Performance of the smallest-variance-first rule in appointment sequencing

TL;DR

This paper analyzes the performance of the smallest-variance-first (SVF) rule in appointment scheduling, providing theoretical bounds and showing asymptotic optimality as the number of patients increases.

Contribution

It offers the first theoretical bounds on SVF's worst-case performance and proves its asymptotic optimality in appointment sequencing.

Findings

SVF has bounded worst-case ratio to optimal in various settings.

SVF's ratio approaches 1 as the number of patients grows large.

This is the first application of approximation ratio analysis in appointment scheduling.

Abstract

A classical problem in appointment scheduling, with applications in health care, concerns the determination of the patients' arrival times that minimize a cost function that is a weighted sum of mean waiting times and mean idle times. One aspect of this problem is the sequencing problem, which focuses on ordering the patients. We assess the performance of the smallest-variance-first (SVF) rule, which sequences patients in order of increasing variance of their service durations. While it was known that SVF is not always optimal, it has been widely observed that it performs well in practice and simulation. We provide a theoretical justification for this observation by proving, in various settings, quantitative worst-case bounds on the ratio between the cost incurred by the SVF rule and the minimum attainable cost. We also show that, in great generality, SVF is asymptotically optimal, i.e., the ratio approaches 1 as the number of patients grows large. While evaluating policies by considering an approximation ratio is a standard approach in many algorithmic settings, our results appear to be the first of this type in the appointment scheduling literature.

Figures

Tables

Table 1. Table 1 . . Summary of main results of Section  3

Theorem	Assumptions on service time distributions	Bound on ${\varrho }_{\omega }$
3.3	dilation ordering and symmetry	2
3.4	dilation ordering	4
3.8	normal distributions	$4\left(\sqrt{2}-1\right)\approx 1.66$
3.11	dilation ordering	between 2 and 4

Table 2. Table 2 . . Parameters of the lognormal distributions fitted by [ 7 ] .

Group	Mean	Standard deviation
Return	$15.50$	$5.038$
New	$19.09$	$6.85$

Table 3. Table 3 . . The values of K ​ ( B , ω ) 𝐾 𝐵 𝜔 K(B,\omega) and K ​ ( B , 1 2 ) 𝐾 𝐵 1 2 K\left(B,\frac{1}{2}\right) for some location-scale families.

Location-scale family	$K(B,\omega )$	$K(B,\frac{1}{2})$
Normal	See Theorem 5.3	$\sqrt{2\pi }\approx 2.51$
Uniform	$\frac{1}{1-\omega }\sqrt{\frac{2}{3\omega }}$	$\frac{4}{3}\sqrt{3}\approx 2.31$
Shifted exponential	$-\frac{\sqrt{2}}{\sqrt{\omega }\ln \left(\omega \right)}$	$2/\ln \left(2\right)\approx 2.89$
Laplace	$\frac{2\sqrt{\omega }}{\min \{\omega ,1-\omega \}\left(1-\ln \left(2\min \{\omega ,1-\omega \}\right)\right)}$	$2\sqrt{2}\approx 2.83$

Table 4. Table 4 . . The optimal sequence, optimal cost, cost of the SVF sequence, and approximation ratio ϱ 1 subscript italic-ϱ 1 \varrho_{1} when patient i 𝑖 i has an exponentially distributed service time with rate n + 1 − i 𝑛 1 𝑖 n+1-i , and mean-based schedules are used.

$n$	optimal sequence	optimal cost	cost for SVF	approximation ratio
3	1,2,3	0.2646	0.2646	1
4	1,2,3,4	0.3098	0.3098	1
5	2,1,3,4,5	0.3388	0.3389	1.0003
6	3,1,2,4,5,6	0.3588	0.3590	1.0007
7	4,2,1,3,5,6,7	0.3735	0.3739	1.0011
8	5,3,1,2,4,6,7,8	0.3847	0.3853	1.0014
9	6,4,2,1,3,5,7,8,9	0.3936	0.3943	1.0017
10	7,5,3,1,2,4,6,8,9,10	0.4008	0.4015	1.0019
11	8,5,3,1,2,4,6,7,9,10,11	0.4067	0.4076	1.0021

Table 5. Table 5 . . The optimal sequence, optimal cost, cost of the SVF sequence, and approximation ratio ϱ 1 subscript italic-ϱ 1 \varrho_{1} when patient i 𝑖 i has a service time that is a discrete approximation to a lognormal distribution with m i = ln ⁡ ( 50 ) + ln ⁡ ( i ) subscript 𝑚 𝑖 50 𝑖 m_{i}=\ln(50)+\ln(i) and s i = 0.33 subscript 𝑠 𝑖 0.33 s_{i}=0.33 , and mean-based schedules are used.

