choi s.-comparison of a branch-and-bound heuristic

7/25/2019 Choi S.-comparison of a Branch-And-bound Heuristic

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Comparison of a branch-and-bound heuristic, anewsvendor-based heuristic and periodic Bailey

rules for outpatients appointment schedulingsystemsSangdo (Sam) Choi1 and Amarnath (Andy) Banerjee2*1

Shenandoah University, Winchester, USA; and2

Texas A&M University, College Station, USA

Appointment-based service systems admit limited number of customers at a specic time interval to make serviceproviders more accessible by reducing customers waiting time and make the costly resources more productive.A traditional approach suggests the Bailey rule, which assigns one or more customers at the initial block and onlyone customer at remaining blocks. We prescribe two heuristic approaches and variations of the traditional Baileyrule to appointment scheduling systems with the objective of minimizing total expected costs of delay and idletimes between blocks. The rst heuristic adopts a branch-and-bound approach using forward dynamic program-

ming and tries to fully enumerate with some restrictions. The second heuristic uses a sequential-inverse news-vendor approach using a starting solution. We conduct numerical tests, which show that both heuristics getnear-optimal solutions in a quicker time than a commercial solver, CPLEX and that the second approach givesnear-optimal solutions far faster than the rst approach. In addition, we suggest the use of aperiodicBailey rule,which can be implemented easily in practice, and provides a close solution to the best result of both heuristics,depending upon cost parameters and service-time variances.

Journal of the Operational Research Society(2016)67(4), 576592. doi:10.1057/jors.2015.79

Published online 4 November 2015

Keywords: health-care management; appointment scheduling; sequential inverse newsvendor; branch-and-boundheuristic; periodic Bailey rule

1. Introduction

Appointment-based service systems, such as care delivery

organization (CDO, eg, hospital, clinic), admit limited

number of customers at a specic time interval (eg, 30-min

time block) to make CDOs more accessible by reducing

crowding in waiting rooms and to effectively utilize costly

resources. CDOs that do not make their outpatient depart-

ments more cost effective may not be able to remain in good

standing nancially in the fast-growing health-care industry

(Cayirli and Veral, 2003). To keep patients waiting longer

than their expectation is undesirable on humanitarian

grounds (Gupta and Denton, 2008), since all patients require

timely care by CDOs. Many researchers (Bailey and Welch,

1952;Fries and Marathe, 1981;Ho and Lau, 1992;Ho et al,

1995; Klassen and Rohleder, 1996; Denton and Gupta,

2003; Robinson and Chen, 2003; Kaandorp and Koole,

2007; Begen and Queyranne, 2011) have proposed out-

patient appointment scheduling systems (OASys) for higher

utilization of resources and timely access of patients to

CDOs, that is, to minimize the sum of idle and waiting times.

A primary objective of an OASys is to nd an optimal

appointment rule for a particular set of performance measures

(eg, delay (or waiting) and idle times) in a CDO environ-

ment. OASys is trading off the interests of CDOs

(eg, minimizing idleness of doctors) and patients (eg, mini-

mizing delayed time of physicians service and/or waiting

time of patients) while matching demand and supply of

CDOs. An OASys can improve resource efciency by

smoothing workow in a CDO, and increase patient satisfac-

tion by providing timely access.

The operating rules for an OASys depend on the health-caresetting. In surgical CDO, service durations are longer and more

variable than primary CDO. Scheduling surgical appointment is

more complex than primary CDO (Gupta and Denton, 2008).

On the other side, scheduling surgeries assigns fewer numbers

of patients (eg, two or three) in a day. Patient service time in

surgical CDO tend to vary more depending on the patients

characteristics. The vast majority of patients of primary CDO

require a similar service that can be performed within axed

time length. OASys deals with a large number of patients

(eg, 50) in a day. We focus on decision factors (eg, block size,

*Correspondence: Amarnath Banerjee, Department of Industrial and Systems

Engineering, Texas A&M University, 3131 TAMUS, 4041 ETB, College

Station, TX 77843-3131, USA.

E-mail:[email protected]

Journal of the Operational Research Socie ty (2016) 67, 576592 2016 Operational Research Society Ltd. All rights reserved. 0160-5682/16

www.palgrave-journals.com/jors/
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block duration) under primary CDO setting in which all patients

are supposed to be homogeneous.

The major factors in designing OASys are block size

(ie, number of assigned patients), begin block, and block

duration (Cayirli and Veral, 2003). Figure 1 depicts two

combinations of block size and begin block with constant block

duration. The block of OASys is the minimal manageable time

unit, in which a certain number of patients (ie, block size) is

scheduled. The begin block is the number of patients at the rst

block in a day. The block duration is the interval between two

successive appointment times, also called job allowance. The

block duration can be divided in two veins: continuous and

discrete (Kaandorp and Koole, 2007). Discrete time increment

is in practice and more realistic than continuous one.

A continuous time increment is not typically found for a clinic,

because a real-valued time (eg, 10:48:23 AM) is not scheduled

in practice. A common practice is to use a function of the mean

of consultation times. For example, if the mean consultation

time is 8 min, block durations can be 20 min (ie, rounded up

from two times of 8 min) or 30 min (ie, rounded up from threetimes of 8 min). Hence, we concentrate on determining the

block size with a xed block interval, assuming that a block

duration is axed and discrete value such as 20 min, 30 min, or

an hour.

Block duration decisions are meaningful for time-tabling of

individual patients, that is, interval times between patients.

We consider an equal-length block duration, which allows

several patients in each block. CDOs tend to divide available

service time into equal-length time slots such that patients

needs can be accommodated in a standard appointment slot. For

certain types of visits that require more time, clinics may assign

multiple appointment slots (Gupta and Denton, 2008).

We assume that the number of blocks is given and xed,

because the total available service time is also given and xed.

Cayirli and Veral (2003)described seven appointment rules

by some combinations of block size, begin block, and block

duration: (1) single block, (2) individual block/xed interval,

(3) individual block/xed interval with an initial block, which is

the original Bailey rule (Bailey and Welch, 1952), (4) multiple

block/xed interval, (5) multiple block/xed interval with an

initial block, which is the generalized Bailey rule, (6) variable

block/xed interval, and (7) individual block/variable interval.

Figure 1 depicts two appointment rules: (5) and (6), respec-

tively. Since we assume equal-length time slots, we exclude the

combination of (7) from our study. We assume that several

patients are assigned into a single block, whereas we do not

assume that a single patient is assigned into a single block.

Hence, we do not consider the following combinations: (1), (2),

and (3). We compare our appointment rules with either (5) or

(6), assuming that (4) is a special case of either (5) or (6).

Bailey and Welch (1952) did the rst study to analyze an

individual-block appointment system at a time when most

hospitals were still using single-block systems in the 1950s.

They recommended that an individual-block/xed interval

appointment system with an initial block of two patients

(ie, (3) ofCayirli and Veral, 2003) lead to a reasonable balance

between patient-waiting and doctor idle times. We callt-Bailey,

which stands for traditional Bailey, if other blocks have one less

block size than the begin block (ie, (5) of Cayirli and Veral,

2003), which is not necessarily two patients. We prescribe two

heuristics to nd a near-optimal begin block and block size and

suggest periodic Bailey rules.

In addition, we propose variations of periodic Bailey rules by

modifying the results from two heuristics. A periodic Bailey

rule, which is a special case of (6) ofCayirli and Veral (2003),has several begin blocks and is repetitive, while t-Bailey has

one begin block. If intervals between initial-blocks are long, we

call p-Bailey-l, which stands for periodic Bailey with long

interval. If intervals are short, we name p-Bailey-s. For

example, consider 8 blocks with begin block of 4 and other

subsequent block sizes are 3 or 4. t-Bailey is 4-3-3-3-3-3-3-3.

If block sizes are 4-3-3-3-4-3-3-3, we callp-Bailey-l because the

interval between two begin blocks, which have 4 block size atrst

and fth blocks, respectively, is long. We call 4-3-4-3-4-3-4-3

p-Bailey-s because intervals between begin blocks are short.

In addition, we can consider p-Bailey-m, which has shorter

interval thanp-Bailey-l and longer interval than p-Bailey-s.

The objectives of the paper are: (1) an optimization model to

determine begin block and block size, (2) two heuristics to

solve the optimization model, (3) suggestions for using practical

p-Bailey rules, and (4) implications from extensive numerical

studies. The optimization model is a stochastic dynamic

programming (SDP) problem, which requires prohibitively

large computational times. We present an equivalent stochastic

integer programming (SIP) problem to utilize a commercial

solver such as CPLEX. We prescribe two heuristics to get near-

optimal solutions in a faster time. The two heuristics are

efcient because it takes a couple of minutes or less and a

couple of hundreds iterations or less for both heuristics to solve

most cases, while it takes more than 10 min and a couple of

Time

Block-size

Time

Block-size

a b

Figure 1 Two exemplar combinations of block size and begin block.(a)Multi block/xed interval with an initial block;(b)variable block/xed interval.

Sangdo Choi and Amarnath BanerjeeOutpatient appointment scheduling system 577


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patientow and found out that the source of the bottleneck is

the lack of availability of beds downstream in the care chain.

