance between the average waiting time in the waiting room and the average examination or visit time is required to achieve a satisfactory level of patient care, which is the focus of this research. In this sense, this paper belongs to meso-level thrusts as discussed by [19]. Visit time and total waiting time are essential in patient satisfaction, albeit not exclusively. As alluded to before, other factors significantly affect patient satisfaction, recommendation likelihood, and as a result, the profitability of the clinic. To justify the reason for only considering the wait time and the visit time, we categorize the important factors addressed in the literature into two main classes. In the first class are the factors whose improvement are uni-directional, whether they are linked to the clinic’s operational costs or not. For example, politeness of the nurses and the degree of courtesy by which the patients are treated should always improve; this improvement is not related to the operational costs of the clinic. The second category includes the factors whose best levels are not clear beforehand. For instance, as we discussed previously, the amount of information conveyed to the patients by the doctors is an essential factor to increase the overall patient satisfaction. However, more time spent by the doctors addressing the patient questions and concerns means higher waiting time for the patients in the waiting room, and that results in less satisfaction. Therefore, a balance is required in the factors belonging to the second category; among these factors, we concentrate on visit time which directly affects a clinic’s profitability, and therefore government intervention may be required. Before proceeding, the terminology used in this paper is introduced. Visit time is the amount of time a patient spends with the doctor. This is the same as examination time. In the queuing theory literature, the amount of time spent by the customer at the server, which is the doctor in a clinic, is represented by service time. Therefore, visit time, examination time and service time are used interchangeably in this paper. Finally, the total amount of time that a patient is waiting in the waiting room is denoted by waiting time. This research is motivated by the healthcare status in Canada where the federal government’s approach is to connect the patients with family doctors, while the timely access to primary care units when urgently required has made walk-in clinics attractive to many Canadians. The walkin clinics, however, are treated as businesses and therefore are not significantly regulated by the regulating body of healthcare services in Canada. Consequently, there is not enough information about the quality of the health services provided by such institutions. 5

In the next section, the literature of healthcare operations management and regulatory intervention approach in healthcare are discussed.

2

Literature Review

The importance of visit time and the required balance between visit time, wait time, and clinic profitability was discussed in Section 1. Based on these arguments, identifying the necessity for potential regulatory intervention can be considered as the main contribution of this paper. In this section, we position this research in regards to two healthcare-related topics in the literature: 1) healthcare and regulation, and 2) healthcare and operations management. Healthcare and Regulation: Regulation is used as a societal risk-control tool [20]. [21] carry out a survey-based research and reveal that government regulation improves operational performance of healthcare providing firms through enhancing nursing facilities’ capabilities. [22] presents four categories of regulatory interventions, namely: 1) managerial and contractual interventions, 2) market or quasi-market systemic change, 3) user voice and choice, and 4) capability interventions. The second and the third categories are based on public choice theory, and as the title suggests, require alternative options for the users of the services, or patients, to work appropriately. However, walk-in clinics are used for urgent or semi-urgent cases when patient do not intend to wait for appointments. Moreover, there is no guarantee that there are multiple walk-in clinics in an area, and therefore competition cannot be the case. Thus, for walk-in clinics, the regulator cannot rely on quasi-market systemic change and user voice regulatory approaches. The fourth approach of [22], capability interventions, includes different types of process and capacity support from the regulator to improve performance. However, without the force stemming from a regulation with minimum standards, which is categorized as managerial interventions or “outside-in” approach [23], and also without the competition offering options to the service users, supporting walk-in clinics in their process does not lead to performance improvements in the longrun. Regulation is needed to dictate balancing patient desires and cost as a “dominant logic” [24], and without that, capability interventions may lead to a lower cost for clinics, but not a better service for the consumers. After a comprehensive review of healthcare regulation, [25] find that

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there is a shift from “self-regulatory” approach to more direct oversight, which was also implied by [26]. This paper concentrates on the first approach, which is a top-down managerial intervention by setting minimum standards. Upon identifying the conditions where managerial interventions are needed, regulator then may proceed by investigating the methods to develop clinics’ capabilities. Operations Management: Operations management and operations research have provided a wide variety of tools to deal with healthcare problems, often called as Healthcare Operations Management (HOM). The purpose of HOM is to provide “affordable and timely access to quality healthcare” [27]. Multiple operations research methods are used to analyze healthcare problems, including, but not limited to, game theory (28, 29), mathematical programming (30, 31), and queuing theory (32, 33, 34). Game Theory is an approach to deal with interactions of multiple rational decision makers. A Stackelberg game between the government and an influenza vaccine manufacturer, leading to a wholesale price contract is considered in [29]. [28] employ game theoretical techniques to model multi-governmental interactions and try to design a contract for international influenza vaccination ordering decisions. Mathematical programming is a method to tackle different aspects in healthcare. These problems include, but no limited to, reducing waiting times, patient scheduling, and capacity planning (35, 36). A three-stage staff-scheduling mathematical model is presented in [37], that tries to maximize the number of admitted patients while balancing the distribution of patients among the staff. Minimizing total waiting and travel time while avoiding unbalanced workload distribution is also studied by [38]. [39] propose an integer-programming model for operating room scheduling, aiming at minimizing hospital-related costs. [30] carry out a novel solution strategy to solve a stochastic integer programming for scheduling and staffing problem and reduce the solution time to one sixth. Later, [31] consider home healthcare staffing and scheduling problem as a two-stage stochastic programming case. They show that performing staffing and scheduling via a stochastic programming model result in significant cost savings in comparison to a deterministic programming model. Finally, restricting the total wait time, [40] propose a stochastic programming model to schedule patients in emergency departments. [41] presents an interesting survey on different types of mathematical programming model in patient scheduling. 7