$n$	optimal sequence	optimal cost	cost for SVF	approximation ratio
3	1,2,3	18.0131	18.0131	1
4	2,1,3,4	32.6526	32.5799	1.0022
5	3,1,2,4,5	50.3769	50.1484	1.0046
6	4,2,1,3,5,6	70.8629	70.4453	1.0059
7	5,3,1,2,4,6,7	93.8700	93.2174	1.0070
8	6,4,2,1,3,5,7,8	119.210	118.302	1.0077
9	7,5,3,1,2,4,6,8,9	146.730	145.538	1.0082
10	8,5,3,1,2,4,6,7,9,10	176.305	174.803	1.0086

Equations

$$W_{i+1}=(W_i+B_{\tau (i)}-x_{\tau (i)}{)}^{+},I_{i+1}=(W_i+B_{\tau (i)}-x_{\tau (i)}{)}^{-},$$

$$C_{\omega }(\tau ,\mathbf{x}):=\omega \sum _{i=1}^n\mathbb{E}{I}_i+(1-\omega )\sum _{i=1}^n\mathbb{E}{W}_i,$$

$${\varrho }_{\omega }(\mathbf{B}):=\frac{C_{\omega }(\mathbf{B},\text{id},\mathbf{\mu })}{\min \{C_{\omega }(\mathbf{B},\tau ,\mathbf{\mu }):\tau \in {\mathsf{S}}_n\}}.$$

$$r_{\omega }(\mathbf{B},\mathbf{x}):=\frac{C_{\omega }(\mathbf{B},\text{id},\mathbf{x})}{\min \{C_{\omega }(\mathbf{B},\tau ,\mathbf{y}):\tau \in {\mathsf{S}}_n,\mathbf{y}\in {\mathbb{R}}_{+}^n\}}.$$

$$W_{k+1}=\max \left\{0,\max \left\{\sum _{i=j}^k{B}_{\tau (i)}-x_{\tau (i)}:j\in \{1,\dots ,k\}\right\}\right\}.$$

$$S_j:=\sum _{i=1}^j{X}_{\tau (k-i+1)}.$$

$$W_{k+1}=\max \{0,S_1,\dots ,S_k\}.$$

$$\sum _{i=1}^n{I}_i+\sum _{i=1}^n{B}_i=\sum _{i=1}^{n-1}{x}_{\tau (i)}+W_n+B_{\tau (n)},$$

$$B_i=\{\begin{array}{ll}{\mu }_i+\frac{1}{a_i}& \text{ with probability }{a}_i^2\\ {\mu }_i-\frac{1}{a_i}& \text{ with probability }{a}_i^2\\ {\mu }_i& \text{ with probability }1-2a_i^2,\end{array}$$

$$\omega \mathbb{E}{I}_2+(1-\omega )\mathbb{E}{W}_2=\omega \mathbb{E}(B_1-{\mu }_1{)}^{-}+(1-\omega )\mathbb{E}(B_1-{\mu }_1{)}^{+}=\omega a_1+(1-\omega )a_1=a_1.$$

$$\frac{1}{2}\mathbb{E}{\left(B_{\tau (1)}-x_{\tau (1)}\right)}^{-}+\frac{1}{2}\mathbb{E}{\left(B_{\tau (1)}-x_{\tau (1)}\right)}^{+}.$$

$$B_i=\{\begin{array}{ll}m+1& \text{with probability }m/(m+1)\\ 0& \text{with probability }1/(m+1),\end{array}$$

$${\varrho }_{\omega }=\frac{\omega \mathbb{E}{W}_n^{\text{scsvf}}+(1-\omega ){\sum }_{i=1}^n\mathbb{E}{W}_i^{\text{scsvf}}}{\omega \mathbb{E}{W}_n^{\text{scopt}}+(1-\omega ){\sum }_{i=1}^n\mathbb{E}{W}_i^{\text{scopt}}}.$$

$$\mathbb{E}{W}_{k+1}^{\text{scsvf}}/\mathbb{E}{W}_{k+1}^{\text{scopt}}⩽\mathbb{E}{W}_{k+1}^{\text{scsvf}}/\mathbb{E}{W}_{k+1}^{{\text{scopt}}^{\prime }}⩽{\varrho }^{\prime },$$

$$\mathbb{E}{W}_{k+1}⩽\mathbb{E}|S_k|=\mathbb{E}|X_{\tau (1)}+X_{\tau (2)}+\cdots +X_{\tau (k)}|.$$

$${\hat{S}}_i=\{\begin{array}{ll}{S}_i& \text{ if }i<T(a)\\ 2a-S_i& \text{ if }i⩾T(a).\end{array}$$

$$\mathbb{P}(W_{k+1}⩾a)=\mathbb{P}(S_k⩾a)+\mathbb{P}({\hat{S}}_k>a)⩽\mathbb{P}(S_k⩾a)+\mathbb{P}(S_k>a)⩽\mathbb{P}(|S_k|⩾a).$$

$$W_{k+1}⩾S_k^{+}=(X_{\tau (1)}+\cdots +X_{\tau (k)}{)}^{+}.$$

$$\mathbb{E}{W}_{k+1}^{{\text{scopt}}^{\prime }}⩾\mathbb{E}(X_1+\cdots +X_k{)}^{+}.$$