Lastly, our research belongs to the area of appointment

scheduling to determine the optimal sizes of each block

(Bailey and Welch, 1952;Fries and Marathe, 1981;Liaoet al,

1993; Dexter and Traub, 2002; Marcon et al, 2003; Chase,

2005;Gupta, 2007).Bailey and Welch (1952)used a manual

Monte-Carlo simulation technique in their search for the best

initial block and appointment interval for clinics with a variety

of number of patients to show that an initial block of two

patients leads to a reasonable balance between patient waiting

time and doctor idle time.Dexter and Traub (2002)used online

and off-line bin-packing techniques to plan elective cases and

evaluated their performances using simulation. Chase (2005)

developed a simulation model for a hospital to reduce wait

times, improve resource utilization, and determine the right

number of downstream resources needed. On the contrary, we

build a stochastic optimization model in order to take into

account uncertainty of service durations, and propose two

heuristics, both of which show near-optimal results numeri-cally. Fries and Marathe (1981) studied the variable-block/

xed-interval appointment system and compared the results

with single-block and multiple-block/xed-interval systems.

Liao et al(1993)constrained customer arrivals to xed lattice

of times with specied numbers of intervals and patients. Both

Fries and Marathe (1981)and Liao et al(1993)used dynamic

programming to fully search the optimal block sizes for the next

period given that the number of patients remaining to be

assigned is known. We mix a dynamic programming and

branch-and-bound approach by eliminating some states that

would not improve at the next stages. Marcon et al(2003)and

Gupta (2007) focused on determining how best to assign

arriving surgery requests to assist in the planning negotiation

among different stakeholders in a surgical suite. On the

contrary, we assume that all patients have the same consultation

distribution under primary CDO. In sum, whereas all of prior

studies try to achieve optimality, we emphasize both optimality

and practicality by proposing optimization models and prescrib-

ing effective and efcient heuristics.

3. Problem formulation

We propose a SDP model and its associated SIP model to

involve uncertainties of service durations. The dynamicapproach is able to involve all meaningful, possible combi-

nations of integer values. However, it is hard to evaluate the

objective function value by the integral operation for a

multi-dimensional space. Hence, we devise an equivalent

SIP model with a number of scenarios to approximate the

objective function value, and use it to compare the objective

function values.

This section comprises of two subsections. We address

notation and assumptions in the rst section, then explain a

prototypical SDP and its associated SIP in the next subsection.

3.1. Notation and assumptions

We employ the following index set, parameters, and random

variables:

k Index for blockkK

cd Delay time penalty

ci

Idle time penaltyh Unit block interval

p Individual random service time

Tk Random completion time up to blockk

Dk Random delay time of blockk

Ik Random idle time of blockk

xk Block size of blockk

We assume that all patients assigned in the same block arrive

on time for the aggregate-level planning purpose. We focus on

devising and evaluating appointment policies, notnding exact

scheduling times for individual patients. Hence, we do not

consider delay and idle time between individual patients in the

same block. We assume that all patients in the same block arriveon time. Consider two blocks and its associated block sizes are

x1 and x2, respectively. Individual idleness associated with the

rst patient in the second block is as follow:

E TI-px1 +

; (1)

where pk is the k-times sum of iid random variable p. There is

no idle time between patients in the same block, because we

assume all patients in the same block arrive on time. Individual

performance measure does not take into account individual idle

times within the same block. Individual waiting times are as

follows:

E p +E p2 + +E px1 - 1 +E px1 -T1

+ +E px1 - T1

+

+ph i

+ +E px1 -T1 +

+px2 - 1

+ 2+ + x1 - 1 +x2E px1-T1

+

++ 2+ + x2 - 1

x1 x1 - 1

2+

x2 x2 - 1

2 +x2E p

x1-T1

+

:

The rst-block individual waiting performance measure is a

polynomial function of block size, x1. The second-block

individual waiting performance measure is a function of block

size,x2 and block-based waiting time, px1 -T1+

: The block-based waiting performance has been countedx2times, whereas

the idleness performance measure, T1 -px1 + , has been

counted once. Mixing individual and block-based performance

measures will be intractable. If there is no patient waiting in the

waiting room when patients in certain block arrive on time, they

expect the rst-come-rst-served rule even though they arrive

on time and the rst checked-in patient is in service without any

delay. Other patients in the same block expect certain amounts

of delay since they recognize other patients are assigned in the

same block. However, if they observe some patients waiting



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from the previous block when they arrive, the unexpected

waiting times may cause a burden to patients. Patients expect

polynomial-type waiting times, such as (x1(x1 1)/2), whereas

Epx1 -T1+ is an unexpected waiting time. Block-based

delay time is more serious and relevant in our analysis.

We consider block-based performance measure rather than

individual patient-waiting or physician idle time. We take into

consideration delay time of the rst patient in each block except

the rst block as well as idle time associated with the last patient

in each block.

Since the main concern of this study is service time, we take

into consideration uncertainties of service durations. We deter-

mine policies for CDO operations at an aggregate level. Hence,

we do not take into consideration no-show for real-time

rescheduling. We assume that random service time for

individual patient is normally distributed as Belin and

Demuelemeester (2007),Belinet al(2009),van Houdenhoven

et al(2007),Hans et al(2008), andChoi and Wilhelm (2012,

2014) did. Total service time for each block is also normally

distributed, since the sum of the normal distributions is thenormal distribution. We assume that the mean of service times,

, is even larger than the standard deviation of service times,

such that |z |for any |z|. We useve levels of coefcient of

variations (CV), up to 0.5. The probability that a service time is

negative is P(z2)=0.0228, when CV=0.5. We regard this

probability as negligible.

We use a constant block interval h, which can be determined

by CDO administrative staffs. We use 30 min for numerical

tests in Section 5. The number of assigned patients in each

block is the primary decision variable in this study, while other

research studies (Ho and Lau, 1992; Ho et al, 1995; Ho and

Lau, 1999) determine block intervals with the same number of

assigned patients.

The primary decision variable in this paper is the number of

patients in blockk(ie, block size),xk. Block start time of blockk

is dened as (k 1)h.p is a random service time for individual

patient with mean ofand variance of2, and total service time

in blockkis the sum ofxkiidp, denoted bypxk;which has mean

ofxkand variance ofxk2. The sum ofxk1 +xk2 ps is the sum

of two random variables pxk1 and pxk2: Hence, the followingequation holds:

pxk1 + xk2 pxk1 +pxk2 : (2)

We dene the random completion time of blockk, Tkusing

the above Property (2) as follows:

T1 px1 (3)

Tk max Tk- 1; k- 1 hf g +pxk (4)

We penalize delay time Dk (Tk kh)+ and idle time

Ik (khTk)+ in the objective function. Especially, the delay

time of the last block can be represented by overtime of the day.

We describe the objective function and formulate the problem

in the next subsection.

3.2. Mathematical models

Wedene the objective function that minimizes the sum of total

expected delay and idle times between blocks. Delay and idle

times are dened recursively and the problem is formulated in a

dynamic fashion. The objective function associated with block

k, fk(x1,,xk), and overall objective function f|K|(X|K|), where

Xk= (x1,,xk), a vector form, can be expressed recursively ina stochastic dynamic fashion as follows:

SDP minfKj j XKj j

(5)

s:t:f1 X1 cdE T1 - h

+

+ ciE h -T1 +

(6)

fk+ 1 Xk+1 fk Xk + cdE max Tk; khf g +p

xk+1

- k+ 1 h+

+ ciE k+ 1 h - max Tk; khf g -pxk+1 +

2 k Kj j - 1: 7

The rst block problem, (6) is an inverse newsvendorproblem, which has been solved by Choi and Ketzenberg

(2014) explicitly. The problem min f1(X1) of (6) is

re-expressed as follows:

INV minx1

cdE px1 - h +

+ ciE h -px1 +

: (8)

The solution to the problem INV is given as follows (Choi

and Ketzenberg, 2014):

x*1 x1 orx1; (9)

wherex1 -z+

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiz22 +4h

p2

!2

; (10)

such that (z) = cd/(cd+ ci). All other objective function

values of the remaining blocks are hard to evaluate

analytically, because of a max operator such as

EmaxfTk; khg +pxk+1 - k+ 1h+ in (7).

To evaluate the objective function value of f|K|(X|K|), we

reformulate the problem SDP as a SIP model with a number of

scenarios. We generate a set of scenarios to represent random

events of service durations, p and associated random times Tk,

Dk, and Ik. We change subscripts and superscripts for these

random variables by adding the probability of each scenario,

q, where is a probability space for service durations.

For example, pki is the individual service duration of each

patienti belonging to blockkunder scenario. With re-dened

random variables, we reformulate the original problem SDP

into the following stochastic non-linear programme to approx-

imate the objective function value.

SIP minX

minD

k

minI

k

Xk2K

X2

qcdDk+ q

ciIk

n o (11)

s:t: Tk-1 +Xxki1

pki+I

k k*h +D

k k 2; 2 (12)

580 Journal of the Operational Research Society Vol. 67, No. 4


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IkD

k0 k2 K; 2 (13)

Dk; I

k; T

k 0 k2 K; 2 (14)

The objective function (11) minimizes total expected costs of

delay and idle times. Constraint (12) denes the delay and idle

times of blockkunder scenario . Constraint (13) ensures

that both delay and idleness do not occur in the same block toavoid inefciency. Constraint (13) is non-linear, and can be

linearized by introducing binary variables, IkandD

k;both of

which enforce the SIP to have thousands of binary variables if

the number of scenario is in thousands. The linearized formula-

tion associated with (13) forkK, , is given as follows:

Ik+D

k 1

IkM*I

k

Ik *I

k

D

kM*D

k

Dk *D

k;

where Mis a big number and is a small number. Constraint

(14) guarantees the non-negativity of all decision variables.

