Abstract
This paper develops a sequential appointment algorithm considering walkin patients. In practice, the scheduler assigns an appointment time for each callin patient before the call ends, and the appointment time cannot be changed once it is set. Each patient has a certain probability of being a noshow patient on the day of appointment. The objective is to determine the optimal booking number of patients and the optimal scheduling time for each patient to maximize the revenue of all the arriving patients minus the expenses of waiting time and overtime. Based on the assumption that the service time is exponentially distributed, this paper proves that the objective function is convex. A sufficient condition under which the profit function is unimodal is provided. The numerical results indicate that the proposed algorithm outperforms all the commonly used heuristics, lowering the instances of noshows, and walkin patients can improve the service efficiency and bring more profits to the clinic. It is also noted that the potential appointment is an effective alternative to mitigate noshow phenomenon.
1. Introduction
It is routine practice for a clinic patient to make an appointment before seeing a doctor. An effective appointment system provides a balance between revenue and cost in terms of the patients’ waiting and clinic overtime. Appointment scheduling problems have received the attention of operations research (OR) scientists since Bailey’s initial work in 1952 [1], but the problem of long waiting times still exists in the outpatient clinic. The noshow behavior of patients has been identified as a key factor resulting in low efficiency in many field study papers. Attempts to cut down on noshows by reminders, telephones, or emails play an important role in mitigating noshow phenomenon, but it cannot eliminate the negative effects entirely. A revolutionary appointment system named the openaccess system has been introduced as an alternative method to reduce noshow behavior by giving some patients appointments on the day they call.
Cayirli and Veral [2] provide a comprehensive review of previous studies and classify the environmental factors influencing the problem formulation, commonly used performance measures, the appointment rules, and the research methodologies. Gupta and Denton [3] provide stateoftheart, new challenges and opportunities for primary care appointment scheduling, specialty clinic appointment scheduling, and elective surgery appointment scheduling. The previous studies can be viewed in three categories. The first category is the appointment capacity planning level, which determines the optimal number of patients to be seen to maximize the total utility or revenue (Kim and Giachetti [4]; Qu et al. [5]; LaGanga and Lawrence [6]); the second category is the appointment scheduling level, which decides the optimal appointment time for each patient or the optimal interval length for a predetermined number of patients in a session (Kaandorp and Koole [7]; Hassin and Mendel [8]; Cayirli et al. [9]); the last category includes the joint decisions of capacity planning and scheduling to search for the optimal number and the corresponding schedule at the same time. (Muthuraman and Lawley [10]; Chakraborty et al. [11, 12]; Zeng et al. [13]; Turkcan et al. [14]). The sequential scheduling technique, which belongs to the last category, is developed in recent years to address overbooking and general stochastic service time.
The most relevant study to sequential appointment scheduling is as follows. Muthuraman and Lawley [10] propose a sequential scheduling approach with exponential service times (Kopach et al. [15] displayed the validity of this assumption) to maximize the revenue minus the cost of overflow and tardiness and prove that the objective function is unimodal. Chakraborty et al. [11] generalized Muthuraman and Lawley’s [10] model for general distributed service time and proved the unimodality of the profit function as well. Each accepted booking request is arranged in one of the predefined intervals in both papers, which is not required here. Gupta and Wang [16] model appointment sequence considering patients’ choice by a Markov decision process. The optimum threshold policies, after which future booking requests are rejected, are derived for single and multiple physicians under the assumption of deterministic service time; this paper includes exponential service time and walkin patients. Erdogan and Denton [17] formulate a twostage stochastic linear program model that minimizes the expected cost of waiting time, idleness, and overtime considering noshows for a given appointment sequence. The optimal solution is obtained by a standard Lshaped algorithm and the upper bounds are derived without calculating the revenue by seeing patients. Zeng et al. [13] demonstrate that the profit function of their sequential appointment scheduling problem for homogenous patients is multimodular, but the multimodularity is not preserved under scenarios of heterogeneous patients, and a local search heuristic is developed to find the local optimal schedules. Turkcan et al. [14] build a stochastic multiobjective model to account for the impact of the fairness measure on profit and presented a series of sequential search algorithms to find the Paretooptimal schedules. Different from Muthuraman and Lawley’s [10] model, Chakraborty et al. [12] design a sequential clinical scheduling method without predefined intervals, providing the patient with an exact appointment time before the call ends. The convexity of the total cost and unimodality of the net profit are derived under the assumption of exponentially distributed service time. Walkin patients are included in none of the above research. As pointed by Cayirli and Veral [2], walkin patients are prevalent for general practitioners who are responsible for the patients’ total care. It is also a common phenomenon for the large hospital in China.