Queuing Theory can be employed to capture the uncertainty in micro-level of healthcare providers’ decisions. [33] capture the uncertainty in organ supply and demand problem using queuing games where arrival of transplant candidates and organs follow a Poisson distribution. [34] investigate a regulatory scheme, called Yardstick competition, where the regulator rewards monopoly health service providers based on their average waiting time, and this way creates an “artificial” competition between local and unrelated providers. In that paper, providers are modeled as M/M/1 queuing systems, and patient utility is a function of service value, waiting time, and price. Although application of queuing theory in healthcare planning is currently being discussed, it is not a new trend. [42] concentrate on the application queuing theory on improving patient satisfaction in pharmacies that face high demands. [43] discuss the capability of queuing theory on balancing waiting time in healthcare systems. [44] carry out an empirical study on the queuing network characteristics of a hospital’s pharmacy in Iran, and suggest reallocation of trained multitask personnel to filling prescription stage to reduce waiting time. [45] generate appointment schedule considering medical staff and patients’ interests in terms of idle time and waiting time, respectively. Imaging test centers are also studied as queuing systems in [32], where the clinic set the price for the patients that are partially covered by insurance. Interested readers may refer to [27] and [47] for review of queuing theory application in healthcare industry. This Paper’s Position: As discussed above, a top-down regulation is needed to form the “dominant logic” of healthcare providers’ plans. Specifically, regulating visit time in walk-in clinics is required to balance clinic’s revenue and quality of service. For this purpose, we are required to consider a modeling framework that preferably returns closed-form solutions, so that the regulator can derive policies based on specific performance indicators in mind. Furthermore, the uncertain nature of arrival and visit times is a significant factor to take into account. Therefore, considering the micro-level decision making context, queuing theory is employed to tackle this problem. To the best of our knowledge, this is the first paper to investigate a government, imposing control over walk-in clinic visit time to elevate patient satisfaction level. It should be noted that [48] shows that, in an outpatient clinic, a typical tradeoff in appointment scheduling is about idle time, wait times and overtime. In this sense, there are similarities and differences between the considered problem and the appointment scheduling problem. One of the main differences is that for the case of appointment scheduling problem the outpatient clinic has 8

access to an anticipated schedule for a specific period of time in the future. The clinic knows, at least to some extent, about the patients of a specific day, their arrival, and their service time. Hence, the goal of appointment scheduling is to set starting time or ordering of patients to find an appointment system for which a particular measure of performance for the clinic is optimized. On the other hand, in the problem studied in this paper, a priori schedule for the clinic does not exist; the patients are admitted to the clinic or rejected from the queue, on the spot. And the goal is to find the scenarios in which the government must intervene to protect the patients’ interests. Furthermore, this work does not focus on optimizing or measuring idle time, which reflects resource utilization. However, recent work suggests that idle time is often not a major concern for most clinics as the clinics are over-utilized [46]. The rest of the paper is organized as follows. Section 3 is devoted to presenting and solving four models which differ in terms of clinic capacity and competition (relation between patient satisfaction and recommendation likelihood). To address the main research question which is to understand where government intervention is required and how, Section 4 is presented. Finally, the research is summarized and the main findings and insights are reviewed in Section 5.

3

Models and Analyses

In this paper, we consider a clinic with a single doctor who serves the admitted patients. Queuing theory is employed to account for the uncertainty in patient arrival time and examination time. The general assumption of the queuing theory literature that assumes exponential distributions for time-between-arrival and service time (visit times in this paper) is followed, resulting an M/M/1 system. One may argue about the validity of an M/M/1 system for this study, considering that different inter-arrival and service time distributions could be the case, or a higher number of doctors could be available in the clinic. For a discussion on sufficiency of an M/M/1 system for this study, please see Appendix A. To provide the governments with a comprehensive set of regulatory guidelines, four scenarios are defined based on the system’s capacity and the relation between patient satisfaction and recommendation likelihood: Scenario 1: The clinic has infinite capacity, meaning that no arriving patient is rejected; 9

sooner or later, all of the referring patients are served. This is the case if the clinic’s capacity is relatively large comparing to the arrival rate of the patients. Furthermore, patient satisfaction is not related to the recommendation likelihood, which is the case where there are no other walk-in clinics in the vicinity; hence, all of the patients refer to this clinic no matter how satisfied or dissatisfied they are. The model associated with this scenario is called Model UM, where U denotes the uncapacitated system and M represents the monopolistic position of the clinic in the region. Scenario 2: The clinic is assumed to be capacitated, which means that if a certain number of admitted patients are already in the clinic, then an arriving patient will be turned away. This is the case if the clinic’s capacity is small relative to the patients’ arrival rate. Furthermore, the monopolistic situation of Scenario 1 is still the case. The model of this scenario is called Model CM, or capacitated monopolistic. Scenario 3: This scenario considers a clinic where all of the arriving patients are admitted, and there is a positive relation between patient satisfaction and recommendation likelihood [1]. In other words, if the patients are satisfied (dissatisfied), there is a higher (lower) chance that they return or advise others to refer to the clinic. This is the case where there are several walk-in clinics in the area, therefore, creating an oligopoly-type market for the clinic. Assuming that all patients are admitted to the clinic, the model associated with this scenario is named Model UO, where U represents an uncapacitated system, and O denotes the oligopoly-type situation. Scenario 4: In this scenario, there is a threshold for the maximum number of admitted patients at any instant of time, and there is a positive relationship between patient satisfaction and likelihood of recommendation. The model associated with this scenario is called Model CO. This scenario is the case for many real-life situations in more populated areas where there are numerous walk-in clinics, but there is a cap on the admitted patients due to limited space and higher patient arrival rates. While a clinic with a limited admission capacity matches many real-life cases, exploring it also gives an idea about the impact of controlling the arrival rate on the clinic’s profitability and patient