The termPxk

i1pki

of (12) should be changed because the

number of summation, xk,kKshould be xed. We reformu-

late (12) into the following with binary variable xki ; k2 K :XNi1

xkipi; k2 K

0xkN xkN- 1 xk2 xk1 1; k2 K

where N is a large number in order to cover all possibilities.

xki 1 if theith patient is assigned to blockk, 0 otherwise. Theith patient should be assigned before the i + 1st patient is

assigned.PN

i1xki xkholds forkK. For example, ifh =30and=8 min, we do not need to consider numbers greater than

6 for N, because 6 or more patients penalize delay times

severely (see Figure 2). We set N=6 for =8 min; N= 4,

= 15 and 20 min, respectively.

If a feasible solution X|K| = (x1,x2,,x|K|) is given, we get

the approximation of objective function valuef|K|(X|K|) using the

SIP model. In the following sections, we use SIP to evaluate

objective function values.

4. Solution approaches

One may solve SDP by either forward or backward induction.

However, it would take prohibitively long owing to the curse of

dimensionality as each block is added. In the next subsections,

we suggest two heuristic approaches to avoid the curse of

dimensionality. The rst approach is a constructive method

(CON), which starts with the rst block and adds next blocks

step by step. The CON approach tries to fully enumerate with

certain restrictions in order to nd near-optimal block sizes. The

second approach is an improving method (IMP), which starts

with a solution to a sequential-inverse newsvendor (SINV)

problem since the solution to the SINV problem provides a

lower limit. Both approaches provide similar objective function

values.

4.1. Preliminary numerical tests

In order to devise efcient methods, we conduct preliminary

numerical tests togure out how to search and evaluate feasible

solutions in a faster and efcient way. Let eitherbxc ordxe beone of the solutions to the problem INV. If we use some other

value, which is greater than dxe or smaller than bxc; theobjective function values at these values are far away from the

optimal.Figure 2shows an exemplar objective function, which

is decreasing-then-increasing, when=8 min,cd: c i=0.5:0.5,

and h= 30 min. The optimal value to the problem INV, x is

3.75, and the optimal block size should be either 3 bxc or4 dxe: The objective function values of the block sizes of1, 2, 5, and 6 are far greater than the block size of 4.

Table 1shows an exemplar case for two blocks with the same

parameter values ofFigure 2. The header row values are the

second block size, and the left-most column gives the rst block

size. The optimal block size up to the second block is (43). All

combinations with values that are greater than 4 dxe orsmaller than 3 bxc provide greater value than the optimalobjective function value (ie, 6.1) by more than 200%. Once the

rst two block sizes are far from the optimality, the block sizes

including these two cannot get close to the optimality as oneadds the third block.

Table 2 shows an exemplar case for three blocks with the

same parameter values ofTable 1andFigure 2. The header row

values are the third block size, and the left-most column gives

the rst and second block sizes. The optimal block sizes

up to the third block is (4-3-4), while (4-4-3) block sizes is

close to the optimal one. All combinations with values other

than 3bxcor 4 dxeare far greater than the optimal objectivefunction value. As we add more blocks at next stages, other

block sizes that do not include 3 or 4 cannot contribute towards

0

5

10

15

20

25

1 2 3 4 5 6

O

bjectivefunctionvalue

Block size

Figure 2 Objective function values for a single block for the caseof= 8 min, CV= 0.1, andc

d:c

i=0.5:0.5.



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minimizing the objective function value. Hence, we summarize

our observation regarding search space with bxc and dxe asfollows:

Observation 1 It is not necessary to search less thanbxc orgreater than dxe: One of the two values, bxc anddxe; isused for feasible solutions.

We use eitherbxc ordxe for block sizes based on Observa-tion 1. We show two extreme cases: all bxcs (Case I) anddxesfor all blocks (Case II):

Case I: (allbxcs): E[Dk] is non-increasing and E[Ik] isnon-decreasing because the high possibility of

idleness of previous blocks may not affect next

blocks. The block with idleness does not have an

inuence on the next blocks.

Case II: (all dxe): E[Dk] is increasing and E[Ik] is non-increasing because the high likelihood of delay of

previous blocks may affect next blocks. It is more

likely for delay to happen in later blocks, like

snow-balleffect.

For any general assignment of block sizes, if E[Dk] for

blockkis the worst (ie,E[Dk] E[Dk] orE[Ik], for any other

blockkK), bxc is a better block size for blockk+1 thandxe:IfE[Ik] for blockkis the worst, the block size of blockk+1 isdxe:

Table 3shows block sizes for 16 blocks and its associated

E[Dk] and E[Ik] with bxc 3 and dxe 4: For the rst twoblocks withdxe 4 (ie,k=1, 2),E[Dk] is increasing and E[Ik]is decreasing. For the blocks from 4 through 7 (9 through 11, 13

through 14), with dxe 4; E[Dk] is increasing and E[Ik] isdecreasing. For the blockk=14,E[Dk] is the worst and the next

block 15 has block size ofbxc 3:Hence, we summarize ourobservation as follows:

Observation 2 If the worst term of the objective function of

SIP, (11) is one of the delay penalties (ie, E[Dk]=max

{E[Dk],E[Ik]for kK}), the block-size of the next block

k+ 1 is bxc: Otherwise, the block-size of the next blockk+ 1isdxe:

The block k having the worst E[Dk] or E[Ik] should be

changed because it has the biggest room to improve. For the

later block sizes after block k, we use a partial series of

block sizes generated by SINV. We shift one block forward.For example, the current block sizes of (4-4-3-4/4-4-4-3/4-

4-4-3/4-4-3-4) in Table 3 has the worst E[Dk] at k= 14.

We replace block 14 with block 15, block 15 with block 16,

respectively. There is a shift of one block from block

15 onwards.

We use bxc for the last block, if E[D|K|]E[I|K|], dxeotherwise.Table 4shows an example of block sizes for blocks

14, 15, and 16 to improve the solution. All previous block sizes

remain the same. The new block sizes are (4-4-3-4/4-4-4-3/4-4-

4-3/4-3-4-3). Hence, we summarize our observation as follows:

Table 1 Objective function values for two blocks when= 8 min,CV= 0.1, andc

d:c

i=0.5:0.5

x1 x2

1 2 3 4 5 6

1 43.9 35.9 27.9 24.1 31.9 40.1

2 36.1 29.9 19.9 16.1 23.9 31.93 27.9 20.1 12.00 8.1 15.9 24.14 22.1 14.1 6.1* 6.3 14.2 22.25 21.9 14.1 14.1 21.9 30.1 37.96 22.1 21.9 30.1 37.9 46.1 53.9

Notes: The superscript * denotes the optimal value. Bold-faced numbers are

for combinations withx1and x2having value of either 3 or 4 only.

Table 2 Objective function values for three blocks when= 8 min,CV= 0.1, andcd:ci=0.5:0.5

(x1,x2) x3

1 2 3 4 5 6

(2, 3) 41.9 34.1 25.8 22.1 29.8 38.1(3, 2) 40.1 33.9 26.1 22.1 29.9 38.1(3, 3) 33.9 26.1 17.9 14.1 22.1 30.1(3, 4) 28.1 20.1 12.1 12.4 20.4 28.2(3, 5) 28.1 20.1 19.8 27.7 35.7 43.7(4, 2) 36.2 28.2 20.2 16.2 23.9 31.9(4, 3) 27.9 20.1 12.2 8.4* 16.2 24.1(4, 4) 24.2 16.2 8.8 12.1 20.4 28.7(4, 5) 24.2 16.8 20.6 28.7 36.2 44.2(5, 3) 32.1 16.9 16.9 19.7 27.7 35.7

Notes: The superscript * denotes the optimal value. Bold-faced numbers are

for combinations withx1and x2having value of either 3 or 4 only.

Table 3 An exemplarE[Dk] andE[Ik] values for 16 blocks when= 8 min, CV= 0.1, andc

d:c

i= 0.5:0.5

k xk E[Dk] E[Ik]

1 4 4.34 2.112 4 7.91 1.273 3 5.19 3.15

4 4 8.79 1.225 4 11.87 0.836 4 14.67 0.657 4 17.52 0.548 3 13.20 1.509 4 16.12 0.7610 4 18.89 0.5211 4 21.64 0.3912 3 17.11 1.2813 4 19.99 0.6114 4 22.61* 0.4515 3 18.17 1.3116 4 21.12 0.64

Note: The superscript * denotes the worstE[Dk].



8/17

Observation 3 It is a good and fast way to use a partial

series ofX from the current solution in order to improve

the current solution. Let k be the worst block having

E[Dk]=max{E[Dk], E[Ik] for k K} or E[Ik]=

max{E[Dk],E[Ik]for kK}. A series of block-sizes from

block k+ 1 will shift one block forward, ie, replace blocks

from k to |K| 1. The last block-size is based on compar-

ison of E[D|K|]and E[I|K|]. If E[D|K|]E[I|K|], the last block-

size isbxc:Otherwise,dxe:

We develop two heuristic methods by combining and

utilizing Observations 1, 2, and 3. The CON heuristic uses

Observation 1 to build the optimal block sizes by adding one

block as steps proceed. The IMP heuristic uses all three

observations to revise a current schedule.