All the previous studies use overbooking to reduce the negative effect of noshows without incorporating walkin patients. Based on previous work, this paper formulates a stochastic overbooking model for an openaccess clinic, which allows for moderate walkin probability on the appointed day. The numerical results demonstrate the correlation between the total profit and the walkin, noshow probability. Anticipating future arrivals can be viewed as an alternative method to reduce noshow. The paper is structured as follows. Section 2 presents the basic assumptions and the model formulation considering walkin patients. Section 3 establishes the theoretical proof that the cost function is convex under the assumption of exponential service time together with a sufficient condition under which the objective evolution is unimodal. Section 4 provides an illustrative example and parameter impact analysis. Section 5 concludes the main finding and provides future directions.
2. Assumptions and Model for the Sequential Appointment Problem with WalkIn Patients
2.1. Assumptions and Parameters
This paper develops a model for a sequential appointment scheduling problem in an openaccess clinic, deciding an appointment time for each booking request and a stopping criterion after which the clinic should stop accepting any requests. To provide consistent diagnoses to the patients, every doctor has their own waiting list in realistic situations. A singleserver queueing system is utilized to describe the appointment system. Some clinics allow the doctors to conclude the session after the examination of the final patient, but openaccess clinics require the doctor to remain available throughout the day to address walkin patients. It is assumed that the service time is exponentially distributed, and the mean is standardized at 1. Let stand for the length of the session, and each booking request will be given an appointment time in or rejected. represents the revenue from an individual patient, and the cost of waiting and overtime per unit of time are denoted by and , respectively. stands for the optimal appointment time assigned to the th callin patient, and is the optimal schedule after patients are arranged, where is the appointment time for the th scheduled patient, , and . The th scheduled patient is not identical to the th callin patient, as illustrated in Figure 1, which describes the scenario that the th callin patient is the th scheduled patient and the patient is attempted to be arranged in . This updated schedule is denoted by . It is assumed that the patients can be categorized into groups according to noshow probabilities, which can be inferred using statistics. Let be the showing probability of the th callin patient if he or she is of type and let be the showing probability at . gives the walkin probability, and it is assumed that . Cayirli et al. [9] propose a procedure to calculate the walkin probability based on historical data. The probability, represented by , that patients actually arrive at for the appointment can be determined from formula (1). The first line represents the probability of no arrivals, which is the joint probability of noshow and no walkin appointments. According to the assumption, it is easy to obtain from formula (1). This is reasonable because too many walkins bring large uncertainty in the daily workload, which is the main deficiency of openaccess appointment system. Consider
Let denote the number of patients in the system at before new arrivals; obviously, ; then . is a random variable subject to Poisson distribution, representing the number of patients seen during time . Define and . Consider
is the total expected arrivals after patients are scheduled; then . If , that is, the th callin patient is the th scheduled patient, can be calculated recursively based on as follows.
denotes the overtime after callins, denotes the waiting time at , and represents the total net profit of schedule , with denoting the total cost.
2.2. Model
The key point of modeling sequential appointment scheduling problem is the transition matrix from to . With the initial condition that , another can be iteratively derived from as follows.
Formula (3) means that when , the probability of patients in the waiting list (including the one in service) at before new arrivals comprises three terms: the first term is the case that patients remain waiting at before new arrivals, no new patients come at , and services are completed during ; subsequent term represents the joint probability that there are patients at , 1 new patient arrives at , and services are completed during ; likewise, the last term has the same implication except that there are 2 new arrivals at and service completion during . By the same means, one can obtain the probability when . Consider
The expected waiting time at after callins is calculated by
Similarly, the expected overtime after patients are scheduled is given by
The sequential appointment scheduling problem is to determine the optimal number of booking requests and how to schedule these patients in a session day towards maximizing the total net profit, which is the difference between the revenue and total cost: where .
Of course, when is fixed, the objective function is equivalent to minimizing the total cost as follows:
3. Sequential Appointment Scheduling Algorithm
3.1. Characteristics of the Objective Function
Proposition 1. The first patient is scheduled at time 0.