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satisfaction. We proceed by presenting the mathematical models for each scenario. An admitted patient will wait in the waiting area until s/he is called on. The notation used in this paper is presented in Table 1. Table 1: Model Notation Parameter Πi Pi

Definition Clinic’s profit in scenario i ∈ {um, cm, uo, co}

Patient satisfaction in scenario i ∈ {um, cm, uo, co}

γ

Patient sensitivity to waiting time

θ

Patient sensitivity to visit time

Wq

Total waiting time in the waiting room

λ

Arrival rate of the patients

α

Patient dissatisfaction caused by being refused

ω

Recommendation coefficient

c

Clinic’s capacity

s

Average visit time

Assumption 1. The average visit time measures the time required by the physicians to provide care to the patients. Also, it is assumed that patient satisfaction is a linearly increasing function of visit time. Assumption 2. If a doctor spends more time than is required with a patient, the patient satisfaction may decrease, concluding that the relation between visit time and patient satisfaction is not strictly increasing. However, we concentrate on the range of visit time where it increases patient satisfaction. The addressed maximum required time is called sˆ in this paper. Assumption 3. It is assumed that when a patient joins the queue (is admitted to the clinic), s/he will never leave until being visited by the doctor. After introducing the notation and underlying assumptions, the model of each scenario is introduced and analyzed. For each scenario, we maximize patient satisfaction. Section 4 investigates the policies the clinics adopt to maximize their revenue, and identifies the scenarios where revenue maximization and patient satisfaction maximization are not aligned.

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3.1

Model UM

This model represents a clinic that acts as a monopoly in the region and admits all of the arriving patients. Such clinic with the addressed assumptions is modeled as an M/M/1/∞ queuing system. Patient satisfaction increases by longer visit time and decreases by larger total waiting time, suggesting a functional form as presented in equation (1).

Pum = θs − γWq ,

(1)

where Wq is the total waiting time in the waiting room and is an increasing function of s. Quality of service and patient utility are also modeled as a linear decreasing function of waiting time in [32] and [34], respectively. Assumption 4. In this paper, for the sake of tractability, we assume steady-state conditions to approximate the outcome. In an M/M/1/∞ queuing system, the total waiting time in the queue is

λs2 1−λs .

Revising equation

(1) by inserting Wq , the patient satisfaction function will be:

Pum = θs − γ

λs2 . 1 − λs

(2)

The goal, then, is to maximize Pum by setting the best average value of visit time s. Proposition 1 performs this task. Proposition 1. For a monopolistic clinic where all of the arriving patients are admitted, the average value of visit time that maximizes patient satisfaction is:

s∗um

=

1−

q λ

γ θ+γ

.

Proof. The optimum average visit time to maximize patient satisfaction is obtained by applying the first order conditions on (2) and confirming its concavity inside (0, 1/λ). Intuitively, the more sensitive the patients are about the amount of time the doctor spends checking them and conveying pertinent information, the higher must be the average visit time 12

to maximize overall patient satisfaction. However, if the average number of incoming patients increases, then the average visit time must be reduced. The same outcome is expected if patient sensitivity to waiting time increases. Substituting s∗um in equation (2) results in:

∗ Pum =

θ + 2γ −

√ 2 γ √ (θ θ+γ

√

λ

+ γ)

.

(3)

Optimum patient satisfaction (3) is a decreasing function of γ, and an increasing function of θ if θ ≥ 3γ. Therefore, the practical intuitions from analyzing Model UM is that highly sensitive patients to visit and service times will be satisfied more than less sensitive patients, if the goal is to maximize patient satisfaction.

3.2

Model CM

We proceed by introducing the scenario where patient satisfaction is not related to recommendation likelihood (monopoly), but there is a cap on the number of admitted patients at any instant of time. It is the case for those walk-in clinics that are the only clinic in a region and their waiting room capacity is small relative to the incoming number of patients. The difference between patient satisfaction in Models UM and CM is that in Model UM, the expected satisfaction values are the same for all patients who arrive at the system. In Model CM, however, not all arriving patients are accepted, and consequently, we need to distinguish the expected satisfaction for an admitted patient and the anticipated satisfaction of all patients. Since this research is conducted from the regulator’s point of view, we concentrate on all of the arriving patients rather than regarding only the admitted patients. The patients who are turned away are assumed to have a fixed level of dissatisfaction α. Therefore, to maximize the expected patient satisfaction, the regulator solves:

M ax Pcm = (1 − πc )(θs − γW + γs) − απc s

where W =

s(1−(c+1)(λs)c +c(λs)c+1 ) (1−λs)(1−(λs)c+1 )(1−πc )

(4)

is the total waiting time in an M/M/1/c queuing system, W − s

denotes the expected waiting time in queue, and πc =

(λs)c (1−λs) 1−(λs)c+1

is the probability that the system

contains c patients, and therefore an arriving patient is turned away [49].