4.2. A constructive heuristic approach

We prescribe a heuristic using forward dynamic program-

ming approach with restricted values. We dene astage a s

we add one more block. We use both dxeand bxc to searchthe solution using Observation 1. We denote the objective

function values associated with blocks from 1 through ka s

follows:

zsk min fk Xrk

r2 Rk; (15)

where Xkr= (x*1, ,x

*

k)r is a solution vector at the kth stage,

and Rkis an index set of all feasible solutions. The number

of total possible candidate block sizes is 2|X|, where |X| is the

number of assigned block sizes so far. We start with large

enough tolerance ratio, kand then decrease it gradually in

order that we search as many as possible at the beginning

stages to minimize the possibility of removing the optimal

appointment rule, and as few as possible at the later stages

to minimize the possibility of searching non-optimal

appointment rules. We determine decreasing tolerance

ratios experimentally. For h = 30, we use 5% up to Stage

10; 3% from Stage 11 to 12; 2% at Stage 13; 1.5% at Stage

14; 1% at Stage 15 for numerical tests. We suggest 10% for

the rst several blocks and 1% for the last couple of blocks.

Algorithm for the constructive heuristic. Initialization: Set

k= 0,R0= {}, |R0| =1, andto an initial value (eg, 10%) as a

tolerance. Solve the single-period inverse newsvendor pro-

blem to getx*.

Step 1: Solve |Rk| 2 problems using the following vec-

tors, where |Rk| is the number of all possible

combinations of feasible solutionsRk:

x*1; ; x*k

r dxef gr2 Rk

x*1; ; x*k

r bxcf gr2 Rk:

Rk+1= {1,, 2s}. Letzk+1s ,sRk+ 1be the objective function

value of problem min fk+ 1(Xk+ 1s ) whereXk+1

s= (x*1,,x*k+ 1)

andz*k+ 1=minszk+1s .

Step 2: If (|z*

k+1zk+1s

|/z*

k+1) k, remove s fromRk+ 1and re-index elements fromRk+ 1.

Step 3: If k= |K|, stop. Otherwise, k= k+1 and go to

Step 1.

A numerical example is provided in Section 5, in which

we compare the CON, IMP, and variations of periodic Bailey

rules to the optimal solution obtained by a commercial

solver, CPLEX.

4.3. An improving heuristic approach

The IMP approach requires a good initial solution. To get a

good one, we reformulate the objective function into the

problem ORG simply as follows:

ORG

minX

Xk2K

cdE Tk- kh +

+ ciE kh-Tk

+

: (16)

Next, we provide a heuristic to solve the problem SDP by

approximation. First, we nd the lower bound of the expected

delay time and the upper bound of the expected idle time. The

expected delay time, E[(Tk kh)+] has the following lower

bound:

E Tk- kh +

E px1 + +xk - kh

+

: (17)

The expected idle time,E[(khTk)+] has the following upper

bound:

E kh -Tk +

E kh-px1 + +xk

+

: (18)

Equations (17) and (18) can be proven by induction easily.

We dene revised penalty costs associated with

block k, denoted by cdk and cik as follows:

cdETk- kh+ cdkEpx1 + +xk - kh+ such that cd< cdk;

andciEkh -Tk+ cikEkh-px1 + +xk+ such thatci > cik:

Table 4 An exemplar revised last 3 block sizes among 16 blocks

k xk E[Dk] E[Ik]

14 3 18.17 1.3115 4 21.12 0.6416 3 14.65 1.88



9/17

We revise the problem ORG as following:

SINV minX

Xk2K

cdkE px1 + +xk - kh +

+ cikE kh-px1 + +xk +

: 19

Each term of SINV is an inverse newsvendor problem.The whole problem is a series of inverse newsvendor

problems. Hence, we name this problem as SINV. The

optimal number of patient x*kcan be expressed as in (20),

assuming that unit block interval is h and service time p , is

normally distributed:

Fp

x*1+ +x*

kh

cdk

cdk + cik zk

; (20)

where Fp

x*1+ +x*

k is the distribution function of px

*1+ +x*

k:

Proposition 1 establishes the exact representation of the

optimal solutionx*k,kK.

Proposition 1 (Choi and Ketzenberg, 2014) x*1 + +x*k is

expressed explicitly as follows:

x*1 + +x*kxk orxk; (21)

wherexk-zk+

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiz2k

2+ 4kh

q2

0@

1A

2

: (22)

Choi and Wilhelm (2014) show that sequential news-

vendor problem can be solved separately and independently if

k |zk|k.

Proposition 2 Let z and zkbe standard normal score values

that satisfy (z)= cd/(cd+ ci) andzk cdk=cdk + cik;

respectively. Then zzk

for kK.

Proof of Proposition 2 Since cd cdk and ci cik; thefollowing inequality holds:

cd

cd+ ci

cdk

cdk + cik: (23)

(z) is an increasing function. Hence, z zk.

Proposition 2 shows that solutions to the problem SINV

provide the lower limit. Each block size by SINV is eitherbxcordxe:Hence, we use Observation 1. Using the lowerlimit, we provide the heuristic as follows:

Algorithm for the improving heuristic. Initialization: Solve

SINV and get the optimal solutions X0= (x*1,,x*

|K|)0 to

SINV. Sets = 1 andX1=X0.

Step 1: Solve SIP with Xs to get the objective

function value.

Step 2: Set j= arg max{k: E[Dk],E[Ik]}.

Step 3: Update new solution as Xs+ 1= (x*1,,x*

j1,

x*

j+1,,x*

|K|,x*

n). IfE[D|K|]E[I|K|], setx*n bxc:

Otherwise, setx*n dxe:Step 4: Compute the objective function value using SIP

with the new solution,Xs+1.

Step 5: If f(Xs+ 1)f(Xs), stop. Otherwise, set s= s + 1

and go to Step 1.

5. Numerical study

We conduct numerical tests to compare the two heuristics by

considering a single-day scheduling scheme. We assume that

OASys schedules 8 h with 30-min time block (ie, 16 blocks a

day). One of the authors has experienced the same time block

whenever he made an appointment with a doctors ofce.

Given that the block duration is xed ash = 30 min, we show

three xed values for: 8, 15, and 20 min. As mean valueincreases, the possible block size (ie, bxc ordxe) decreasesstep-wise. For example, if = 4 min, the possible block size

is either 7 or 8; = 5 min, 5 or 6; = 6 min, 4 or 5,

respectively. Preliminary numerical tests show that one

patient is very likely to be allocated for all blocks, if

22 min, and that two patients are very likely to be

allocated for all blocks, if 13 18. We vary with ve

levels of variance, corresponding to ve levels of CV (0.1,

0.2, 0.3, 0.4, 0.5) so as to use a relative measure of varia-

tion independent of magnitude of . We consider nine

levels of cost ratio cd:c i= (0.01:0.99, 0.09:0.91, 0.37:0.63,

0.435:0.565, 0.5:0.5, 0.63:0.37, 0.91:0.09, 0.99:0.01) to

cover general cases, in each of which CDO administrators may

value delay (or waiting) and utilization differently. One cost is

100 times of the other when 0.01:0.99 or 0.99:0.01. One cost

is about 10 times of the other when 0.09:0.91 or 0.91:0.09.

One cost is 70% higher than the other when 0.37:63 or

0.63:0.37, equivalently 1:1.7 or 1.7:1. One cost is 30% higher

than the other when 0.435:0.565 or 0.565 or 0.435, equiva-

lently 1:1.3 or 1.3:1. Two costs have the same weight when

0.5:0.5 or equivalently 1:1. We change CV and cost ratio in

order to nd out the impact of variance and weight of delay

and idle time. We show the case for CV=0.1 and cd:

ci= 0.5:0.5 for numerical examples and explanations in the

next two subsections. In the last subsection, we analyze resultsof all cases.

We compare two heuristics (CON and IMP) with four

periodic Bailey rules. We prescribe four periodic Bailey rules:

t-Bailey, p-Bailey-l, p-Bailey-m, and p-Bailey-s, since the

Bailey rules are easy to implement.

5.1. A numerical example for the constructive heuristic

First, we build an appointment schedule by the CON heuristic

approach.Figure 3depicts the procedure up to Stage 6 using the



10/17

constructive method, which is similar to branch-and-bound

heuristics of mixed integer programming.

Initial step

k 0:R0

fg: R0

j j 1; 0:1:x* 3 o r 4:

Iteration 1

We solve two SIP problems with X1r, rR1= {1, 2} as

follows:

z11 min f1 X11

7:29 where X11 3 :

z21 minf1 X21

6:35 where X21 4 :

z*1 6:35:

Since (|z*1z11|/z*1)= 0.148, remover=1 andR1= {1}.

