Optimal appointment scheduling in continuous time: The lag order approximation method

We study appointment scheduling problems in continuous time. A finite number of clients are scheduled such that a function of the waiting time of clients, the idle time of the server, and the lateness of the schedule is minimized. The optimal schedule is notoriously hard to derive within reasonable computation times. Therefore, we develop the lag order approximation method, that sets the client’s optimal appointment time based on only a part of his predecessors. We show that a lag order of two, i.e., taking two predecessors into account, results in nearly optimal schedules within reasonable computation times. We illustrate our approximation method with an appointment scheduling problem in a CT-scan area.     

Outpatient appointment scheduling given individual day-dependent no-show predictions

This paper examines the combined use of predictive analytics, optimization, and overbooking to schedule outpatient appointments in the presence of no-shows. We tackle the problem of optimally overbooking appointments given no-show predictions that depend on the individual appointment characteristics and on the appointment day. The goal is maximizing the number of patients seen while minimizing waiting time and overtime. Our analysis leads to the definition of a near-optimal and simple heuristic which consists of giving same-day appointments to likely shows and future-day appointments to likely no-shows. We validate our findings by performing extensive simulation tests based on an empirical data set of nearly fifty thousand appointments from a real outpatient clinic. The results suggest that our heuristic can lead to a substantial increase in performance and that it should be preferred to open access under most parameter configurations. Our paper will be of great interest to practitioners who want to improve their clinic performance by using individual no-show predictions to guide appointment scheduling.     

A stochastic appointment scheduling system on multiple resources with dynamic call-in sequence and patient no-shows for an outpatient clinic

This research focuses on the stochastic assignment system motivated by outpatient clinics, especially the physical therapy in rehabilitation service. The aim of this research is to develop a stochastic overbooking model to enhance the service quality as well as to increase the utilization of multiple resources, like therapy equipment in a physical therapy room, with the consideration of patients’ call-in sequence. The schedule for a single-service period includes a fixed number of blocks of equal length. When patients call, they are assigned to an appointment time for that block, and an existing appointment is not allowed to be changed. In each visit, a patient might require more than one resource and a probability of no-show. Two estimation methods were proposed for the expected waiting and overtime cost with multiple resources: Convolution Estimation Method and Joint Cumulative Estimation Method for the upper and lower bound value; respectively. A numerical example based on a physical therapy room was used to show that this stochastic model was able to schedule patients for better profitability compared with traditional appointment systems based on four prioritization rules. The workload in each appointment slot was more balanced albeit more patients were assigned to the first slot to fill up the empty room.     

Improving appointment scheduling for medical screening

We investigate the optimization of appointment scheduling forbreast cancer screening, using the fact that a woman's attendanceprobability can be predicted. The methodology used applies tomedical screening in general. The results of the mathematical investigation presented in thispaper include a new formula for the cost of a screening session,a probabilistic model of rebooking appointments, a model ofattendance probability as a function of previous performance,and a heuristic cost optimization procedure. Breast Test Wales have improved efficiency by introducing heavilyoverbooked sessions for patients who are unlikely to attend.We use simulation modelling and insights from probability theoryto confirm the gain achieved by the Wales procedure and to assessthe further gain achievable by optimization of appointment scheduling.It is found that a gain in throughput of at least 10% can beobtained by optimizing appointment scheduling for screeningsessions, in particular by inviting patients in decreasing orderof attendance probability, and by overbooking near the end ofthe session. This avoids the need to set up dedicated sessionsfor poor attenders. Another possibility is to book patientswho change their appointment time, and who are therefore verylikely to attend, into dedicated sessions. The provision of appointment scheduling software with a built-insimulation and optimization module along the lines describedin this paper could enable radiographers to tailor appointmentscheduling for each area and so to schedule appointments veryefficiently.     

Matching daily healthcare provider capacity to demand in advanced access scheduling systems

Advanced access scheduling, introduced in the early 1990s, is reported to significantly improve the performance of outpatient clinics. The successful implementation of advanced access scheduling requires the match of daily healthcare provider capacity with patient demand. In this paper, for the first time a closed-form approach is presented to determine the optimal percentage of open-access appointments to match daily provider capacity to demand. This paper introduces the conditions for the optimal percentage of open-access appointments and the procedure to find the optimal percentage. Furthermore, the sensitivity of the optimal percentage of open-access appointments to provider capacity, no-show rates, and demand distribution is investigated. Our results demonstrate that the optimal percentage of open-access appointments mainly depends on the ratio of the average demand for open-access appointments to provider capacity and the ratio of the show-up rates for prescheduled and open-access appointments.     

