Choi S.-comparison of a Branch-And-bound Heuristic
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Transcript of 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/
http://dx.doi.org/10.1057/jors.2015.79mailto:[email protected]://www.palgrave-journals.com/jorshttp://www.palgrave-journals.com/jorsmailto:[email protected]://dx.doi.org/10.1057/jors.2015.79 -
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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.
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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)
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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].
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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
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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
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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.
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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.
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(, ); 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
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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, 0.1) 0.5:0.5 CPLEX 4-4-3-4/3-4-3-4/4-3-4-3/4-4-3-4 23.9 - 168 379 676.3Constructive 4-3-4-4/3-4-3-4/3-4-4-3/4-3-4-3 24.4 1.9 108 11.7Improving 4-3-4-4/3-4-4-3/4-3-4-4/3-4-3-4 24.2 1.0 6 1.1p-Bailey-s 4-3-4-3/4-3-4-3/4-3-4-3/4-3-4-3 24.6 2.7 1 0.1
(8, 0.1) 0.565:0.435 CPLEX 4-3-4-4/3-4-3-4/3-4-3-4/3-4-3-4 23.5 - 70 392 448.3Constructive 4-3-4-3/4-3-4-3/4-3-4-3/4-3-4-3 23.6 0.2 52 5.9Improving 4-3-4-3/4-3-4-3/4-3-4-3/4-3-4-3 23.6 0.2 6 1.1p-Bailey-s 4-3-4-3/4-3-4-3/4-3-4-3/4-3-4-3 23.6 0.2 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) 0.435:0.565 CPLEX 4-3-4-3/4-3-4-3/4-3-3-4/3-4-3-4 40.5 - 42 920 381Constructive 4-3-4-3/4-3-4-3/4-3-3-4/3-4-3-4 40.5 0.0 167 17.1Improving 4-3-4-3/4-3-4-3/4-3-3-4/3-4-3-4 40.8 0.9 8 0.9p-Bailey-s 4-3-4-3/4-3-4-3/4-3-4-3/4-3-4-3 41.2 1.7 1 0.1
(8, 0.3) 0.5:0.5 CPLEX 4-3-4-3/3-4-3-3/4-3-4-3/3-4-3-4 40.1 - 40 549 481Constructive 4-3-4-3/3-4-3-3/4-3-4-3/3-4-3-4 40.1 0.0 182 19.1Improving 4-3-4-3/4-3-4-3/4-3-3-4/3-4-3-4 40.9 1.9 8 0.7p-Bailey-s 4-3-4-3/4-3-4-3/4-3-4-3/4-3-4-3 41.6 3.6 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, 0.1) 0.435:0.565 CPLEX 2-1-2-1/2-1-1-2/1-2-1-1/2-1-2-1 52.7 - 284 708 1213Constructive 2-1-2-1/1-2-1-2/1-2-1-1/2-1-2-1 52.9 0.3 68 6.1Improving 2-1-2-1/1-2-1-1/1-2-1-2/1-1-2-1 55.2 4.6 10 1.1p-Bailey-m 2-1-1-2/1-1-2-1/1-2-1-1/2-1-1-1 58.5 10.9 1 0.1
(20, 0.1) 0.5:0.5 CPLEX 2-1-1-2/1-1-2-1/1-2-1-2/1-1-2-1 56.4 - 323 056 1344Constructive 2-1-1-2/1-1-2-1/1-2-1-2/1-1-2-1 56.4 0.0 48 4.2Improving 2-1-1-2/1-1-2-1/1-2-1-1/2-1-1-1 58.2 3.2 11 1.7p-Bailey-m 2-1-1-1/2-1-1-1/2-1-1-1/2-1-1-1 62.7 11.2 1 0.1
(20, 0.1) 0.565:0.435 CPLEX 2-1-1-2/1-1-2-1/2-1-1-2/1-1-2-1 58.2 - 207 610 1030Constructive 2-1-1-1/2-1-1-1/2-1-1-2/1-1-2-1 59.0 1.3 42 3.9Improving 2-1-1-1/2-1-1-1/2-1-1-2/1-1-2-1 59.0 1.3 10 1.5p-Bailey-m 2-1-1-1/2-1-1-1/2-1-1-1/2-1-1-1 60.7 4.3 1 0.1
(20, 0.2) 0.435:0.565 CPLEX 2-1-1-2/1-1-2-1/1-2-1-1/2-1-2-1 64.1 - 233 618 1164Constructive 2-1-1-2/1-1-2-1/1-2-1-1/2-1-2-1 64.1 0.0 42 4.1Improving 2-1-1-1/2-1-1-1/2-1-1-1/1-1-2-1 65.5 0.9 11 1.1p-Bailey-m 2-1-1-2/1-1-2-1/1-2-1-1/2-1-1-1 64.7 0.9 1 0.1
(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) 0.435:0.565 CPLEX 2-1-1-2/1-1-2-1/1-1-2-1/1-2-1-1 74.9 - 134 261 865Constructive 2-1-1-2/1-1-1-2/1-1-2-1/1-1-1-1 75.3 0.6 144 15.1Improving 2-1-1-2/1-1-1-1/2-1-1-1/1-2-1-1 75.4 0.7 16 2.0p-Bailey-m 2-1-1-1/2-1-1-1/2-1-1-1/2-1-1-1 76.2 1.7 1 0.1
(20, 0.3) 0.5:0.5 CPLEX 2-1-1-1/1-2-1-1/1-2-1-1/1-2-1-1 73.6 - 120 681 755Constructive 2-1-1-1/1-2-1-1/1-2-1-1/1-2-1-1 73.6 0.0 342 35.1Improving 2-1-1-1/1-2-1-1/1-1-2-1/1-1-1-1 74.8 1.6 10 1.1p-Bailey-m 2-1-1-1/2-1-1-1/2-1-1-1/2-1-1-1 74.7 1.3 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) 0.435:0.565 CPLEX 1-2-1-1/1-1-2-1/1-1-1-2/1-1-1-1 86.9 - 97 895 1192Constructive 1-2-1-1/1-1-2-1/1-1-1-2/1-1-1-1 86.9 0.0 162 17.1Improving 2-1-1-1/1-1-1-1/1-1-1-2/1-1-1-1 88.8 2.2 10 0.9p-Bailey-m 2-1-1-1/2-1-1-1/2-1-1-1/2-1-1-1 87.7 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
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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.
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Received 30 October 2014;
accepted 27 August 2015 after three revisions
592 Journal of the Operational Research Society Vol. 67, No. 4