Iteration 2

We solve two SIP problems withX2r,rR2as follows:

z12 minf2 X12

13:3 where X12 4; 3 :

z22 minf2 X22

15:51 where X22 4; 4 :

z*213:3: Since jz*2 -z

22 j

z*2

0:166;

remove r2:R2 1f g:

Iteration 3

We solve two SIP problems withX3s ,rR3as follows:

z13 minf3 X13

20:7 where X13 4; 3; 3 :

z23 minf3 X23

21:5 where X23 4; 3; 4

z*320:6:R3 1; 2f g:

Iteration 4

We solve four SIP problems with X4r,rR4as follows:

z14 minf4 X14

27:6 where X14 4; 3; 3; 3 :

z24 minf4 X24

28:1 where X24 4; 3; 3; 4 :

z34 minf4 X34

29:2 where X34 4; 3; 4; 3 :

z44 minf4 X44

32:4 where X44 4; 3; 4; 4 :

z*427:6: Sincez*4 -z

r4

z*4

>0:1forr 3; 4;

remove r3; 4 andR4 1; 2f g:

Iteration 5

We solve four SIP problems with X5r, rR5as follows:

z15 minf5 X15

34:7 where X15 4; 3; 3; 3; 3 :

z25 minf5 X25

35:1 where X25 4; 3; 3; 3; 4 :

z35 minf5 X35

35:8 where X35 4; 3; 3; 4; 3 :

z45 minf5 X45 38:8 where X

45 4; 3; 3; 4; 4 :

z*534:7: Since j z*5 - z

4r

5 j

z*5>0:1;

remove r 4 andR5 1; 2; 3f g:

Iteration 6

We solve four SIP problems with X6r,rR6as follows:

z16 minf6 X16

41:9 whereX16 4; 3; 3; 3; 3; 3 :

z26 minf6 X26

42:1 whereX26 4; 3; 3; 3; 3; 4 :

z36 minf6 X36 42:5 whereX36 4; 3; 3; 3; 4; 3 :

z46 minf6 X46

45:1 whereX46 4; 3; 3; 3; 4; 4 :

z56 minf6 X56

43:1 whereX56 4; 3; 3; 4; 3; 3 :

z66 minf6 X66

44:6 whereX66 4; 3; 3; 3; 4; 4 :

z*534:7: Sincez*5 -z

r5

z*5

>0:05 forr 4; 6;

remove r 4; 6 andR6 1; 2; 3; 4f g:

Root

7.29 6.35

13.3 15.51

20.66 21.50

27.56 28.11 29.15 32.40

34.68 35.08 35.86 38.82

41.95 42.06 42.48 45.14 43.13 44.59

candidate node

pruned node

best node

Legend of node type

Figure 3 An illustrative example of CON heuristic with=8 andCV=0.1 up to Stage 6.



11/17

We skip the remaining steps and the nal solution is follows:

4-3-4-4/3-4-3-4/3-4-4-3/4-3-4-3. We splitX into 2-h intervals

(ie, four blocks) to present in a clear way using a separator (/).

5.2. A numerical example for the improving heuristic

In this subsection, we build an appointment schedule by the

IMP approach.

Initial step

We solve SINV to get the initial solution X0 as follows:

X0 4-3-4-4=4-3-4-4=3-4-3-4=4-3-4-3 :

Table 5shows the expected delay and idleness time of the

initial solution X0. The rst column gives the block index; the

second, the optimal block size; the third, the expected delay

time,E[Dk]; the last one, the expected idle time, E[Ik].

Iteration 1

f(X0)= 206.5 andj=13. We change all the next blocks after

the 13th block, x*13, x*

14, x*

15, and x*

16. Since E[D|K|]>E[I|K|],

x*

n=3. X1

= (4-3-4-4/4-3-4-4/3-4-3-4/3-4-3-3). We use bold-face for the change. f(X1)=195.9.

Iteration 2

For the next iteration, we skip detail objective function

values. j= 8 and X2= (4-3-4-4/4-3-4-3/4-3-4-3/4-3-3-3).

f(X2)=174.6.

Iteration 3

j= 5 and X3= (4-3-4-4/3-4-3-4/3-4-3-4/3-3-3-3). f(X

3) =

155.9.

Iteration 4

j= 12 and X4= (4-3-4-4/3-4-3-4/3-4-3-3/3-3-3-4). f(X

4)=

149.1.

Iteration 5

j= 10 and X5= (4-3-4-4/3-4-3-4/3-3-3-3/3-3-4-3). f(X5)=

141.1.

Iteration 6

j= 8 and X6= (4-3-4-4/3-4-3-3/3-3-3-3/3-4-3-3). f(X6)=

132.1.

Iteration 7

j= 6 and X7= (4-3-4-4/3-3-3-3/3-3-3-3/4-3-3-4). f(X7)=

125.3.

Iteration 8

j= 4 and X8= (4-3-4-3/3-3-3-3/3-3-3-4/3-3-4-3). f(X8)=

117.5.

Iteration 9

j= 3 and X9

= (4-3-3-3/3-3-3-3/3-3-4-3/3-4-3-3). f(X9

)=114.8.

Iteration 10

j= 11 and X10= (4-3-3-3/3-3-3-3/3-3-3-3/4-3-3-4). f(X

10)=

113.8.

Iteration 11

j= 16 and X11= (4-3-3-3/3-3-3-3/3-3-3-3/4-3-3-3). f(X

11)=

113.2.

Iteration 12

j= 8 and X12= (4-3-3-3/3-3-3-4/3-3-3-3/4-3-3-3). f(X12)=

114.9. We stop and chooseX11 as the best solution.

5.3. Analysis of comparative results

Table 6 summarizes optimal block sizes of cases with

=15 min, ve levels of CV, and nine cost ratios. The rst

column gives parameter values (, CV); the second one, cost

ratio; the third one, optimal solution to INV,x1in real number; the

fourth one, possible optimal block size for the second or later blocks;

the fth one, optimal block sizes; the sixth one, total assigned

patients; the seventh one, objective function value; the eighth one,

number of iterations; the last one, running time in seconds.

One may expect two patients to be assigned on average without

an exact computational study because ofh =30 and =15 min.

As the weight of cd increases and other factors are unchanged,

CDO would be likely to assign fewer number of patients.

Computational results verify this insight. Comparing two extreme

cost ratios (ie, 0.01:0.99 and 0.99 and 0.01), the former has

double or greater total assigned patients than the latter case. The

optimal objective function value increases then decreases as the

weight ofcd increases. Cases with extremely different cost ratios

(eg, 0:01:0.99, 0.09:0.91, 0.91:0.09, and 0.99:0.01) have eitherE

[Dk]=0 orE[Ik]=0 forkK. Hence, objective function values

of these extreme cases are smaller than other cases, because the

coefcient of the positive term out ofE[Dk] orE[Ik] is very small.When ci is 100 times of cd (ie, 0.01:0.99), rst block size is

relatively larger than the other block sizes. We do not use x*1for

all block sizes, butx*2for the second or later block size for this

extreme case. To nd heuristic block sizes, we use optimal

solution to INV for all cases except the case of 0.01:0.09. Next

two tables summarize the cases of=8 and 20 min in order to

show a variety of numerical results.

Table 7 summarizes numerical results of cases with =8,

ve levels of CV, three cost ratios (cd:ci=0.435:0.565, 0.5:0.5,

and 0.565:0.435). The rst column gives parameter values

Table 5 The numerical results for the expected delay and idlenesstime when the sequence is given as X0

Block index x*k E[Dk] E[Ik]

1 4 4.4 2.12 3 2.8 4.23 4 6.6 1.6

4 4 9.8 1.15 4 12.8 0.76 3 9.1 2.17 4 12.3 0.98 4 15.2** 0.79 3 11.3 2.110 4 14.5 0.911 3 10.8 2.212 4 14.1 0.913 4 16.8* 0.714 3 12.9 1.815 4 16.1 0.816 3 12.4 2.1

Note: The superscript * denotes the worst, and the superscript ** denotes the

second worst.



12/17

(, ); the second one, cost ratio; the third one, solution

methods; the fourth one, solution (block sizes); the fth one,objective function value; the sixth one, gap between the optimal

objective function value and the corresponding solution

methods objective function value; the seventh one, number of

iterations; the last one, running time. We provide the best

periodic Bailey among those for solution methods.

Table 8summarizes numerical results of cases with =20-

min, ve levels of CV, three cost ratios as Table 7. Column

information is the same asTable 7.

Figure 4depicts the impact of, CV, and cost ratio on block

size. It shows cumulative allocated patients for the three

congurations: (1) =8 min, CV=0.1; (2) =8 min, CV=

0.5; and (3) =20 min, CV=0.5. We analyze the impact ofon block size by comparing (1) with (3). As increases, the

number of allocated patients decreases. We analyze the impact

of CV on block size by comparing (1) with (2). For each

conguration, we analyze the impact of cost ratio on block size.

The cost ratio apparently affects block size. Figure 4shows

that for each conguration with certain and CV, as the ratio,

cd to ci increases, fewer numbers of patients are allocated

cumulatively. The cumulative line for the highest ratio ofcd to

ci is the lowest and the cumulative line for the lowest ratio is the

highest for each conguration with the same and CV.