$$\mathbb{E}{W}_{k+1}^{\text{scsvf}}⩽\mathbb{E}|X_1+\cdots +X_k|=2\mathbb{E}(X_1+\cdots +X_k{)}^{+}⩽2\mathbb{E}{W}_{k+1}^{{\text{scopt}}^{\prime }}.$$

$$\mathbb{E}[X_i-X_i^{\prime }|X_i]=X_i,$$

$$\mathbb{E}{W}_{k+1}^{\text{scsvf}}$$

$$⩽2\mathbb{E}|X_1+\cdots +X_k|=4\mathbb{E}(X_1+\cdots +X_k{)}^{+}⩽4\mathbb{E}{W}_{k+1}^{{\text{scopt}}^{\prime }}.$$

$$W_{n+1}={\left(W_n+X_{\tau (n)}\right)}^{+},$$

$$\mathbb{E}{W}_{k+1}⩽\mathbb{E}{\left(X_{\tau (1)}+\cdots +X_{\tau (k)}\right)}^{+}+\mathbb{E}{\left(X_{\tau (1)}+\cdots +X_{\tau (k-1)}\right)}^{+}.$$

$$\displaystyle\mathbb{E}W_{k+1}\geqslant\frac{1}{2}\bigg{(}\mathbb{E}\left(X_{\tau(1)}+\cdots+X_{\tau(k)}\right)^{+}+\>\mathbb{E}\left(X_{\tau(1)}+\cdots+X_{\tau(\ell)}\right)^{+}\>$$

$$\displaystyle+\>\mathbb{E}\left(X_{\tau(\ell+1)}+\cdots+X_{\tau(k)}\right)^{+}\bigg{)}.$$

$$\mathbb{E}{W}_{k+1}^{\text{scsvf}}⩽\frac{1}{\sqrt{2\pi }}\left({\Sigma }_k+{\Sigma }_{k-1}\right).$$

$$\mathbb{E}{W}_{k+1}⩾\frac{1}{2}\left({\tilde{\Sigma }}_k+{\tilde{\Sigma }}_{\ell }+\sqrt{{\tilde{\Sigma }}_k^2-{\tilde{\Sigma }}_{\ell }^2}\right).$$

$${\Sigma }_k+{\tilde{\Sigma }}_{\ell }+\sqrt{{\Sigma }_k^2-{\tilde{\Sigma }}_{\ell }^2}$$

$$\frac{1}{2}{\Sigma }_{k-1}^2⩽{\tilde{\Sigma }}_{\ell }^2⩽\frac{1}{2}{\Sigma }_{k-1}^2+{\sigma }_k^2.$$

Full text

Performance of the smallest-variance-first rule

in appointment sequencing

Madelon A. de Kemp, Michel Mandjes, Neil Olver

Abstract. A classical problem in appointment scheduling, with applications in health care, concerns the determination of the patients’ arrival times that minimize a cost function that is a weighted sum of mean waiting times and mean idle times. One aspect of this problem is the sequencing problem, which focuses on ordering the patients. We assess the performance of the smallest-variance-first (svf) rule, which sequences patients in order of increasing variance of their service durations. While it is known that svf is not always optimal, it has been widely observed that it performs well in practice and simulation. We provide a theoretical justification for this observation by proving, in various settings, quantitative worst-case bounds on the ratio between the cost incurred by the svf rule and the minimum attainable cost. We also show that, in great generality, svf is asymptotically optimal, i.e., the ratio approaches 1 as the number of patients grows large. While evaluating policies by considering an approximation ratio is a standard approach in many algorithmic settings, our results appear to be the first of this type in the appointment scheduling literature.

Subject classification. Health care: appointment scheduling. Scheduling: stochastic appointment sequencing. Optimization: approximation algorithm.

Area of review. Stochastic Models.

Affiliations. M. de Kemp and M. Mandjes are with Korteweg-de Vries Institute, University of Amsterdam. N. Olver is with the Department of Econometrics & Operations Research, Vrije Universiteit Amsterdam, and is also affiliated with CWI, Amsterdam. The research for this paper is partly funded by the NWO Gravitation Programme Networks, Grant Number 024.002.003 (de Kemp, Mandjes) and NWO Vidi grant 016.Vidi.189.087 (Olver).

1. Introduction

Setting up appointment schedules plays an important role in health care and various other domains. The main challenge lies in efficiently running the system, but at the same time providing the customers an acceptable level of service. The service level can be expressed in terms of the waiting times the customers are facing, and the system efficiency in terms of the service provider’s idle time. The problem of generating an optimal schedule is generally formulated as minimizing a cost function (or simply “cost”) that is a weighted average of the expected idle time and the expected waiting times. As most literature on this topic focuses on applications in health care, we refer throughout this paper to customers as patients, and to the server as the doctor.