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To find the optimum average visit time, the critical points of an n-degree polynomial must be calculated. According to Abel-Ruffini theorem [50], obtaining the closed-form solutions of a general polynomial function with higher than 5 degrees is not analytically possible. Remark 1. The primary goal of this research is identifying the cases where revenue maximization policies are not aligned with patient care. Therefore, what is necessary to find by analyzing these models is not the exact closed-form solution of the optimum average visit time, but the region where it can be. Specifically, the scenarios with the optimum solutions at extreme points are of interest from the regulator’s point of view. Therefore, we need to identify the regions that contain the optimum average visit time. Propositions 2 and 3 shed light on this matter. Proposition 2. The maximum value of Pcm is at s∗cm > 0. Proof. One can calculate that lims→0

∂Pcm ∂s

= θ, which is positive. Moreover, Pcm is continuously

differentiable on s > 0. Therefore, there is an  > 0 where Pcm (s = ) > Pcm (s = 0), concluding that the maximum of Pcm is achievable at a value of s greater than zero. Proposition 3. If sˆ is large enough and c ≥ Proof. One can calculate that lims→∞

∂Pcm ∂s

θ+γ−αλ , γλ

s∗cm is smaller than sˆ.

= (θ + γ)/λ − α − γc. If it is negative and assuming

that sˆ is large enough, and also considering that lims→0

∂Pcm ∂s

> 0 and Pcm (s = 0) = 0, there is an

s∗cm ∈ (0, sˆ) where P (s∗cm ) > Pcm (s = 0) and P (s∗cm ) > Pcm (s = sˆ).4 Propositions 2 and 3 establish the interiority of the revenue maximizing visit time. They show that if the capacity is greater than a specific threshold, it is never optimal for the clinic to move very close to the extreme points. The addressed threshold decreases in δ, meaning that it becomes easier to avoid the service time extreme point for a higher average number of arriving patients. Since obtaining a closed-form solution is not possible, we define six cases (scenarios) to examine the impact of α, γ and θ on the optimum patient satisfaction and the optimum average visit time. For each of these parameters, two levels (high and low, being represented by subscripts h and l ) are considered. The low level of α is zero, meaning that the dissatisfaction caused by turning away 4

In Proposition 3, the underlying assumption is that sˆ is larger than the greatest critical point. If it is not the case, Pcm (ˆ s) must be compared with any local maximum point greater than 0 and smaller than sˆ.

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patients is negligible. For γ and θ, the high and low levels are set to 4 and 15, arbitrarily. It is important to add that since maximizing total patient satisfaction requires a balance between visit time and wait time, we disregard the cases where (θ = θh ) ∧ (γ = γh ) at the same time. The described cases are introduced in Table 2. Table 2: Numerical cases for Model CM Cases

Parameters α

γ

θ

Case 1

0

4

4

Case 2

0

4

15

Case 3

0

15

4

Case 4

10

4

4

Case 5

10

4

15

Case 6

10

15

4

The maximum patient satisfaction and optimum average visit time are reported in Table 10 in Appendix B. The observations, however, are summarized in Table 3 below. Table 3: Impacts of α, θ and γ in Model CM Parameters

s∗cm

P (s∗cm )

α

&

&

&

&

θ γ

%

%

The % (&) sign means by increasing the parameter on the first column, the variable at the top of that column increases (decreases). The most important intuition obtained from Model CM is that higher sensitivity to interpersonal skills of the doctor means higher patient satisfaction. Therefore, the fact that doctor’s interpersonal interaction is one of the most important patient satisfaction factors [8, 4] is an opportunity for the clinics to improve customer satisfaction. Next, we proceed by analyzing the scenarios where the incoming rate of patients depends on overall patient satisfaction, which can be the case where there exists competition.

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3.3

Model UO

In this model, we assume that there is a positive linear relationship between the referring patients and patient satisfaction. In other words, the patients who have referred to the clinic and come out satisfied will probably come back and recommend the clinic to others. Hence, the arrival rate itself is a function of patient satisfaction, which assumed to be a linear function. Thus, based on these assumptions, patient satisfaction and ωλ are the same. The clinic is assumed to be uncapacitated, and therefore can be modeled as an M/M/1/∞ queuing system. In this sense, we have:

ωλ = θs − γ

λs2 1 − λs

(5)

and ω is related to the probability that the customers recommend the clinic; the higher they are satisfied, the higher is the recommendation chance and vice versa. ω can be seen as an indicator of competition, as dissatisfied patients refer to other clinics in the region for their next visit. Proposition 4 concerns the arrival rate of the clinic. Proposition 4. If all of the arriving patients are admitted to a clinic with local competition, the arrival rate would be:

λ=

s2 (θ + γ) + ω −

p (s2 (θ + γ) + ω)2 − 4ωθs2 2ωs

(6)

Proof. Solving equation (5) for λ results in two positive values:

λ=

s2 (θ + γ) + ω ±

p (s2 (θ + γ) + ω)2 − 4ωθs2 2ωs

Since the system is uncapacitated, only the value that satisfies λs < 1 is acceptable, which is the smaller one. Considering the closed form of the patient arrival rate, the optimum value of s is addressed in Proposition 5. Proposition 5. The optimum average service time for Model UO is s∗uo =

√

p ω/ θ + γ

16

Proof. Maximizing λ is equivalent to maximizing patient satisfaction in equation (5). There are two critical points for (6), one of which is positive. Considering that equation (6) is continuously differentiable on s 6= 0, the second derivative at the positive critical point is: −

√ √ (θ + γ)2 (1 − γ/ θ + γ) <0 √ ω ωγ

which concludes that the single obtained critical point is the global maximizer of λ. Interestingly, higher recommendation likelihood means higher average visit time. Furthermore, opposite to what is observed in Model UM, a higher θ yields a lower s∗uo . Therefore, not only the presence of recommendation likelihood (which can be the result of competition) impacts the optimum service time and consequently patient satisfaction, but also it alters the impact of other model parameters on optimum visit time and patient satisfaction. Substituting the optimum visit time from Proposition 5 in arrival rate presented in Proposition √ √ 1− γ 4, the optimum arrival rate and patient satisfaction would be λ = √ω and Puo = ω(1 − γ), respectively. It is important to note that for Model UO, arrival rate and patient satisfaction at the optimum point do not depend on patient sensitivity to visit time. Furthermore, higher sensitivity to wait time leads to lower patient satisfaction.