Table 6 Optimal block sizes for the case of=15 min, the different cost ratios, and variance

(, CV) cd:c

ix1 x

*k, k 2 Block size Total patients Objective value Iteration Time (s)

(15, 0.1) 0.01:0.99 2.35 2, 3 3-2-2-2/2-2-2-2/2-2-2-2/2-2-2-2 33 2.5 2865 360.09:0.91 2.19 1, 2 3-2-2-2/2-2-2-2/2-2-2-2/2-2-2-2 33 21.4 2176 200.37:0.63 2.04 1, 2 2-2-2-2/2-2-2-2/2-2-2-2/2-2-2-2 32 24.0 1538 32

0.435:0.565 2.02 1, 2 2-2-2-2/2-2-2-1/2-2-2-2/2-2-2-2 31 25.7 6019 133

0.5:0.5 2.0 1, 2 1-2-2-2/2-2-2-2/1-2-2-2/2-2-2-2 30 32.8 4775 1240.565:0.435 1.97 1, 2 2-2-2-2/2-1-2-2/1-2-2-1/2-2-2-2 30 28.4 16 031 2130.63:0.37 1.95 1, 2 2-2-2-2/2-1-2-2/2-2-2-1/2-2-2-2 29 27.7 7295 1950.91:0.09 1.81 1, 2 2-1-2-1/2-1-2-1/2-1-2-1/2-1-2-1 24 17.0 2473 990.99:0.01 1.69 1, 2 1-1-1-1/1-1-1-1/1-1-1-1/1-1-1-1 16 2.4 15 64

(15, 0.2) 0.01:0.99 2.77 2, 3 3-2-2-2/2-2-2-3/2-2-2-2/2-2-2-2 34 4.5 3997 330.09:0.91 2.41 1, 2 2-2-2-2/2-2-2-2/2-2-2-2/2-2-2-2 32 21.0 3447 280.37:0.63 2.09 1, 2 2-2-2-2/2-2-2-1/2-2-2-2/2-2-2-2 31 41.3 11 490 216

0.435:0.565 2.04 1, 2 2-2-2-2/2-1-2-2/2-2-2-1/2-2-2-2 30 42.6 16 024 2380.5:0.5 2.0 1, 2 2-2-2-1/2-2-2-1/2-2-2-2/1-2-2-2 29 44.2 22 579 240

0.565:0.435 1.95 1, 2 2-2-2-1/2-2-2-2/1-2-2-2/1-2-2-2 29 43.6 15 157 2370.63:0.37 1.91 1, 2 2-2-2-1/2-2-1-2/2-1-2-2/2-1-2-2 28 42.8 17 160 2830.91:0.09 1.65 1, 2 1-1-1-1/1-1-1-1/1-1-2-1/1-1-1-1 17 21.6 1499 1020.99:0.01 1.44 1, 2 1-1-1-1/1-1-1-1/1-1-1-1/1-1-1-1 16 2.4 21 79

(15, 0.3) 0.01:0.99 3.26 2, 3 3-3-2-2/2-2-2-3/2-2-2-2/2-2-2-2 35 7.0 4196 31

0.09:0.91 2.65 1, 2 2-2-2-2/2-2-2-2/2-2-2-2/2-2-2-2 32 30.7 81 1040.37:0.63 2.14 1, 2 2-2-2-2/1-2-2-2/1-2-2-2/1-2-2-2 29 57.5 6692 145

0.435:0.565 2.07 1, 2 2-2-2-2/1-2-2-2/1-2-2-2/1-2-2-2 29 58.3 26 527 2480.5:0.5 2.0 1, 2 2-2-2-1/2-2-1-2/2-1-2-2/1-2-2-2 28 58.0 45 972 297

0.565:0.435 1.93 1, 2 2-2-2-1/2-2-1-2/2-1-2-2/1-2-2-2 28 56.7 27 618 2210.63:0.37 1.86 1, 2 2-2-1-2/2-1-2-2/1-2-2-1/2-2-1-2 27 54.3 44 712 2460.91:0.09 1.51 1, 2 1-1-1-1/1-1-1-1/1-1-1-1/1-1-1-1 16 21.6 68 920.99:0.01 1.22 1, 2 1-1-1-1/1-1-1-1/1-1-1-1/1-1-1-1 16 2.4 42 91

(15, 0.4) 0.01:0.99 3.81 2, 3 3-3-2-2/2-2-2-2/3-2-2-2/3-2-2-2 35 9.2 4722 280.09:0.91 2.91 1, 2 2-2-2-2/2-2-2-2/2-2-2-2/2-2-2-2 32 41.9 4005 250.37:0.63 2.19 1, 2 2-2-2-2/1-2-2-2/1-2-2-2. 1-2-2-2 29 69.9 40 718 246

0.435:0.565 2.09 1, 2 2-2-1-2/2-1-2-2/2-1-2-2/2-1-2-2 28 72.2 60 462 3040.5:0.5 2.0 1, 2 2-2-1-2/2-1-2-1/2-2-1-2/2-1-2-2 27 69.9 36 836 355

0.565:0.435 1.91 1, 2 2-1-2-2/1-2-1-2/1-2-2-1/2-1-2-2 26 68.2 32 490 2570.63:0.37 1.82 1, 2 2-2-1-2/1-2-1-2/1-2-1-2/1-2-1-2 25 64.3 32 673 319

0.91:0.09 1.37 1, 2 1-1-1-1/1-1-1-1/1-1-1-1/1-1-1-1 16 21.8 10 319 910.99:0.01 1.04 1, 2 1-1-1-1/1-1-1-1/1-1-1-1/1-1-1-1 16 2.6 57 93

(15, 0.5) 0.01:0.99 4.45 2, 3 4-3-2-2/2-3-2-2/2-2-2-2/2-2-3-2 37 11.1 9484 110.09:0.91 3.19 1, 2 3-2-2-2/2-1-2-2/2-2-2-2/2-2-2-2 32 48.9 3330 1610.37:0.63 2.25 1, 2 2-2-1-2/2-1-2-2/1-2-1-2/2-1-2-2 27 82.5 65 823 409

0.435:0.565 2.12 1, 2 2-2-1-2/1-2-2-1/2-2-1-2/1-2-1-2 26 82.7 67 848 3380.5:0.5 2.0 1, 2 2-2-1-2/1-2-1-2/1-2-2-1/2-1-2-2 26 83.3 20 203 196

0.565:0.435 1.88 1, 2 2-2-1-2/1-2-1-2/1-2-1-2/1-2-1-2 25 78.9 22 661 2560.63:0.37 1.78 1, 2 2-1-2-1/1-2-1-2/1-2-1-2/1-2-1-2 24 73.3 17 762 2710.91:0.09 1.25 1, 2 1-1-1-1/1-1-1-1/1-1-1-1/1-1-1-1 16 22.2 9614 980.99:0.01 0.89 1, 2 1-1-1-1/1-1-1-1/1-1-1-1/1-1-1-1 16 3.6 67 101



13/17

Table 7 Comparative results when=8 min with different cost ratios, variance, and heuristic rules

(, CV) cd:c

iMethod Block size Objective value Gap (%) Iterations Time (s)

(8, 0.1) 0.435:0.565 CPLEX 4-4-3-4/3-4-4-3/4-3-4-4/3-4-3-4 23.0 - 166 971 776.2Constructive 4-4-3-4/3-4-4-3/4-4-3-4/3-4-4-3 23.3 1.3 113 13.8Improving 4-4-3-4/3-4-3-4/4-3-4-4/3-4-3-4 23.4 1.7 4 0.7p-Bailey-s 4-3-4-3/4-3-4-3/4-3-4-3/4-3-4-3 25.5 10.8 1 0.1



(8, 0.2) 0.435:0.565 CPLEX 4-3-4-3/4-3-4-3/4-3-4-3/4-3-4-4 30.3 - 77 067 412Constructive 4-3-4-3/4-3-4-3/4-3-4-3/4-3-4-3 30.4 0.3 189 15.2Improving 4-3-4-3/4-3-4-3/4-3-4-3/4-3-4-3 30.4 0.3 9 1.1p-Bailey-s 4-3-4-3/4-3-4-3/4-3-4-3/4-3-4-3 30.4 0.3 1 0.1

(8, 0.2) 0.5:0.5 CPLEX 4-3-3-4/3-4-3-4/3-4-3-4/3-4-3-4 30.6 - 75 403 650Constructive 4-3-3-4/3-4-3-4/3-4-3-4/3-4-3-4 30.6 0.0 167 14.3Improving 4-3-4-3/4-3-4-3/4-3-4-3/4-3-4-3 30.7 0.3 8 1.3

p-Bailey-s 4-3-4-3/4-3-4-3/4-3-4-3/4-3-4-3 30.7 0.3 1 0.1(8, 0.2) 0.565:0.435 CPLEX 4-3-4-3/4-3-4-3/4-3-4-3/4-3-4-3 30.1 - 51 061 310

Constructive 4-3-4-3/4-3-4-3/4-3-4-3/4-3-4-3 30.1 0.0 165 13.9Improving 4-3-4-3/4-3-4-3/4-3-3-4/3-4-3-4 30.3 0.7 6 0.5p-Bailey-s 4-3-4-3/4-3-4-3/4-3-4-3/4-3-4-3 30.1 0.0 1 0.1



(8, 0.3) 0.565:0.435 CPLEX 4-3-3-4/3-3-3-4/3-3-3-4/3-3-4-3 39.0 - 32 941 435Constructive 4-3-3-4/3-3-3-4/3-3-3-4/3-3-4-3 39.0 0.0 191 19.2

Improving 4-3-4-3/3-4-3-4/3-3-4-3/3-4-3-4 39.7 2.0 16 1.3p-Bailey-m 4-3-3-3/4-3-3-3/4-3-3-3/4-3-3-3 39.4 1.1 1 0.1

(8, 0.4) 0.435:0.565 CPLEX 4-3-4-3/3-4-3-3/3-4-3-4/3-3-4-3 48.7 56 765 338Constructive 4-3-4-3/3-4-3-3/3-4-3-4/3-3-4-3 48.7 0.0 122 10.5Improving 4-3-4-3/3-4-3-3/4-3-3-3/4-3-3-4 49.3 1.2 15 1.8p-Bailey-m 4-3-3-3/4-3-3-3/4-3-3-3/4-3-3-3 51.7 6.1 1 0.1

(8, 0.4) 0.5:0.5 CPLEX 4-3-3-3/4-3-3-3/4-3-3-4/3-3-3-4 48.8 - 30 473 386Constructive 4-3-3-3/4-3-3-3/4-3-3-4/3-3-3-4 48.8 0.0 196 20.5Improving 4-3-4-3/3-3-3-3/4-3-4-3/3-4-3-4 49.4 1.2 11 1.5p-Bailey-m 4-3-3-3/4-3-3-3/4-3-3-3/4-3-3-3 49.3 1.0 1 0.1