The problem of scheduling appointments can be split into two parts: one needs to determine the amount of time scheduled for each appointment, and one needs to determine in which order the patients should arrive. These problems are usually referred to as the scheduling problem and sequencing problem, respectively. This paper will primarily focus on the sequencing problem (but also includes results on the combined sequencing and scheduling problem), in a context with a single doctor seeing a sequence of patients. We impose the common assumptions that the service times of the patients form a sequence of independent random variables, while they arrive punctually at the scheduled times (which we will refer to as epochs). In this setting, a variety of techniques is available that determines for a given order the optimal arrival epochs; see, e.g., [3, 26] and references therein. However, much less is known about the efficient computation of “good” sequences. Already for a relatively modest number of patients, the number of possible sequences is huge, thus seriously complicating the search for an optimal order. An appointment scheduling review paper from 2017 [2] states that the optimal sequencing problem is one of the main open problems in the area:

“[…] one of the biggest challenges for future research is to find optimal (or near-optimal) solutions to more realistic appointment sequencing problems.”

A number of papers consider the sequencing (or combined sequencing and scheduling) problem and develop various stochastic programming models for it [4, 11, 28, 30]. However, the resulting optimization problems are very difficult to solve. Variants of the problem have been shown to be NP-hard [25, 30], indicating that this difficulty is inherent.

In a popular alternative approach one considers simple heuristics for the sequencing problem. The most frequently used heuristic is to order the patients by the variance of their service times, from smallest to largest. Throughout this paper we refer to this sequence as the svf (smallest-variance-first) sequence. The intuition for using the svf sequence is that an unusually long service time in the beginning could cause many later patients to have to wait, and the svf sequence aims to reduce the risk of this occurring. An additional appeal lies in the fact that svf is simple to implement, as it only requires knowledge of the variances of the service times. Simulation experiments have revealed that the svf rule typically performs remarkably well. In some cases it can be formally shown to be optimal; for instance (imposing some distributional assumptions) in the case of two patients [15, 40]. Recently, however, Kong et al. [25] provided instances showing svf need not be optimal, even for cases involving relatively simple service times (e.g., uniform or lognormal). Despite the svf sequence appearing promising in simulations, little is known about its theoretical performance, or of any other simple heuristic for that matter.

In this paper, we propose a new direction of research for the sequencing problem: finding sequences that provably perform well. Instead of finding an optimal sequence, such research aims at finding performance bounds on easily-computed sequences. Considering previous research, the svf sequence is the obvious candidate for such an easily-computed and well-performing sequence, and will therefore be our focus. The precise quantity of interest to us will be the ratio of the cost of the schedule coming from the svf sequence, and the cost of the schedule coming from the optimal sequence.

Our main goal in this paper is to prove upper bounds on this ratio – known as the approximation ratio – in various settings. This direction of study is very standard in the algorithmic community when considering intractable (NP-hard) problems, for example in machine scheduling (see [17, 18, 36] and references therein). However, it has not been studied in the appointment sequencing context. Note that for typical problem instances the svf sequence could perform significantly better than suggested by an upper bound on the approximation ratio, as the bound must also hold for worst-case instances.

1.1. Main contributions

In the first part of the paper we concentrate exclusively on the effect of the sequence, using the simplest choice of schedule: each patient is assigned a slot of length equal to its mean service time. In other words, the arrival time of any patient is set equal to the sum of the mean service times of all preceding patients. This is certainly not the optimal solution to the scheduling problem, in the sense of minimizing the cost function introduced above, but it has the advantage of being very simple and easily applicable, and also completely independent of the choice of tradeoff in the cost function between doctor idle time and patient waiting time. Owing to these attractive properties, this “mean-based” type of schedule is a commonly used approach in practice, as was stated in, e.g., the survey paper [2] and in [13].

We start, in Section 2, by arguing that without any restrictions on the service-time distributions, no bound on the approximation ratio of svf is possible, both under mean-based schedules and under optimally-spaced schedules (i.e., schedules in which the arrival epochs are chosen so as to minimize the cost function). We do so by constructing an example involving only two patients, but in which the service-time distributions are rather artificial. Then we present the ordering assumption on the service-time distributions that will be required for most of our results. Importantly, a large class of distributions meets this assumption, including e.g. the exponential distribution, but also the lognormal distributions that are frequently used in the health care context (such as the ones identified in [7]).

For the mean-based scheduling rule, we have performed an extensive set of numerical experiments for exponential and lognormal service time distributions, to gain insight into the performance of the svf rule for typical problem instances. These experiments indicate that the svf sequence performs very well for these distributions, with approximation ratios being below 1.01. However, an example at the start of Section 3 indicates that there are instances that are covered by the ordering assumption where the approximation ratio is 1.52, implying that it is not possible to prove a better bound than 1.52 if this assumption is in place.

Section 3 focuses on the mean-based scheduling rule. Under the ordering assumption we prove that the approximation ratio of svf is at most 2 for symmetric service-time distributions, and at most 4 in general. In other words, we show that for all instances (i.e., for all numbers of patients and all service-time distributions satisfying the assumption imposed) the svf cost is at most four times the optimal cost. We also consider two special cases:

• •

Service times are evidently nonnegative, but one could consider the situation that normal distributions are used as an approximation of the actual distributions of service times. In Section 3.2, we prove that then the approximation ratio is at most $4(\sqrt{2}-1)\approx 1.6569$. While we do not believe that our result here is sharp, it indicates that the performance of svf for well-behaved service-time distributions is most likely substantially better than suggested by the bounds 2 and 4 mentioned above.