Proof. Suppose that the first patient is scheduled at time ; then formula (3) can be simplified as From (4), (5), and (8), the waiting time and overtime can be calculated as Differentiating the total cost so far with respect to , . Then the total cost is monotonically increasing with respect to , and, to minimize the total cost, the first patient is scheduled to time 0, .
Next, it is proved that the total cost is convex function for . Central to the proof are the firstorder and the secondorder derivatives of and . Firstly, it is shown that the waiting time at time is decreasing convex function of . Secondly, it is noted that the waiting time at time is increasing convex function of . Thirdly, it is proved that the waiting time at time () is increasing convex function of by induction. Finally, the overtime is demonstrated to be an increasing convex function of .
Theorem 2. For schedule , the cost function is convex of , if the patient is arranged at .
Proof. It is easy to get that satisfies , ; ; , . The system parameters corresponding to are , , , , , , and , for . Because the system states prior to are not influenced by the new arrival, the cost will not change. The cost function since (, ) is classified to four parts.
(1) The waiting time at time (, ) is monotonically decreasing convex function of .
For , from formula (3),
Likewise,
From (10), (11), and (12), it is simple to derive
Using (10) and (11), the firstorder and secondorder derivatives of are calculated by
Then, from formula (4), the result is established by
(2) The waiting time at time (, ) is monotonically increasing convex function of .
For , formula (3) is reduced to
Using formula (3), further result is expressed as
Accordingly, when ,
By (17), (18), and (19), it is noted that
Using (17), (18), and (19), the firstorder and secondorder derivatives of are given by
Again, from formula (4), the result is established by
(3) The waiting time at time , , is monotonically increasing convex function of .
First, the following two equations are established by induction:
For the base case, when , applying (3), (17), (18), and (19), it is found that
Thus, .
Combining equations (13) and (25),
According to the assumption, ; hence,
In the same way, , ;
Due to the limited space and tedious computation process, the concrete steps are omitted.
Suppose that (24) holds for ; the next step is the proof that they also hold for .
Consider ; from equations (3) and (17) and the hypothesis, it is easy to get
When , from and (17), it can be obtained that . Using (3) and (17) and the hypothesis, one can get
Differentiate with respect to ; then .
As for , similar to the previous discussion,
Differentiate on both sides; .
Equation (24) hold from induction.
Calculating the first and second derivatives of following equations (17) and (24),
Hence, from formula (4), the result is followed by
(4) The overtime is a monotonically increasing convex function with respect to .
From (3), , .
From formula (5), , .
All in all, the new generated cost function is convex with respect to .
Based on Proposition 1 and Theorem 2, the optimal can be easily found by any algorithm designed for convex optimization. For schedule , best are found when the booking request arrives. The optimal appointment time for this patient is the one with the lowest cost. The appointment time also maximizes the profit increase considering that the revenue per patient is fixed. The next thing to do is deciding when the scheduler should stop accepting new appointments. Theorem 3 provides a sufficient condition for rejecting new requests.
Theorem 3. If scheduling a patient who is sure to come lowers the profit, the clinic should reject all future appointments.
Proof. Define to denote the optimal updated schedule from after accepting a patient who is sure to show up for the appointment; represents the event that patients actually arrive at time .
If scheduling a patient who is sure to come lowers down the profit, which can be expressed as
then
This means that
It is equivalent to .
The results follow from the above inequality.
Theorem 3 provides a stopping criterion for rejecting new arrivals and a sufficient condition that the objective function is unimodal.
3.2. Sequential Appointment Scheduling Algorithm
Based on Proposition 1 and Theorems 2 and 3, each callin request is scheduled to a time to maximize the profit increase. If scheduling a patient who is sure to come or scheduling a patient of any type lowers the profit, the algorithm terminates. The concrete steps are as shown in Algorithm 1.

The proposed algorithm first initializes the input parameter, and the first patient is arranged at time 0 based on Proposition 1. Every new booking request is attempted to schedule to , ; the minimum in each is obtained based on the convexity of the cost function. is the selected time with minimum cost; if , then accept this patient and arrange him or her at ; otherwise, check whether set “testset” is empty. If , testify whether scheduling a patient who is sure to come lowers down the profit. The algorithm will terminate according to Theorem 3 if holds. Put into set “testset” until , if . The algorithm stops with any further acceptances lowering the profit.