(8, 0.4) 0.565:0.435 CPLEX 4-3-3-3/3-4-3-3/3-3-3-3/3-3-3-3 46.9 - 16 175 214Constructive 4-3-3-3/3-4-3-3/3-3-3-3/3-3-3-3 46.9 0.0 201 21.2Improving 4-3-4-3/3-3-4-3/3-3-4-3/3-3-4-3 47.1 0.6 11 1.3p-Bailey-l 4-3-3-3/3-3-3-3/4-3-3-3/3-3-3-3 46.9 0.1 1 0.1

(8, 0.5) 0.435:0.565 CPLEX 4-3-3-3/4-3-3-3/3-3-4-3/3-3-3-4 57.9 - 26 088 165.2

Constructive 4-3-3-3/3-4-3-3/3-3-4-3/3-3-3-4 58.8 1.7 253 31.8Improving 4-3-3-3/3-4-3-3/3-3-4-3/3-3-3-4 58.8 1.7 13 2.3t-Bailey 4-3-3-3/3-3-3-3/3-3-3-3/3-3-3-3 60.0 3.6 1 0.1

(8, 0.5) 0.5:0.5 CPLEX 4-3-3-3/3-3-3-3/3-3-3-3/3-3-3-3 56.4 - 20 813 134.7Constructive 4-3-3-3/3-3-3-3/3-3-3-3/3-3-3-3 56.4 0.0 224 22.2Improving 4-3-3-3/3-3-3-3/3-3-3-3/4-3-3-3 56.5 0.4 11 1.9t-Bailey 4-3-3-3/3-3-3-3/3-3-3-3/3-3-3-3 56.4 0.0 1 0.1

(8, 0.5) 0.565:0.435 CPLEX 3-3-3-3/3-3-3-3/3-3-3-3/3-3-3-3 122.2 - 8836 61.1Constructive 3-3-3-3/3-3-3-3/3-3-3-3/3-3-3-3 122.2 0.0 58 6.9Improving 3-4-3-3/3-3-3-3/3-3-4-3/3-4-3-3 131.8 7.9 10 1.8t-Bailey 4-3-3-3/3-3-3-3/3-3-3-3/3-3-3-3 122.6 0.3 1 0.1



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Table 8 Comparative results when=20 min and the different cost ratios, variance, and heuristic rules

(, CV) cd:c

iMethod Block size Objective value Gap (%) Iterations Time (s)





(20, 0.2) 0.5:0.5 CPLEX 2-1-1-2/1-1-2-1/1-1-2-1/1-2-1-1 64.3 - 163 595 966Constructive 2-1-1-2/1-1-2-1/1-1-2-1/1-2-1-1 64.3 0.0 76 7.9Improving 2-1-1-2/1-1-2-1/1-1-2-1/1-2-1-1 64.3 0.0 15 1.9

p-Bailey-m 2-1-1-1/2-1-1-1/2-1-1-1/2-1-1-1 67.0 4.2 1 0.1(20, 0.2) 0.565:0.435 CPLEX 2-1-1-1/2-1-1-1/2-1-1-1/2-1-1-1 64.9 - 223 639 1258

Constructive 2-1-1-1/2-1-1-1/2-1-1-1/2-1-1-1 64.9 0.0 52 5.8Improving 2-1-1-1/2-1-1-1/1-2-1-1/2-1-1-1 65.5 0.9 10 2.1p-Bailey-m 2-1-1-1/2-1-1-1/2-1-1-1/2-1-1-1 64.9 0.0 1 0.1



(20, 0.3) 0.565:0.435 CPLEX 1-1-1-1/1-1-1-1/1-1-1-1/1-1-1-1 70.6 - 60 640 504Constructive 1-1-1-1/1-1-1-1/1-1-1-1/1-1-1-1 70.6 0.0 60 7.1

Improving 2-1-1-1/1-1-1-1/1-1-1-1/1-1-1-1 71.3 1.0 16 1.9t-Bailey 2-1-1-1/1-1-1-1/1-1-1-1/1-1-1-1 71.3 1.0 1 0.1


(20, 0.4) 0.5:0.5 CPLEX 2-1-1-1/1-1-1-1/2-1-1-1/1-1-1-1 83.8 - 45 337 694Constructive 2-1-1-1/1-1-1-2/1-1-1-1/1-1-1-1 84.1 0.4 214 20.5Improving 2-1-1-1/1-1-2-1/1-1-1-1/1-1-1-1 83.8 0.0 13 1.2p-Bailey-l 2-1-1-1/1-1-1-1/2-1-1-1/1-1-1-1 83.8 0.0 1 0.1

(20, 0.4) 0.565:0.435 CPLEX 1-1-1-1/1-1-1-1/1-1-1-1/1-1-1-1 74.4 - 24 287 373Constructive 1-1-1-1/1-1-1-1/1-1-1-1/1-1-1-1 74.4 0.0 34 3.1Improving 1-1-1-1/1-1-1-1/1-1-1-1/1-1-1-1 74.4 0.0 11 0.9t-Bailey 2-1-1-1/1-1-1-1/1-1-1-1/1-1-1-1 75.9 2.0 1 0.1

(20, 0.5) 0.435:0.565 CPLEX 2-1-1-1/1-1-1-1/1-1-1-1/1-1-1-1 97.4 - 40 416 535

Constructive 2-1-1-1/1-1-1-1/1-1-1-1/1-1-1-1 97.4 0.0 334 30.7Improving 2-1-1-1/1-1-1-1/1-1-1-1/1-1-1-1 97.4 0.0 8 0.9t-Bailey 2-1-1-1/1-1-1-1/1-1-1-1/1-1-1-1 98.7 0.0 1 0.1

(20, 0.5) 0.5:0.5 CPLEX 1-1-1-1/1-1-1-1/1-1-1-1/1-1-1-1 89.3 - 21 558 268Constructive 1-1-1-1/1-1-1-1/1-1-1-1/1-1-1-1 89.3 0.0 42 4.1Improving 2-1-1-1/1-1-1-1/1-1-1-1/1-1-1-1 89.8 0.6 9 1.2t-Bailey 2-1-1-1/1-1-1-1/1-1-1-1/1-1-1-1 91.5 0.6 1 0.1

(20, 0.5) 0.565:0.435 CPLEX 1-1-1-1/1-1-1-1/1-1-1-1/1-1-1-1 79.7 - 185 105Constructive 1-1-1-1/1-1-1-1/1-1-1-1/1-1-1-1 79.7 0.0 32 2.9Improving 1-1-1-1/1-1-1-1/1-1-1-1/1-1-1-1 79.7 0.0 7 0.9t-Bailey 2-1-1-1/1-1-1-1/1-1-1-1/1-1-1-1 84.6 6.2 1 0.1



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CDO administrative staffs may use a relative importance

measure of both penalties, even though they do not have exact

cost information about delay and idleness. If they want to utilize

physicians better (ie, cd ci), they are willing to assign more

patients (ie, dxe) from the begin block. If more patients areassigned in the earlier blocks, subsequent block sizes are less

likely to bedxe:On the contrary, if they want to avoid crowdingin the waiting room (ie, cd ci), they are willing to assign less

patients. If less patients are assigned in the earlier blocks,

subsequent block sizes are more likely to bedxe:Variability also plays an important role in determining block

sizes. Figure 4(see Congurations (1) and (2)) shows that the

cumulative line for the higher CV is below and that the total

allocated patients in a day is smaller as CV is higher. Higher CV

affects block sizes of the later blocks. CDO administrative staffs

are likely to assign less patients bxcin the later blocks, if CVis high. The cost ratio may affect block sizes more than CV

does, when CV is low.

The rst block size is always dxe; which is the optimalsolution to INV problem, except when cd< ci and variability is

high (CV=0.5). The second or subsequent block sizes dependsupon variance and cost ratio. Ifcd< ci and variance is small,dxeis the best for the second block. Otherwise,bxcis optimal. Therst block size has an impact on the subsequent blocks. Hence,

bxc is optimal even though dxe is optimal for individual INVproblems. When variance is small, more dxe block sizes areassigned. It is trivial to assign moredxeblock sizes whencd> ci

than whencd< ci.

We nd that either CON or IMP provides the highest quality

solutions for all cases, which is the same or very close to the

solutions obtained by CPLEX. We expect that CON provides

the best block sizes for all cases since it searches a large number

of feasible block sizes. To get the optimal block sizes, we

should use a larger tolerance rather than the current values.

Preliminary tests show that current values not only remove most

unrealistic values that are far from the optimum, but there is also

a possibility of removing the optimal block sizes as well. The

numerical studies show that the best solution using the current

values is close to the optimal solution. However, there are

limitations for CON and IMP heuristics. The CON heuristic has

a limitation that it may be hard to nd good tolerance ratiosat

stages. The IMP heuristic also has a limitation of neighbour-

hood search approach that there might be no break-through

solution from current solution to global optimal solution.

t-Bailey can be the best block size rule for certain case in

which variance is large. For example, t-Bailey is the best when

=8 min,cd: ci=1:1 and CV=0.5; and when =20 min,cd:

ci=1:1.3 and CV=0.5.p-Bailey-s rule can be used for the case

of small variance, whereas p-Bailey-l rule can be used for the

case of larger variance.