Literature review on multi-appointment scheduling problems in hospitals

This paper presents a review of the literature on multi-appointment scheduling problems in hospitals. In these problems, patients need to sequentially visit multiple resource types in a hospital setting so they can receive treatment or be diagnosed. Therefore, each patient is assigned a specific path over a subset of the considered resources and each step needs to be scheduled. The main aim of these problems is to let each patient visit the resources in his or her subset within the allotted time to receive timely care. This is important because a delayed diagnosis or treatment may result in adverse health effects. Additionally, with multi-appointment scheduling, hospitals have the opportunity to augment patient satisfaction, allowing the patient to visit the hospital less frequently. To structure the growing body of literature in this field and aid researchers in the field, a classification scheme is proposed and used to classify the scientific work on multi-appointment scheduling in hospitals published before the end of 2017. The results show that multi-appointment scheduling problems are becoming increasingly popular. In fact, multi-appointment scheduling problems in hospitals are currently gaining progressively more momentum in the academic literature.     

Optimized pump scheduling in water distribution systems

A method based on nonlinear programming for determining the optimal operation of general water distribution systems containing multiple sources and reservoirs is presented. The problem is formulated and solved so that, given the forecasted demands for the coming 24 hours, the initial and final conditions in the reservoirs, the unit and maximum demand electricity charge, and the constraints in the hydraulic properties of all system components, an optimized pumping schedule is found. An optimization algorithm which employs the generalized reduced gradient method and the nonlinear sensitivity analysis has been developed for a basic scheduling problem in which only unit charges are considered. The maximum demand charge, which is weighted by varying degrees from day to day, is incorporated into the scheduling problem. The algorithm uses a feasible initial solution as the starting solution and iterates so that all the interim solutions are feasible.The work described here was undertaken at Brunel University and the University of Durham, UK. The authors are grateful to SERC for dunding the project and also thank Professor Uri Shamir, Department of Civil Engineering, Israel Institute of Technology for his review and many useful comments.     

Stochastic radiotherapy appointment scheduling

Central European Journal of Operations Research - When scheduling the starting times for treatment appointments of patients in hospitals or outpatient clinics such as radiotherapy centers,...     

Computing redundant resources for the resource constrained project scheduling problem

Several efficient lower bounds and time-bound adjustment methods for the resource constrained project scheduling problem (RCPSP) have recently been proposed. Some of them are based on redundant resources. In this paper we define redundant functions which are very useful for computing redundant resources. We also describe an algorithm for computing all maximal redundant functions. Once all these redundant functions have been determined, we have to identify those that are useful for bounding. Surprisingly, their number is reasonable even for large resource capacities, so a representative subset of them can be tabulated to be used efficiently. Computational results on classical RCPSP instances confirm their usefulness.     

Cyclical scheduling and multi-shift scheduling: Complexity and approximation algorithms

We consider the multiple shift scheduling problem modelled as a covering problem. Such problems are characterized by a constraint matrix that has, in every column, blocks of consecutive 1s, each corresponding to a shift. We focus on three types of shift scheduling problems classified according to the column structure in the constraint matrix: columns of consecutive 1s, columns of cyclical 1s, and columns of k consecutive blocks. In particular, the complexity of the cyclical scheduling problem, where the matrix satisfies the property of cyclical 1s in each column, was noted recently by Hochbaum and Tucker to be open. They further showed that the unit demand case is polynomially solvable. Here we extend this result to the uniform requirements case, and provide a 2-approximation algorithm for the non-uniform case. We also establish that the cyclical scheduling problem’s complexity is equivalent to that of the exact matching problem—a problem the complexity status of which is known to be randomized polynomial (RP). We then investigate the three types of shift scheduling problems and show that, while the consecutive ones version is polynomial and the k-block columns version is NP-hard (for k≥2), for the k-blocks problem we give a simple k-approximation algorithm, which is the first approximation algorithm determined for the problem.     

Active and stable project scheduling

The paper proposes a new classification of schedules for resource-constrained project scheduling problems with minimum and maximum time lags between project activities and regular and different types of nonregular objective functions. The feasible region of the scheduling problems represents a (generally disconnected) union of polytopes. In addition to the well-known concepts of active and semiactive schedules, pseudoactive and quasiactive as well as stable, semistable, pseudostable, and quasistable schedules are introduced. The (quasi-, pseudo-, semi-)active schedules are related to different types of left-shifts of sets of activities and correspond to minimal points of certain subsets of the feasible region. The (quasi-, pseudo-, semi-)stable schedules do not allow oppositely directed shifts and correspond to extreme points of certain subsets of the feasible region. The different sets of schedules contain optimal schedules for project scheduling problems which differ in their objective functions. The correspondence between those sets of schedules and vertices of specific polyhedral subsets of the feasible region can be exploited for analyzing schedule generation schemes which have been developed recently for finding solutions to the different classes of project scheduling problems.     