3.4

Model CO

In this section, we explore the impact of limiting the clinic’s capacity on the overall patient satisfaction if there is a relationship between patient satisfaction and patient arrival rate. The described clinic is modeled as an M/M/1/c queuing system where the arrival rate is obtained by solving the following equation: ωλ = (1 − πc )(θs − γW + γs) − απc ,

(7)

with the same πc and W as discussed for Model CM. The closed-form of λ cannot be obtained from (7), and therefore the same analysis as appeared for Model CM are used here. 12 cases are defined based on high and low levels of ω, α, γ, and θ. The pairs {•l , •h } for ω, α, γ, and θ are {0.5, 1}, {0, 10}, {4, 15}, and {4, 15}, respectively. The 12 resulting cases appear in Table 4.

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Table 4: Numerical cases for Model CO Cases

Parameters ω

α

γ

θ

Case 1

1

10

4

4

Case 2

1

10

4

15

Case 3

1

10

15

4

Case 4

1

0

4

4

Case 5

1

0

4

15

Case 6

1

0

15

4

Case 7

0.5

10

4

4

Case 8

0.5

10

4

15

Case 9

0.5

10

15

4

Case 10

0.5

0

4

4

Case 11

0.5

0

4

15

Case 12

0.5

0

15

4

The numerical cases appear in Table 11 in Appendix B, and the intuitions are summarized in Table 5 below. Table 5: Impacts of α, θ and γ in Model CO Parameters

s∗co

P (s∗co )

α

&

&

&

&

θ γ ω

& %

% &

Intuitively, similar to Model CM, sensitivity to waiting time and being turned away means lower overall patient satisfaction. Furthermore, higher sensitivity to visit time leads to lower optimum visit time and higher patient satisfaction. Therefore, as was the case for Model CM, higher sensitivity to visit time, which is equivalent to doctor’s interpersonal skills in conveying information and hearing concerns, is an opportunity to improve overall satisfaction. Another important but counter-intuitive observation from Table 5 is that a higher θ results in a lower s∗co . At the first glance, one may expect a higher average visit time for patients with higher average sensitivity to visit time. However, in this specific scenario, higher visit time means fewer

18

number of patients are admitted. To put it differently, improving the overall patient satisfaction cannot be directly measured by increasing θs − γWq . Therefore, higher sensitivity to visit time does not necessarily lead to higher visit time, where higher satisfaction means more patients and probably more rejections due to limited capacity.

4

Regulatory Implications

In section 3, we discussed the policies to maximize patient satisfaction, and how the parameters that define the surroundings of the issue can affect the optimal strategies. Since the walk-in clinic’s revenue is based on the fee-for-service model, the clinic may intend to maximize short-term revenue rather than patient satisfaction, and that may invite government intervention. Therefore, this section is devoted to maximizing the clinic’s revenue under the assumptions of each model to decipher if regulation is needed to protect the patients, and how these regulations should be.

4.1

Model UM

In this scenario, the clinic is not worried about the dissatisfied patients as there are no competitors in the vicinity. Therefore, more admissions mean more revenue for the clinic. However, since the clinic has no control over the patient arrival rate and the sustainable service rate is already more than λ, the output rate is always λ, and consequently, as long as all of the arriving patients are admitted and served, the clinic neither benefits nor loses by modifying the average visit time. Thus, the clinic may adopt the average visit time s∗um to maximize patient satisfaction without hurting its revenue, concluding that government intervention is not necessary.

4.2

Model CM

In this scenario, revenue is made by serving the admitted patients, which is λ(1 − πc ). Proposition 6 concerns the revenue-maximizing average visit time. Proposition 6. If there is a cap on the admitted patients at any instant of time, the clinic maximizes its revenue at srcm < 1/λ, where superscript “rcm” denotes the revenue maximizer under Scenario CM.

19

Proof. We have: ∂Πcm (c + 1)λc+1 sc (1 − (λs)c ) − cλc sc−1 (1 − (λs)c+1 ) . = ∂s (1 − (λs)c+1 )2 The denominator is always positive. The nominator is positive if c − (c + 1)λs − (λs)c+1 > 0. This is a polynomial function with only one positive root in (0, 1/λ), at which it is decreasing. Proposition 6 implies an interior solution for the revenue maximization function. Since the closed-form solution of the patient satisfaction maximization problem cannot be found, there is no analytical base for comparing the results. Therefore, we define numerical examples to conclude the necessity of regulating the clinics. Six cases based on low and high levels of θ, γ and λ are defined, which appear in Table 6. The average visit time that maximizes patient satisfaction is s∗cm , and the average visit time that maximizes profit is srcm . What matters in this analysis is the difference between these two variables for it shows the necessity of regulation. This difference is denoted by ∆s = s∗cm − srcm . Table 7 illustrates the impact of θ, γ and λ on ∆s (detailed results appear in Table 12 in Appendix B). Table 6: Numerical cases for regulatory implications under scenario CM Cases

Parameters λ

γ

θ

Case 1

5

4

4

Case 2

5

4

15

Case 3

5

15

4

Case 4

15

4

4

Case 5

15

4

15

Case 6

15

15

4

Table 7: Impacts of θ, γ and λ on ∆s in Model CM Parameters

θ

γ

λ

∆s

+, %

+, &

+, &

In Table 7, +/− sign means positivity/negativity of ∆s, and & / % implies that ∆s decreases/increases if that specific parameter increases. First, the optimum average visit time to