• •

In Section 3.3 we bridge the gap between the upper bound of 2 for symmetric distributions and the general upper bound of 4, by developing a method that isolates the effect of asymmetry. For the lognormal distributions fitted to real data in [7], this method results in an approximation ratio of at most 3.43.

The problem of finding the optimal sequence becomes increasingly difficult as the number of patients grows larger, and thus the need for considering heuristics such as the svf rule is more important when a substantial number of patients is involved. In Section 4, we show for mean-based schedules that as the number of patients grows large, the approximation ratio tends to 1. This result requires only a very weak assumption on the service-time distributions (the ordering assumption is not needed here). The important practical implication of this result is that svf is close to optimal in settings where the number of patients is substantial.

In Section 5 we shift our attention to the performance of svf when optimally-spaced schedules (i.e., with arrival epochs minimizing the cost function), rather than mean-based schedules, are being used. In this setting, our numerical experiments indicate that the svf sequence tends to be optimal for exponential and lognormal service time distributions; we do however present an example satisfying the ordering assumption in which svf is no longer optimal.

We proceed by proving bounds on the approximation ratio of the svf rule for the combined sequencing and scheduling problem. Here, we wish to compare a heuristic for this combined problem to the overall optimal schedule, over all possible sequences and schedules. Observe that the simple mean-based scheduling rule may in general lead to high cost, because waiting times could easily propagate. We therefore consider a simple alternative scheduling rule, suggested by Charnetski [9]: the slot assigned to a patient is equal to its mean service time, plus some multiple $\alpha$ of the standard deviation of its service time (where this $\alpha$ is optimized). Again under some assumptions, we show that this scheduling rule, combined with the svf sequencing rule, yields a cost that is (relative to the optimal cost) off by at most a constant factor. Because of its frequent use in health care contexts [7, 23], we pay special attention to the case of lognormally distributed service times. Using a slightly different scheduling heuristic (the interarrival time being a multiple of the mean service time), we find an upper bound on the approximation ratio. Applying this result to the data in Çayırlı et al. [7], we find an upper bound of 2.90 in the case that in the cost function the waiting and idle times are equally important.

We finish Section 5 by giving an example to demonstrate that for optimally-spaced schedules the svf sequence is no longer asymptotically optimal as the number of patients tends to infinity, in contrast to the result for mean-based schedules that was presented in Section 4.

1.2. Further related work

We proceed by providing an account of the related literature, without attempting to give a full overview; for more extensive reviews on the appointment scheduling and sequencing literature, we refer the reader to, e.g., Ahmadi-Javid et al. [2], Çayırlı and Veral [6], and Gupta and Denton [16].

As we mentioned, Kong et al. [25] gave examples showing that svf is not in general optimal. In some very specific cases, optimality of svf has been demonstrated. For only two patients, the svf sequence is optimal when the service times are both exponentially distributed or both uniformly distributed [40], or more generally, when the two service times are comparable according to a certain convex ordering [15]. For three patients, Kong et al. [25] find sufficient conditions for the svf sequence to be optimal, when the time scheduled for each appointment is equal to the mean service time. (We have verified that this result can be extended to four patients using the same methods.)

Kemper et al. [20] analyze a sequential optimization approach, meaning that the arrival time of a patient is optimized without taking into account its impact on later patients. They show that under this rather different notion of optimality, and if the service times come from the same scale-family, then svf does provide the best ordering.

One line of research focuses on comparing various sequencing heuristics (including svf) through simulation. Denton et al. [12] consider a model similar to ours, and discuss the effectiveness of a number of simple sequencing heuristics using simulation, based on real surgery data. The svf heuristic performed best of all the heuristics they considered. Mak et al. [28] consider a model where waiting time costs may be different for different patients; by relying on tractable approximations, they also find that svf performs well. Klassen and Rohleder [23] and Rohleder and Klassen [34] consider an appointment scheduling model where not all patient information is known in advance; rather, patients must be scheduled as they call in to make an appointment (and so without information about patients who call later). Once again, it was empirically found that it worked best to put patients with low-variance service times early in the schedule.

A number of papers model variants of the combined sequencing and scheduling problem as stochastic integer or linear programming problems. Solving these programs is very challenging however, and generally exact results were only obtained for small instances. Works along these lines include Denton and Gupta [11], Mancilla and Storer [30] and Berg et al. [4]. For larger instances, it was necessary to resort to heuristics such as svf for the sequencing problem. We mention Vanden Bosch and Dietz [38] who propose instead a local search heuristic to iteratively improve the sequence by finding pairs of patients who can be swapped to reduce the cost.