In most cases,t-Bailey orp-Bailey rule performs worse than

IMP and CON. When variance is large, three rules have

different solutions of block sizes but objective function values

are close to each other. When variance is small, t-Bailey rulegives the worst objective function value, which is 1.7~ 1.9

times of the two heuristic rules.

We see how sensitive the optimal solution is to , CV, and

cost ratios (cd: c

i).Figure 5depicts two iso-solutions for each

case of (= 8 min, CV= 0.1,cd:ci=1:1.3) or (=8, CV= 0.1,

cd: ci= 1.3:1), respectively. Iso-solutions are the same optimal

block sizes for different congurations. We take 8 blocks into

account, because iso-solutions is unique for 16 blocks. Iso-

solutions look off-diagonal, which means that as increases,

CV should be smaller for iso-solutions. If cd ci, the smaller

0

20

40

60

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

(8, 0.1, 0.5:0.5)

(8, 0.1, 0.565:0.435)

(8, 0.1, 0.435:0.565)

(8, 0.5, 0.5:0.5)

(8, 0.5, 0.565:0.435)

(8, 0.5, 0.435:0.565)

(20, 0.1, 0.5:0.5)

(20, 0.1, 0.565:0.435)

(20, 0.1, 0.435:0.565)

, CV, cost ratio

Block

#ofpatients

Figure 4 Cumulative allocated patients for the three

congurations (, CV)= (8, 0.1); (8, 0.5); and (20, 0.1) with threecost ratios 0.435:0.565, 0.5:0.5, and 0.565:0.435.

7.8 7.9 8.0 8.1 8.2 8.37.7 8.4 8.57.6

7.8 7.9 8.0 8.1 8.2 8.37.7 8.4 8.57.6

0.1

0.2

0.3

0.1

0.2

0.3

CV

CV

a

b

Figure 5 An illustrative example of iso-solutions for the case of(= 8 min, CV= 0.1, c

d: c

i= 1:1.3) or (8, 0.1, 1.3:1), respectively.

(a) Iso-solutions when cd

: ci=1:1.3; (b) iso-solutions when

cd:ci=1.3:1



16/17

block size (ie, bxc) is assigned at most times, because thesmaller block size (ie, bxc) affect less on later blocks. The caseofcd ci is less sensitive to iso-solutions than the case ofcd ci.

Two opposite forces are conicting, ifcd ci. The rst force is

to allocate more patients ifcd ci, the other force is to allocate

less patients ifcd ci. Generally, it is very likely to allocate less

patients because earlier blocks affect later blocks. Hence, if

cd ci, two opposite forces interact with each other and iso-

solutions are more robust than sensitive. If cd ci, the same

directional force to allocate less patients works severely and iso-

solutions are more sensitive than the case ofcd ci. When CV

is small, we have more iso-solutions. There is no iso-solution

for CV= 0.3 or larger. It is intuitive that iso-solutions become

increasingly more sensitive with increasing values of CV.

6. Conclusions and discussions

This paper provides two heuristic methodologies (CON and

IMP) and three periodic Bailey rules to appointment sche-duling. CON tries to search as many solution options as

possible and to obtain near-optimal block size, but requires

more running time and iterations than IMP. IMP uses SINV

model to obtain the initial solution and revises by updating a

partial series of the current solution. IMP heuristic takes a

few iterations only. CON heuristic searches many feasible

block size rules near optimality. The best result by CON

depends upon tolerance ratios. One of the practical limita-

tions is the difculty in nding good tolerance ratios.

Empirically, IMP provides the best block size rule in a

few steps. IMP requires a good initial block size rule.

We suggest periodic Bailey rules considering the relativeimportance between delay and idle times. Periodic Bailey

rule is easy to implement and straightforward to interpret.

We recommend that CDO administrators use IMP to get a

base solution rst and nd a periodic Bailey rule that is close

to IMP.

There are several venues for future research. Uncertainties in

patient arrival can be explored for further investigation. No-

show possibility also can be considered for patient arrival

uncertainty. Different classes of patients can be involved in

the appointment scheduling. For example, new patients may

take more time for the rst visit or completing their personal

prole. The current model assumes a single server, which

may be expressed by a representative server even thoughthere are several identical servers. A CDO may involve a

multi-server appointment system. Finally, a multi-phase

appointment system can be explored where the appointment

system can handle the needs of patients to see multiple

caregivers in a sequential manner.

AcknowledgementsThe authors would like to thank anonymous reviewersfor their helpful comments and suggestions. Funding was partially providedby the Vice President of Academic Affair at Shenandoah University,Adrienne Bloss.

References

Bailey NTJ and Welch JD (1952). Appointment systems in hospital

outpatient department.Lancet259(6718): 11051108.

Begen MA and Queyranne M (2011). Appointment scheduling with

discrete random durations. Mathematics of Operations Research

36(2): 240257.

Belin J and Demuelemeester E (2007). Building cyclic master surgery

schedules with leveled resulting bed occupancy. European Journal ofOperational Research176(2): 11851204.

Belin J, Demuelemeester E and Cardoen B (2009). A decision support

system for cyclic master surgery scheduling with multiple objectives.

Journal of Scheduling12(2): 147161.

Bruin AM, Koole GM and Visser MC (2005). Bottleneck analysis of

emergency cardiac in-patient ow in a university setting: An

application of queuing theory.Clinical and Investigative Medicine

28(6): 316317.

Cardeon B, Demeulemeester E and Belin J (2009). Sequencing surgical

cases in a day-care environment: An exact branch-and-price approach.

Computers & Operations Research36(9): 26602669.

Cayirli T and Veral E (2003). Outpatient scheduling in health care:

A review of literature. Production and Operations Management

12(4): 519549.

Chase M (2005). Beginning patient ow modeling in Vancouvercoastal health.Clinical and Investigative Medicine28(6): 323325.

Choi S and Ketzenberg M (2014). An inverse newsvendor model to set

the optimal number of customers in a capacitated environment.

Working paper, Shenandoah University.

Choi S and Wilhelm WE (2012). An analysis of sequencing surgeries

with durations that follow the lognormal, gamma, or normal distribu-

tion. IIE Transactions on Healthcare Systems and Engineering

2(2): 156171.

Choi S and Wilhelm WE (2014). An approach to optimize master

surgical block schedules.European Journal of Operational Research

235(1): 138148.

Denton B and Gupta D (2003). A sequential bounding approach for

optimal appointment scheduling.IIE Transactions 35(11): 10031016.

Dexter F and Traub RD (2002). How to schedule elective surgical cases

into specic operating rooms to maximize the efciency of use of

operating room time.Anesthesia and Analgesia94(4): 933942.

Fries B and Marathe V (1981). Determination of optimal variable-

sized multiple-block appointment systems. Operations Research

29(2): 324345.

Green LV, Savin S and Wang B (2006). Managing patient service in a

diagnostic medical facility.Operations Research54(1): 1125.

Gupta D (2007). Surgical suites operations management. Production

and Operations Management16(6): 689700.

Gupta D and Denton B (2008). Appointment scheduling in health care:

Challenges and opportunities.IIE Transactions40(9): 800819.

Hans E, Wullink G, van Houdenhoven M and Kazemier G (2008).

Robust surgery loading. European Journal of Operational Research

185(3): 10381050.

Ho C and Lau H (1992). Minimizing total cost in scheduling outpatientappointments.Management Science38(12): 17501764.

Ho C and Lau H (1999). Evaluating the impact of operating conditions on

the performance of appointment scheduling rules in service systems.

European Journal of Operational Research112(3): 542553.

Ho C, Lau H and Li J (1995). Introducing variable-interval appointment

scheduling in service systems. International Journal of Operations &

Production Management15(6): 5969.

Kaandorp G and Koole G (2007). Optimal outpatient appointment

scheduling.Health Care Management Science10(3): 217229.

Klassen K and Rohleder T (1996). Scheduling outpatient appointments

in a dynamic environment.Journal of Operations Management14(2):

83101.



17/17

Liao C, Pegden C and Rosenshine M (1993). Planning timely arrivals to a

stochastic production or service system.IIE Transactions25(5): 6373.

Mancilla C and Storer R (2009). Theses and Dissertation, Lehigh

University, Department of Industrial and Systems Engineering, 14

November 2009.

Marcon E, Kharraja S and Simmonet G (2003). The operating theatre

scheduling: An approach centered on the follow-up of the risk of no

realization of the planning. International Journal of Production

Economics85(1): 8390.Pham D and Klinkert A (2008). Surgical case scheduling as a generalized

job shop scheduling problem. European Journal of Operational

Research185(3): 10111025.

Robinson L and Chen R (2003). Scheduling doctors appointment:

Optimal and empirically-based heuristic policies. IIE Transactions

35(3): 295307.

Wang P (1993). Static and dynamic scheduling of customer arrivals to a

single-server system.Naval Research Logistics40(3): 345360.

Wang P (1997). Optimally scheduling N customer arrival times

for a single-server system. Computers & Operations Research 24(8):

703716.

Weiss E (1990). Models for determining estimated start times and case

orderings in hospital operating rooms.IIE Transactions 22(2): 143150.

van Houdenhoven M, van Oostrum JM, Hans EW, Wullink G and

Kazemier G (2007). Improving operating room efciency by applyingbin-packing and portfolio techniques to surgical case scheduling.

Anesthesia and Analgesia105(3): 707714.

Received 30 October 2014;

accepted 27 August 2015 after three revisions


choi s.-comparison of a branch-and-bound heuristic

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