20

maximize patient satisfaction was always higher than that of the revenue maximization problem. Furthermore, lower arrival rate resulted in more significant difference between s∗cm and srcm , which increases the necessity of government intervention. Therefore, for greater θ and smaller γ and δ, regulating the minimum visit time increases the overall patient satisfaction. In Scenario UM, we have concluded that government intervention is not required as revenue maximization goal is aligned with patient satisfaction maximization. In Scenario CM, however, a cap on the number of admitted patients at any instant of time is imposed and resulted in the necessity of regulation. Next, we examine an alternative way of regulating the walk-in clinics which is regulating the number of admitted patients (capacity) to help the regulator reach its goal. Remark 2. In order to regulate an action, it needs to be observable and controllable. While the number of admitted patients is observable and controllable by the clinic, it is not observable, and therefore not controllable by the authorities. One way of enforcing a regulation based on admission capacity is to have random audits. Otherwise, the results of such analysis can be used by the clinic to increase its profit while increasing patient satisfaction. For this purpose, we analyze numerical instances to investigate the impact of capacity on the difference between the visit time that maximizes patient care, and the visit time that maximizes clinic’s revenue in Model CM. Accordingly, an analysis is performed by employing the six combinations of α, θ and γ, as demonstrated in Table 2 (Results are reported in Table 10), with the assumption that the minimum possible visit time increment is 0.01 of time unit.5 Results are depicted in Figure 1. For large α (cases 4, 5 and 6), increasing the capacity increases s∗cm until it reaches a certain value. If α is small or 0, however, a higher capacity (up to a certain threshold) means a lower visit time, which converges to the same s∗cm as of the large α. The latter observation is expected as a large c means fewer patients are refused, and therefore πc converges to 0. Furthermore, the optimum visit time to maximize clinic’s revenue increases as c increases. Therefore, in all cases, the optimum visit times s∗cm and srcm either converge or reach a break-even point. In other words, regulating the number of admitted patients at any instant of time is an effective way of pushing the revenue maximizing clinic toward increasing patient satisfaction. 5

λ is excluded from the analysis. Let λ = 10.

21

·10−2

·10−2 4

4

0.15

3

3

s

s

s

0.1

2

2

5 · 10−2

1

1 0 0

10

20

30

0

40

10

20

30

40

0

10

·10−2

·10−2

20

30

40

c: α=0,γ=15,θ=4

b: α=0,γ=4,θ=15

a: α=0,γ=4,θ=4

·10−2 4

4

4

s

3

s

s

3

2

2 2

1

1 0

10

20

30

40

0

10

20

30

40

0

10

20

30

c

c

c

d: α=10,γ=4,θ=4

e: α=0,γ=4,θ=1

f: α=10,γ=15,θ=4

40

Figure 1: Comparing optimal visit times of Model CM for numerical cases presented in Tables 2 and 10: s∗cm : (

), srcm : (

)

The optimum visit time for the clinic does not change by modifying α, γ and θ. Moreover, from Figure 1, one can observe that 1) the dissatisfaction of refused patients does not affect s∗cm once it reaches a certain threshold, and 2) the higher the proportion of θ to γ, the higher is the point (in terms of clinic’s admission capacity) where srcm meets s∗cm . Thus, it can be suggested that the government can set a lower or an upper limit for the number of admitted patients in order to increase patient satisfaction. If visit time is more important than waiting time for the patients, setting an upper limit would most probably be the case; whereas if waiting time is more important than visit time for the patients, setting a lower limit would potentially yield a higher patient satisfaction.

4.3

Model UO

In this scenario, the clinic admits all arriving patients. Higher patient satisfaction results in greater patient arrival rate, and subsequently, more revenue. Therefore, the clinic adopts the strategy to maximize patient satisfaction in order to maximize its revenue. Thus, the revenue is maximized at: s∗uo =

√ p ω/ θ + γ, 22

and all the analyses regarding patient satisfaction presented in Section 3.3 also holds for clinic’s revenue maximization. In conclusion, government intervention is not needed in this scenario.

4.4

Model CO

Where there is a cap on the possible number of patients in the waiting room, and having more dissatisfied patients lowers the arrival rates, the clinic maximizes λ(1 − πc ), where λ should be obtained by solving equation (7). However, since obtaining the closed-form of the arrival rate is not possible, we use numerical examples to realize if regulation is required. For this purpose, six cases based on different values of γ, ω and θ are defined to investigate the impact of each of these parameters on s∗co and srco . Since necessity of regulation is emphasized here, we focus only on ∆s = s∗co − srco . Considering high and low levels of all parameters, the pairs {•l , •h } for ω, γ, and θ are {0.5, 1}, {5, 20} and {5, 20}, respectively. Note that since λ is related to patient satisfaction, γ and θ affect the revenue maximization decision of the clinic. The difference between γ and θ are set to higher values to highlight the difference between s∗co and srco . The cases are introduced in Table 8 and the observations are summarized in Table 9. The results appear in Table 13 in Appendix B. Table 8: Numerical cases for regulatory implications under scenario CO Cases

Parameters ω

γ

θ

Case 1

1

5

5

Case 2

1

5

20

Case 3

1

20

5

Case 4

0.5

5

5

Case 5

0.5

5

20

Case 6

0.5

20

5

Table 9: Impacts of θ, γ and ω on ∆s in Model CO Parameters

θ

γ

ω

∆s

+, %

+, &

+, %

In all of the cases, the average visit time to maximize patient satisfaction is equal or greater

23

than the average visit time that maximizes the clinic’s revenue. The necessity of regulation is more significant where θ and ω are larger. However, if the patients are, on average, very sensitive to expected wait time, the best social scenario is to reduce visit time; and consequently, the social optimum gets closer to revenue maximization policy. Similar to all other scenarios where we conclude that regulation may be needed, a minimum visit time is one simple-to-implement regulation form. Alternatively, as was the case for Model CM, regulating the number of admitted patients at any instant of time may be a potential method to increase patient satisfaction in a revenue-maximizing clinic. For this purpose, the cases presented in Table 4 are utilized and impact of c on s∗co and srco in Model CO is depicted in Figure 2.