There are also a number of papers which take a robust optimization approach [24, 29, 31]. Here, instead of working with explicitly given service-time distributions, the goal is to find a schedule minimizing the worst-case expected cost given only that the distributions meet certain constraints (such as certain given moments). The main advantage of robust optimization is that only the constraints are needed rather than full distributional information. Note the difference with our approach: robust optimization minimizes the cost of solution for the worst-case distributions satisfying the constraints, while our approach bounds the approximation ratio. The bound on the approximation ratio indicates the performance compared to optimal for any problem instance. In contrast, a robust optimization solution might not be anywhere near optimal for a typical problem instance, but will be good for the worst-case instances. Robust optimization results and bounds on approximation ratios could thus be seen as complementary results.

Among the works on robust optimization, that by Mak et al. [29] is most relevant to us, as they prove (under mild assumptions) that in their framework svf is optimal. In their setup, the joint distribution of the service times could be any distribution matching known moments for individual service times (e.g., means and variances). However, the worst-case distributions corresponding to the optimal interarrival times are typically highly correlated; these results do not carry over to a setup in which independence is assumed. Kong et al. [24] consider a model in which not only the means and variances but also the covariances are specified. However, only the scheduling problem is discussed, and sequencing is not taken into account. Mittal et al. [31] discuss another robust model, in which each service time can take any value in a given interval. They observe that a $(1+\epsilon)$-approximation algorithm for the combined scheduling and sequencing problem can be obtained, for any $\epsilon>0$.

Finally, we would like to point out the relation with the area of machine scheduling (see the book by Pinedo [32] for more background). The main difference between machine scheduling and appointment scheduling is that in the former the arrival times of the jobs (being the patients in our setting) are given, while in the latter these are decision variables. The machine scheduling problem that is most closely related to the setup that we consider, can be found in Guda et al. [14]. In the problem considered there, the due dates and sequence of jobs are to be decided, in order to minimize a weighted average of expected earliness and tardiness around the due dates. It is shown that the svf rule is optimal, under a specific assumption on the service times of the jobs. It should be noted, though, that in the model considered in Guda et al. [14] all jobs are present from the start, implying that there is no idle time. Compared to our model, this greatly simplifies the evaluation of the cost function, thus facilitating finding an optimal solution.

2. Model and preliminaries

Throughout this paper we consider a problem instance with $n$ patients, numbered 1 up to $n$. We denote the service time of patient $i$ in this problem instance by $B_{i}$, which has mean $\mu_{i}$ and variance $\sigma_{i}^{2}$; we assume that $B_{1},\ldots,B_{n}$ are independent random variables.

As pointed out in the introduction, one should distinguish between the scheduling problem and the sequencing problem. The sequencing problem, on which we primarily focus, is to decide which patient is assigned which appointment slot (given a certain scheduling rule). The sequence is denoted by a permutation $\tau\in\mathsf{S}_{n}$ (where $\mathsf{S}_{n}$ denotes the set of all permutations on $\{1,\dots,n\}$). The value $\tau(i)$ will denote the index of the patient that is assigned to appointment slot $i$. The scheduling problem is to decide the interarrival times between patients, given the sequence in which they arrive. We use $x_{j}$ to denote the interarrival time between patient $j$ and the next patient, i.e., the length of the appointment slot reserved for patient $j$. The vector $\bm{x}=(x_{1},x_{2},\ldots,x_{n})$ will be referred to as the schedule.

2.1. Performance metrics

We proceed by introducing the cost function we work with in this paper, which is based on the patients’ waiting times and the doctor’s idle times. Let $W_{i}$ denote the waiting time of the patient in appointment slot $i$. Let $I_{i}$ be the idle time before the start of appointment slot $i$ after the previous patient has been served. Given a sequence $\tau$ and interarrival times $x_{i}$, the waiting times and idle times satisfy the Lindley recursions [27]:

[TABLE]

where we use the compact notation $x^{+}:=\max\{0,x\}$ and $x^{-}:=\max\{0,-x\}$.

We use a parameter $\omega\in(0,1)$ to indicate the relative importance of idle time and waiting time. As a cost function, we seek to minimize

[TABLE]

a weighted average of the expected total idle time and expected total waiting time. Observe that this cost function depends on the sequence $\tau$, on the schedule $\bm{x}$, and on the patient service-time distributions $\bm{B}=(B_{1},\ldots,B_{n})$. We generally suppress the dependence on $\bm{B}$, but we may write $C_{\omega}(\bm{B},\tau,\bm{x})$ if we wish to be explicit. As an aside, we mention that an approach to estimate $\omega$ in a practical context can be found in [33].

Throughout this paper, without loss of generality we let the patients be indexed such that $\sigma_{1}^{2}\leqslant\sigma_{2}^{2}\leqslant\dots\leqslant\sigma_{n}^{2}$. The svf sequence is then the sequence given by the identity permutation id given by $\textnormal{id}(i)=i$, which we compare with the optimal sequence, i.e., the sequence that minimizes (2). To compare these sequences, we study the ratio between the cost functions under the svf sequence and the optimal sequence. If this ratio is close to 1, then this is evidence that the svf sequence performs well.