V-shop scheduling

This paper deals with V-shop scheduling where the route by which a job passes through the machines is: M₁ → M₂ → ? → M_m?1 → M_m → M_m?1 → ? → M₂ → M₁. Flowshop scheduling is a special case of V-shop. The two-machine, minimum schedule length V-shop scheduling problem is proved to be binary NP-complete. Efficient solvable algorithms are presented for some simple special cases of NP-complete V-shop problems; specifically n/2/V/C_max with a dominating machine and n/m/V, t_ij = 1/(C_max ∩ ΣC_i). n/m/V/C_max, m?2, with an increasing series of dominating machines is discussed.     

Outpatient appointment scheduling in presence of seasonal walk-ins

This study investigates appointment systems (AS), as combinations of access rules and appointment-scheduling rules, explicitly designed for dealing with walk-in seasonality. In terms of ‘access rules’, strategies are tested for adjusting capacity through intra-week, or monthly seasonality of walk-ins, or their combined effects. In terms of ‘appointment rules’, strategies are tested to determine which particular slots to double-book or leave open in cases where seasonal walk-in rates exceed or fall short of the overall yearly rate. In that regard, this study integrates capacity and appointment decisions, which are usually addressed in an isolated manner in previous studies. Simulation optimization is used to derive heuristic solutions to the appointment-scheduling problem, and the findings are compared in terms of in-clinic measures of patient wait time, physician idle time and overtime. The goal is to provide practical guidelines for healthcare practitioners on how to best design their AS when seasonal walk-ins exist.     

Stochastic dominance and parameter estimation: The case of symmetric stable distributions

Stochastic dominance rules can be applied to the selection of statistical estimators. This paper applies the procedure to estimators of location parameters of stable distributions: the mean and the median. It was found that the preference of one estimator over another depends on the characteristic exponent and on the sample size. Furthermore for some combinations of characteristic exponents and sample size we found that the stochastic dominance rule yields no preference implying that depending on one's utility function one may prefer the mean over the median or vise versa. This result differs from the common mean squared error criterion.     

A numerically stable algorithm for two server queue models

In this paper, we consider a queueing system in which there are two exponential servers, each having his own queue, and arriving customers will join the shorter queue. Based on the results given in Flatto and McKean, we rewrite the formula for the probability that there are exactlyk customers in each queue, wherek = 0, 1,…. This enables us to present an algorithm for computing these probabilities and then to find the joint distribution of the queue lengths in the system. A program and numerical examples are given.     

Two-day appointment scheduling with patient preferences and geometric arrivals

We consider an appointment system where the patients have preferences about the appointment days. A patient may be scheduled on one of the days that is acceptable to her, or be denied appointment. The patient may or may not show up at the appointed time. The net cost is a convex function of the actual number of patients served on a given day. We study the optimal scheduling policy that minimizes the long-run average cost and study its structural properties. We advocate an index policy, which is easy to implement, performs well in comparison with other heuristic policies, and is close to the optimal policy.     

Delay performance in stochastic processing networks with priority service

We evaluate the delay performance of an open multi-class stochastic processing network of multi-server resources with preemptive-resume priority service. We show that the stationary distribution of aggregate queue lengths has product form. For each service class we derive explicit expressions for the following stationary performance measures: The mean and, under feedforward routing, the Laplace transform of the delay distribution at each resource. We show that these measures are the same as if the resources were operating in isolation.     

Asymptotic scheduling

We deal with the following scheduling problem: a finite set of jobs is given and each job consists in the execution of an infinite number of tasks. A task is a sequence of operations and each operation requires a specific machine. A machine can process only one operation at a time and preemption is not allowed. Performance measures of the processing system involve fixing a time horizon T, counting the number of tasks completed within T for each job and maximizing a specified function of these numbers to estimate the throughput of the schedule. Whilst computing the throughput for a given T is in general an extremely difficult problem, it is shown in this paper that the limit, as T tends to infinity, of the average throughput (i.e. the throughput divided by T) can be easily computed via Linear Programming under fairly mild conditions. This quantity, which may be called the asymptotic throughput, can be used to assess a bound on performance measures of real systems. Buffers play a crucial role and buffer sizes can be taken care of in assessing the system performance. Mathematics Subject Classification (2000):90B35, 90C05, 90C27, 90C90     

Some Remarks on the Class of Continuous (Semi-) Strictly Quasiconvex Functions

We introduce the notion of variational (semi-) strict quasimonotonicity for a multivalued operator T  : X⇉X ^* relative to a nonempty subset A of X which is not necessarily included in the domain of T. We use this notion to characterize the subdifferentials of continuous (semi-) strictly quasiconvex functions. The proposed definition is a relaxation of the standard definition of (semi-) strict quasimonotonicity, the latter being appropriate only for operators with nonempty values. Thus, the derived results are extensions to the continuous case of the corresponding results for locally Lipschitz functions.