4. Numerical Analysis
To demonstrate the effectiveness of the proposed algorithm, a randomly generated callin sequence is analyzed to display the trends of the objectives. The average performance of 1000 random sequences is used to compare with the commonly used heuristics in practice. The impacts of noshows and the walkin probability on the performance are investigated. The basic parameters for the numerical experiments are as follows: the length of the session is (typically, the clinic is open for 4 hours in the morning and the afternoon; the average service time is five minutes, standardized at 1), the revenue per patient is , and the waiting cost and overtime cost per unit of time are 0.2 and 1.5, respectively; (an overtime cost larger than the marginal revenue per patient avoids an infinite increase in patients; otherwise, the clinic will always benefit from serving additional patients through extending the working time.) Suppose that ; then the corresponding noshow probability is [0.5, 0.4, 0.3, 0.2]. It is assumed that each patient arrives independently, and walkin patients are independent of patients with appointments. The basic walkin probability is 0.25. The minimum in the interval is derived by the Matlab 7.8 (product of MathWorks Company, Natick, Massachusetts, USA) optimization solver “fmincon,” which implements the sequential quadratic programming algorithm.
4.1. Illustrating Example
Figure 2 presents the changing trend of the objective function and the appointment time allocated to each patient for a specific callin sequence. The abscissa represents the patient type in the callin sequence; for example, 4, 4, 1 means that the first three patients belong to types 4, 4, 1; the left ordinate represents the profit, and the right ordinate represents the appointment time. The curve displays the profit changes during the callin process. The dot is the appointment time provided by the algorithm for each booking request. Ten additional patients are used to display the trend of the profit after the stopping criterion is achieved. It is shown that the optimal schedule accepts 34 patients, and the profit function monotonically increases initially and decreases later. After the terminating criterion, the profit decreases quickly, and half of the 10 additional patients are arranged at the end of the clinic session. Intuitively speaking, it is of no use incorporating them into the schedule when the system is congested except increasing the waiting time of prescheduled patients. These extra patients are served in overtime when the system is overloaded. Theorem 3 provides sufficient conditions that the objective function is unimodal, but the algorithm stops if any type of patient lowers the profit according to the above procedure. 192 of 1000 random generated callin sequences satisfy the conditions in Theorem 3, and no better solutions are observed for the 10 more patients in the other sequences. The statistics of performance of interest are presented in Table 1, and it is noted that the proposed approach is very stable with small fluctuations. The average of 100 random generated callin sequences is adopted in a subsequent experiment.
4.2. Comparison with Commonly Used Heuristics
Cayirli et al. [18] sum up some appointment rules, including the IBFI (individual block/fixed interval), 2BEG (individual block, fixed interval rule with an initial block of two patients), OFFSET (individual block/variable interval rule, where initial patients are scheduled earlier, and the rest of the patients are scheduled later), DOME (individual block/variable interval rule, where initial patients are scheduled earlier, patients through are scheduled later, and the rest are scheduled earlier), 2BGDM (combination of the 2BEG and the DOME rules), MBFI (multiple block/fixed interval rule calls two patients at a time with the interval length equal to twice the mean service time), and MBDM (combination of the MBFI and DOME rules). These rules are listed in Table 2 to compare with the proposed policy. No patients are scheduled before the beginning of the session because unpunctual patients are not included. , , , , and are used for Cayirli’s rule, and the other parameters are identical to the basic case. The optimal scheduled number and the corresponding profit of all the heuristics are obtained to compare with the algorithm. The numerical results show that the heuristics accept more patients, but they lower the net expected profit significantly. IBFI performs best in all the heuristics considered, but it still leads to a fall in profit by 28%.
4.3. Impacts of NoShow Probability
The impact of noshows is analyzed from two aspects: the mean and the variance. Four groups of noshow probability are used to display the results:(1)impact of mean: [0.65, 0.55, 0.45, 0.35], [0.35, 0.25, 0.15, 0.05];(2)impact of variance: [0.4, 0.367, 0.333, 0.3], [0.65, 0.45, 0.25, 0.05].
Table 3 summarizes the performances for these four groups of the noshow probability. It is noted that a large noshow probability can incur more uncertainty, bringing about a decrease in profit. An interesting result which is contrary to the intuition is that when the mean of the noshow probability is fixed, a small fluctuation lowers the profit. The patients with large showup and noshow probabilities can bring more profits because of small uncertainty. That is to say, the clinic will benefit from reducing the average noshow probability and controlling the difference between the maximum and minimum noshow probabilities.