24

0.23 0.36

0.26

0.36

s

s

s

0.23 0.24

0.35

0.23

0.22 0.35 0

10

20

30

40

0

a: ω=1,α=10,γ=4,θ=4

10

20

30

40

0

b: ω=1,α=10,γ=4,θ=15

10

20

30

40

c: ω=1,α=10,γ=15,θ=4 0.24

0.45

0.4

0.24

0.4

s

s

s

0.5

0.3 0.23 0.35 0.2 0

10

20

30

40

0

d: ω=1,α=0,γ=4,θ=4

10

20

30

0

40

10

20

30

40

f: ω=1,α=0,γ=15,θ=4

e: ω=1,α=0,γ=4,θ=15 0.17

0.25 0.16 0.25

s

s

s

0.16 0.16

0.24 0.16

0.15 0.24

0

10

20

30

40

0

10

20

30

0

40

h:

g: ω=0.5,α=10,γ=4,θ=4

10

20

30

40

i: ω=0.5,α=10,γ=15,θ=4

ω=0.5,α=10,γ=4,θ=15 0.4 0.32 0.17 0.3

s

s

s

0.3 0.17

0.28 0.16 0.2

0.26

0.16 0

10

20

30

40

0

10

20

30

40

0

10

20

30

40

c

c

c

j: ω=0.5,α=0,γ=4,θ=4

k: ω=0.5,α=0,γ=4,θ=15

l: ω=0.5,α=0,γ=15,θ=4

Figure 2: Comparing optimum visit times of Model CO, with data presented in Table 4: s∗co : ( (

), srco :

)

In all cases, s∗co starts from a higher point that srco . However, by increasing the capacity of the system, both visit time values converge to the same value, confirming that increasing the number of admitted patients in the clinic at any instant of time is an assuring method to align the revenue maximization goal with maximizing patient satisfaction. In cases 2, 5, 8 and 11 where θ is large comparing to γ, the point where s∗co and srco match the same value occurs for higher values of c,

25

comparing to the other cases where θ ≤ γ. Therefore, confirming that setting a minimum capacity for the clinic is an effective way to save patient satisfaction, it is shown that for higher sensitivity to doctor’s interpersonal skills, the minimum required capacity is larger.

5

Conclusion

This paper studies the problem of government intervention for the sake of improving patient satisfaction in walk-in clinics. By investigating the literature on patient satisfaction in walk-in clinics and emergency departments, we identified several factors that significantly affect overall satisfaction. Among these top factors, doctors’ interpersonal skills (translated into visit time) and waiting time are selected and the reason to select them is justified. Doctors’ interpersonal skills are reflected in the clarity and comprehensiveness of information conveyed by the doctor. Patient waiting time depends on the amount of time the doctors spend with each patient (i.e., visit time). Since total waiting time and doctors’ performance conflict with each other, and they may affect the revenue of the clinic, optimizing and regulating visit time is the topic of discussion in this paper. By contrasting the revenue maximization policies with patient satisfaction maximization policies, the cases where government intervention may be needed to avoid compromising patient satisfaction were identified. We considered four scenarios based on the capacity of the clinic and the relation between satisfaction and clinic recommendation likelihood, which could be seen as the presence of competition. For all these scenarios, the patient satisfaction function was maximized. It is concluded that higher sensitivity to visit time leads to higher patient satisfaction, unless there is local competition and arrival rate is relatively small compared to clinic’s capacity. For this specific case, patient’s sensitivity to visit time does not matter. Furthermore, higher sensitivity to waiting time means lower patient satisfaction at all cases. Next, we investigated the revenue maximization policies and showed that when the patients’ arrival rate is low relative to the clinic’s capacity (Models UM and UO), the revenue maximization policy is in line with patient satisfaction maximization policy; therefore, government intervention is not required. Furthermore, through multiple numerical examples, it was observed that revenue maximization policies lead to higher average visit times if the clinic’s capacity is small relative

26

to the arrival rate of patients (Models CM and CO). Thus, a regulation in the form of minimum required visit time is a potential way to reduce compromising patient satisfaction to generate higher revenue. Extending the analysis from visit times to clinic’s capacity, we discussed that how modifying the cap on the number of admitted patients at any time can close the gap between the patient satisfaction maximizer and revenue maximizer visit times. More specifically, we concluded that if the clinic faces local competition and also it cannot admit all patients, the government can put a lower limit on the number of admitted patients to align patient satisfaction and revenue maximization goals. However, we emphasized that due to controllability problem, clinic’s capacity may not be regulable. Finally, it was demonstrated that where the arrival rate of patients is large relatively to the clinic’s capacity, higher patient sensitivity to visit time makes the government intervention more vital. The research has some limitations which can be considered for future studies. First, the model can be analyzed by real data obtained from surveys, which will confirm the findings of this paper by investigating the suggestions made in the article. Also, in the proposed models the sensitivity values are considered as average values. However, it can be the case that heterogeneity of the patients questions the robustness of the results obtained in this research. Therefore, stochastic or robust models can be used to analyze the impact of heterogeneity in sensitivity on the best service time. A possible extension of this work is looking at the difference between perception of wait time and actual wait time. In this sense, perception comes from two sources: 1) learning from the past experiences, which is related to actual wait time; 2) expectation of wait time from queue size. The latter, however, requires simulation analysis for validation. Conducting a thorough empirical study on the impact of all of the qualitative factors (such as friendliness of the clinic’s staff members) on the patients’ perception of the wait time is an inspiring future research direction. Furthermore, although generalizing the arrival and visit time beyond what is discussed in this paper makes the problem intractable [48], and while appendix A demonstrates that such relaxation does not change the discussed intuitions, it is mathematically interesting to study such general queues. 27

CRediT author statement Mostafa Pazoki: Conceptualization, Methodology, Formal Analysis, Writing Original Draft, Writing Review & Editing, Visualization Hamed Samarghandi: Methodology, Formal Analysis, Writing Original Draft, Writing Review & Editing, Funding Acquisition

28

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31

Appendix A

Why M/M/1?