In this paper we specifically focus on two settings. In the first place we consider the performance of svf for mean-based schedules, which are given by $\bm{x}=\bm{\mu}$. We then consider the approximation ratio

[TABLE]

We will write just $\varrho_{\omega}$ when the service-time distributions under consideration are unambiguous. In the second place, we consider the performance of svf for optimally-spaced schedules. In this context we compare the svf sequence id along with a given schedule $\bm{{x}}$ with the optimal combination of sequence and schedule (i.e., a sequencing $\tau$ and schedule ${\bm{{y}}}$ that minimize the cost function). This means that here we consider the approximation ratio

[TABLE]

Once again, we will omit $\bm{{x}}$ and $\bm{B}$ when their choice is unambiguous.

The main objective of this paper is to prove, under specific assumptions, upper bounds on $\varrho_{\omega}$ and $r_{\omega}$. Such an upper bound then guarantees that the svf sequence always has a cost function of at most such an upper bound times the optimal cost function. We also show, under a mild condition, that $\varrho_{\omega}(\bm{B})$ converges to 1 as the number of patients tends to infinity, thus proving that the svf sequence is asymptotically optimal when mean-based schedules are used.

Remark 2.1**.**

Service times are inherently nonnegative, but our framework (based on the Lindley recursions (1)) carries over to situations where the $B_{i}$ are allowed to take negative values. This might be useful if the true distributions of service times can be approximated using distributions that can take negative values (with some small probability), for example normal distributions. If such distributions that can take negative values form a good fit to the data in some application, the theoretical performance of the svf rule for these distributions gives a meaningful indication for the performance of the svf rule in this application.

2.2. Preliminaries

We need the following well-known results concerning the waiting and idle times. It follows by iterating the Lindley recursion (1) that $W_{k+1}$ is the maximum of a random walk with steps $B_{\tau(k)}-x_{\tau(k)},B_{\tau(k-1)}-x_{\tau(k-1)},\dots,B_{\tau(1)}-x_{\tau(1)}$, that is,

[TABLE]

In the setting of mean-based schedules $\bm{x}=\bm{\mu}$, we introduce the notation $X_{i}:=B_{i}-\mu_{i}$, and the random walk

[TABLE]

Note that $S_{j}$ also depends on $k$ and $\tau$, but we keep this dependence implicit to ease the notation. We then find for the mean-based schedule that

[TABLE]

Note that (1) implies $W_{i+1}-I_{i+1}=W_{i}+B_{\tau(i)}-x_{\tau(i)}$. Summing over $i$ we find the identity

[TABLE]

which corresponds to the total time until all $n$ patients have been served (the “session end time”). For a given schedule, this relation can be used to express the expected total idle time in terms of the expected waiting time of the last patient. Therefore, we can focus on the waiting times, and derive results for the idle time from (5).

The following example shows that, if one does not impose any conditions on the service-time distributions, one can construct examples in which the approximation ratio is unbounded.

Example 2.2**.**

Suppose we have two patients, and the service time of patient $i$ is given by

[TABLE]

for some values $\mu_{i}>0$ and $a_{i}\leqslant 1/\sqrt{2}$. Then $\mathbb{E}B_{i}=\mu_{i}$, and $\textnormal{Var}B_{i}=2$, and so either of the two possible sequences could be considered the svf sequence. We take the svf sequence to be given by $\tau(i)=i$; one can of course slightly perturb the distributions so that this is the unique svf ordering.

Suppose we use the mean-based schedule given by $\bm{x}=\bm{\mu}$. The cost function for the svf sequence $\tau(i)=i$ is then

[TABLE]

In the same way, the cost function for the other sequence is equal to $a_{2}$. If we take $a_{1}$ to be bigger than $a_{2}$, we conclude $\varrho_{\omega}=a_{1}/a_{2}$, which can be arbitrarily large. As a consequence, it is necessary to impose assumptions on the service-time distributions in order to bound $\varrho_{\omega}$. The construction can easily be extended to one corresponding to any larger number of patients, by introducing additional patients with deterministic service times. Therefore, this example also shows that some assumption is necessary for the asymptotic result in Section 4.

The example applies also when an optimally-spaced, rather than mean-based, scheduling rule is used. Fixing $\omega=\frac{1}{2}$, for two patients with service times as in (6), the cost function is

[TABLE]

By Lemma C.4, the $x_{\tau(1)}$ that minimizes this cost function is given by $x_{\tau(1)}=\mu_{\tau(1)}$, which is the mean of $B_{\tau(1)}$. So the situation is unchanged, and also for the optimal scheduling rule no bound on the approximation ratio can be found without imposing further assumptions.

Example 2.2 shows that it will be necessary to impose assumptions on the service-time distributions in order to be able to bound the approximation ratio. Notice that the three-point distributions used in the example can be considered as artificial, in the sense that they substantially differ from distributions that are frequently used in the context of health care. One would like to use an assumption under which the approximation ratio can be bounded, but which does not rule out relevant, commonly used service-time distributions.