4.4. Impacts of Weight of Patient Type
This subsection investigates the influence of the weight of patient type on the total profit. The noshow probability is the same as in the basic case. The four types of weight are analyzed as follows: (1) 4 : 3 : 2 : 1; (2) 1 : 2 : 3 : 4; (3) 1 : 2 : 2 : 1; (4) 2 : 1 : 1 : 2.
Table 4 summarizes the statistics for these four groups of weight. Consistent with the previous results, (2) brings more profit than (1) by cutting down the weight given to patients with a high noshow probability. The average noshow probability of (3) and (4) is the same, but the large fluctuation produces more profits.
4.5. Impacts of WalkIn Patients
Table 5 displays the performances for six walkin probabilities: 0, 0.1, 0.2, 0.3, 0.4, and 0.5. It is observed that a large walkin probability with more uncertainty leads to a decrease in the profit. Walkin appointments are an alternative approach to compensate for noshows, but overbooking is more efficient. This implies that appointment scheduling is an effective approach to smooth the patient flow in the clinic; the administrator of the clinic should make some constraint on the proportion of walkin patients, allowing the clinic to provide diagnosis to more patients and gain more profits as well.
4.6. Impacts of Anticipating Future Arrivals
According to Theorem 2 and the experimental results about noshow probability, it is noted that the clinic can benefit from restricting the patients with large noshow probability. This motivates a heuristic that only patients with small noshow probability are accepted during the initial portion of the booking horizon and appointment requests with large noshow probability are considered only in the later portion when not enough future arrivals are anticipated. It is easy to see that the proof of Theorems 2 and 3 hold when future arrivals are considered; so the proposed algorithm can be implemented for this scenario. The numerical experiment generates a callin sequence including 80 patients: type 4 will be accepted for the first 20 patients, types 3 and 4 will be accepted for the next 20 patients, and so on. Figure 3 displays the evolution of objective for a given callin sequence denoted by the horizontal coordinate. 10 more patients are scheduled to illustrate the evolution of the profit after the stopping criteria are achieved. In this special case, it is seen that the optimal number of patients is 34 with the expected profit of 24.02. The statistic results of 100 randomly generated callin sequences show that 33.7 patients are scheduled on average with the mean profit of 24.01. Compared with performances from Table 1, it is noted that rejecting some patients with high noshow probabilities will schedule fewer patients but will increase the expected revenue in the meanwhile. This means that future arrivals can be also viewed as an alternative approach to substitute for patients with high noshow probabilities. If the clinic has enough potential booking requests, patients with large noshow probabilities are not accepted.
5. Conclusion
This paper develops a stochastic programming model for a sequential appointment scheduling problem with walkin patients. An algorithm terminating with the myopic optimal number of patients scheduled and the corresponding appointment time is proposed based on the convexity of cost function. A sufficient condition regarding the unimodality of the profit function is established. The numerical results show that this algorithm outperforms the commonly used heuristics in terms of the profit. Numerical results implicate that large noshow probabilities are key factors influencing the profit. Lowering down mean and variance of the noshow probabilities can bring extra profit for the clinic. Although walkin patients can fill in gaps resulted in by noshow, overbooking is preferable in terms of total profit. Rejecting patients with large noshow probabilities when the clinic has enough potential appointments is an alternative approach to mitigate the negative effects of noshow behaviors.
Future directions can introduce other types of variability in the appointment system. An immediate direction is the inclusion of generally distributed service time to see to what extent the previous conclusion holds. Future researches should take into account other behaviors such as patients’ preference, unpunctuality, and service interruption which also have significant impacts on the optimal schedule. As seen from this paper, reducing noshow behavior benefits both the clinic and patients. Intuitively speaking, the indirect waiting time between the call to make an appointment and the actual appointed day may influence the noshow probability. More accurate models should contain multiperiods to account for this effect quantitatively. It is noted that restricting walkin patients will increase the profit of the clinic, but several field studies favor the socalled openaccess appointment system, which provides sameday service to the patients. This paradox stimulates further studies to decide the length of time intervals left vacant for sameday appointments and when these intervals start simultaneously.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This paper is financially supported by the National Natural Science Foundation of China (NSFC Project 71021061).