Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2014 / Article

Research Article | Open Access

Volume 2014 |Article ID 564832 |

Chongjun Yan, Jiafu Tang, Bowen Jiang, "Sequential Appointment Scheduling Considering Walk-In Patients", Mathematical Problems in Engineering, vol. 2014, Article ID 564832, 12 pages, 2014.

Sequential Appointment Scheduling Considering Walk-In Patients

Academic Editor: Baocang Ding
Received26 Sep 2013
Accepted26 Feb 2014
Published06 Apr 2014


This paper develops a sequential appointment algorithm considering walk-in patients. In practice, the scheduler assigns an appointment time for each call-in 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 no-show 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 no-shows, and walk-in 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 no-show 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 no-show behavior of patients has been identified as a key factor resulting in low efficiency in many field study papers. Attempts to cut down on no-shows by reminders, telephones, or emails play an important role in mitigating no-show phenomenon, but it cannot eliminate the negative effects entirely. A revolutionary appointment system named the open-access system has been introduced as an alternative method to reduce no-show 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 state-of-the-art, 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 walk-in patients. Erdogan and Denton [17] formulate a two-stage stochastic linear program model that minimizes the expected cost of waiting time, idleness, and overtime considering no-shows for a given appointment sequence. The optimal solution is obtained by a standard L-shaped 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 Pareto-optimal 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. Walk-in patients are included in none of the above research. As pointed by Cayirli and Veral [2], walk-in 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 no-shows without incorporating walk-in patients. Based on previous work, this paper formulates a stochastic overbooking model for an open-access clinic, which allows for moderate walk-in probability on the appointed day. The numerical results demonstrate the correlation between the total profit and the walk-in, no-show probability. Anticipating future arrivals can be viewed as an alternative method to reduce no-show. The paper is structured as follows. Section 2 presents the basic assumptions and the model formulation considering walk-in 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 Walk-In Patients

2.1. Assumptions and Parameters

This paper develops a model for a sequential appointment scheduling problem in an open-access 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 single-server 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 open-access clinics require the doctor to remain available throughout the day to address walk-in 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 call-in 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 call-in patient, as illustrated in Figure 1, which describes the scenario that the th call-in 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 no-show probabilities, which can be inferred using statistics. Let be the showing probability of the th call-in patient if he or she is of type and let be the showing probability at . gives the walk-in probability, and it is assumed that . Cayirli et al. [9] propose a procedure to calculate the walk-in 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 no-show and no walk-in appointments. According to the assumption, it is easy to obtain from formula (1). This is reasonable because too many walk-ins bring large uncertainty in the daily workload, which is the main deficiency of open-access 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 call-in patient is the th scheduled patient, can be calculated recursively based on as follows.

denotes the overtime after call-ins, 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 call-ins 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 first-order and the second-order 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 first-order and second-order 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 first-order and second-order 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 call-in 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.

Step  1. , , , ;
Step  2. Suppose the n + 1st patient is of type ;
  If , reject the patient, go to Step  2;
   Else for , , compute
   , ,
   , ;
Step  3. If , accept the patient, ,
     , go to Step  2;
   Else reject this patient, go to Step  4;
Step  4. If , ;
   If , stop;
   Else go to Step  5;
Step  5. Scheduling a patient who is sure to come to
    , respectively, compute