The main purpose of this paper is to identify the cases where government intervention in terms of regulating visit time is required, in order to balance patient satisfaction and clinic’s profitability. For this matter, four scenarios based on the impact of patient satisfaction on patient arrival, and system’s capacity are designed, where inter-arrival and service times are exponentially distributed. We claim that, the important factor in determining the necessity of regulation is the structure of the system, rather than the assumptions about inter-arrival and service time’s distributions. To prove this claim, we compare the results of Scenario UM (M/M/1 queue) with the result of an M/G/1 queue. In a clinic modeled as an M/G/1 queue, average patient satisfaction is

Puma = θs − λWq ,

(8)

where subscript uma refers to the alternative modeling of scenario UM for an M/G/1 system. Let s and σ 2 be the average and variance of the service time. In an M/G/1 system, the waiting time (Wq ) is: λ(s2 + σ 2 ) . 2(1 − λs)

(9)

Substituting (9) in (8) and taking the second derivative result: ∂ 2 Puma 3γλ2 3γλ3 (2s(1 − λs) + λ(σ 2 + s2 )) = − − . ∂s2 (1 − λs)2 (1 − λs)4

(10)

Knowing that 1 − λs > 1 in a stable queuing system, a meticulous observer notices that (10) is negative, concluding that (8) is concave and continuous. Therefore, first order condition can be used to obtain the optimum service (visit) time:

∂Puma = 0 ⇒ s∗uma = ∂s

1−

q

γλ(1+λ2 σ 2 ) θ

λ

For M/M/1 queue, we discussed that since patient satisfaction is the main goal, regulation is not required. With the same token, the same is concluded for an M/G/1 system. Further, observe the similarity between s∗uma and s∗um where s∗um

=

1−

q λ

32

γ θ+γ

.

Both s∗uma and s∗um increase in θ and decrease in λ and γ. Therefore, what is modified by shifting from M/M/1 to M/G/1 is only the exact form of the optimum visit time; neither their behavior towards changing the parameters nor the regulation implications are changed. Moreover, it is shown that a more general system G/G/1 can be also approximated by an M/M/1 system, known as Kingman’s formula: E(WqG/G/1 ) = WqM/M/1

c2a + c2s , 2

where ca and cs are coefficient of variation for inter-arrival and service times, respectively. Thus, M/M/1 is a good approximator of M/G/1, G/G/1 and G/M/1 if the exact values are not required for the analysis. Then, by inductive reasoning, we conclude that considering exponential inter-arrival and service times is sufficient for the purpose of this research, extending the results of this reasoning to CM, UO and CO scenarios.

B

Reported Results

All reported results are rounded to 3 or 5 decimal points.

Table 10: Results of numerical examples for Model CM Parameters

s∗cm

P (s∗cm )

4

0.049

0.115

4

15

0.107

1.004

0

15

4

0.018

0.039

Case 4

10

4

4

0.046

0.111

Case 5

10

4

15

0.084

0.862

Case 6

10

15

4

0.018

0.039

Cases

α

γ

θ

Case 1

0

4

Case 2

0

Case 3

33

Table 11: Result of numerical examples for Model CO Parameters

s∗cm

P (s∗cm )

4

0.355

0.829

4

15

0.239

2.36

10

15

4

0.229

0.486

1

0

4

4

0.358

0.831

Case 5

1

0

4

15

0.262

2.454

Case 6

1

0

15

4

0.229

0.486

Case 7

0.5

10

4

4

0.25

1.171

Case 8

0.5

10

4

15

0.165

3.3

Case 9

0.5

10

15

4

0.162

0.687

Case 10

0.5

0

4

4

0.253

1.175

Case 11

0.5

0

4

15

0.186

3.471

Case 12

0.5

0

15

4

0.162

0.687

Cases

ω

α

γ

θ

Case 1

1

10

4

Case 2

1

10

Case 3

1

Case 4

Table 12: Optimum average visit time and clinic’s revenue for scenario CM Parameters

s∗cm

srcm

Π(srcm )

4

0.058

0.008

5

4

15

0.108

0.008

5

5

15

4

0.022

0.008

5

Case 4

15

4

4

0.019

0.001

15

Case 5

15

4

15

0.035

0.001

15

Case 6

15

15

4

0.007

0.001

15

Cases

λ

γ

θ

Case 1

5

4

Case 2

5

Case 3

Table 13: Optimum average visit time and clinic’s revenue for scenario CO Parameters

s∗co

srco

Π(srco )

5

0.31628

0.31627

0.927

5

20

0.20256

0.20140

2.767

1

20

5

0.20000

0.20000

0.528

Case 4

0.5

5

5

0.22363

0.22363

1.31

Case 5

0.5

5

20

0.14276

0.14198

3.910

Case 6

0.5

20

5

0.14142

0.14142

0.746

Cases

ω

γ

θ

Case 1

1

5

Case 2

1

Case 3

34