Not all service time distributions might be sufficiently comparable through their variance alone. We will need the concepts of a convex ordering and dilation ordering on random variables. More background on the convex ordering, dilation ordering and related concepts can be found in [35].

Definition 2.3**.**

The random variable $A$ is said to be smaller in the convex order than the random variable $B$ if $\mathbb{E}\phi(A)\leqslant\mathbb{E}\phi(B)$ for all convex functions $\phi:\mathbb{R}\to\mathbb{R}$ for which the expectations exist. This will be denoted by $A\leqslant_{\text{\rm cx}}B$. If $A-\mathbb{E}A\leqslant_{\text{\rm cx}}B-\mathbb{E}B$, then $A$ is said to be smaller than $B$ in the dilation order, denoted as $A\leqslant_{\text{\rm dil}}B$.

Note that $A\leqslant_{\text{\rm cx}}B$ implies $A\leqslant_{\text{\rm dil}}B$, and $A\leqslant_{\text{\rm dil}}B$ implies $\textnormal{Var}A\leqslant\textnormal{Var}B$. Throughout this paper, unless stated otherwise, we will make the following assumption on the service time distributions.

Assumption 2.4** **(dilation ordering).

We have $B_{1}\leqslant_{\text{\rm dil}}B_{2}\leqslant_{\text{\rm dil}}\dots\leqslant_{\text{\rm dil}}B_{n}$.

We remark that this is the condition under which [15] proves optimality of svf for two patients. Note also that this assumption implies $\textnormal{Var}(B_{1})\leqslant\dots\leqslant\textnormal{Var}(B_{n})$. Examples of instances satisfying this assumption include all $B_{i}$ having exponential distributions (by Theorem 3.A.18 in [35]), and all $B_{i}$ having lognormal distributions such that both $\mathbb{E}[\ln B_{1}]\leqslant\dots\leqslant\mathbb{E}[\ln B_{n}]$ and $\textnormal{Var}(\ln B_{1})\leqslant\dots\leqslant\textnormal{Var}(\ln B_{n})$, as proved in Appendix D. The following lemma, taken from [35], is useful when checking whether given random variables satisfy a convex order.

Lemma 2.5**.**

The random variables $A$ and $B$ satisfy $A\leqslant_{\text{\rm cx}}B$ if and only if there exists a coupling $\widehat{A}\stackrel{{\scriptstyle\text{\rm d}}}{{=}}A$ and $\widehat{B}\stackrel{{\scriptstyle\text{\rm d}}}{{=}}B$ such that \mathbb{E}\big{[}\widehat{B}\big{|}\widehat{A}\,\big{]}=\widehat{A}.

3. Bounds on performance under mean-based schedules

In this section we study $\varrho_{\omega}$, the approximation ratio under the mean-based schedule. To gain some insight as to what approximation ratio we can expect for typical problem instances, we start by discussing the output of a series of numerical experiments, described in more detail in Appendix E. It should be kept in mind that, as finding the optimal sequences in principle requires comparing all $n!$ candidates and is therefore computationally very expensive, such experiments can only be done for relatively small values of $n$. We recall that this is also the main reason why we study approximation ratios: computing the optimal sequence for larger instances is prohibitively slow, which explains why it is useful to gain insight into the performance of heuristics.

We considered exponential and lognormal service-time distributions. The exponential distributions have the advantage of an efficiently computable cost function using the machinery developed in [39]. For exponentially distributed service times the main conclusion is that the svf sequence typically performs within $1\%$ of optimal ($\varrho_{\omega}\leqslant 1.01$). Lognormal distributions are more realistic from a practical perspective [7, 23]. No exact computational scheme being available, we resorted to a discrete-time approximation to evaluate the cost; see Appendix E for more background on the implementation. In the experiments we performed the approximation ratios were uniformly below 1.01.

The numerics discussed in the previous paragraph suggest that svf typically performs very well. However, the service-time distributions featuring in the next example lead to a much higher approximation ratio.

Example 3.1**.**

Suppose the service time of patient $i$ is given by

[TABLE]

for $i<n-1$, and $B_{n-1}\stackrel{{\scriptstyle\text{\rm d}}}{{=}}B_{n}\stackrel{{\scriptstyle\text{\rm d}}}{{=}}KB_{1}$, for some parameters $m$ and $K$. We set $\omega=1$. We compare the svf sequence with the sequence $\tau$ given by $\tau(1)=n-1$, $\tau(n-1)=1$ and $\tau(i)=i$ for $i\neq 1,n-1$. For $n=10$ patients, $m=10$ and $K=10$ this results on a lower bound of $1.29$ on the approximation ratio $\varrho_{1}$. We can, however, obtain considerably higher approximation ratios: for example for $n=50\,000$ patients, $m=500$ and $K=5\,000$ we find that the approximation ratio is larger than $1.52$. Further experiments revealed that similar examples for even bigger values of $n$ only marginally increase this lower bound. Note that this problem instance satisfies Assumption 2.4, the dilation ordering assumption. This means that it is impossible to prove a better bound on $\varrho_{\omega}$ than $1.52$ without imposing further assumptions.