Activities Planning and Resource Assignment on Multi-place

Hospital System

Exact and Approach Methods Adapted from the Bin Packing Problem

Michel Gourgand, Nathalie Grangeon and Nathalie Klement

LIMOS CNRS UMR 6158, Universit

e Blaise Pascal, Complexe Scientiﬁque des C

ezeaux, 63173 Aubi

ere Cedex, France

Keywords:

Bin Packing, Resources Assignment, Exact Method, Metaheuristics.

Abstract:

In France, the Hospital Community of Territory (HCT) has been deﬁned since the settlement of the pricing

by activity (T2A) in 2004, and the new governance. This new community allows the pooling of the hospital

resources between any places in the same territory. It aims at increasing the continuity of healthcare. A medi-

cal exam needs human resources and one material resource, both of them compatible with the exam. Human

resources are supposed available all the time. In this paper, material resources are the critical resources. The

objective is to create a decision support tool which plans the exams on each material resource. Some con-

straints have to be respected: compatibilities between exams and resources, resources opentime. We propose a

mathematical model. It deals with an exams planning taking into account the assignment of material resources

in the community over a given horizon planning. The problem is solved in an exact way and with approach

methods applying bin packing heuristics and single solution based metaheuristics. Experimentation tackles

with randomly generated instances.

1 INTRODUCTION

Nowadays, the main objective of hospital system is

to improve its efﬁciency. Indeed, some hospitals can

have patients waiting a few months or more to get an

appointment. On the other hand, other hospitals have

almost no waiting time. These two kinds of places

can be very close geographically speaking. In France,

a new organization was created a few years ago, the

Hospital Community of Territory (HCT). It is a group

of distinct places which aim at improving their ef-

ﬁciency by putting means from different places to-

gether. The idea developed in this paper is that by dis-

patching medical exams on several places, the global

efﬁciency of all the places will be improved.

After the speciﬁcation of our problem, the formu-

lation of the mathematical model is detailed. This

problem can be compared to the bin packing prob-

lem. This comparison is explained in the state of

the art. This problem is ﬁrst solved in an exact way,

but because it is NP-hard, approach methods are also

considered by applying some single solution based

metaheuristics (kangaroo algorithm and simulated an-

nealing algorithm) combined to a bin packing heuris-

tic. The last section compares the results of the exact

method to those of the approach methods. We will

conclude with some further works.

2 SPECIFICATION

First, the considered problem is detailed in a generic

way and then the analogy with the ﬁeld of medical

imaging is done.

2.1 Problem Statement

The problem studied in this paper can be compared

to a tasks assignment problem over a multi-place sys-

tem.

Each task needs a resource to be treated and a time

slot, a period when it will be done. We make the

hypothesis that human resources are available all the

time, only the material resources are discussed, they

are the critical resources. Each task has a known pro-

cess time. A task has to be completed before a due

date. Each task has got a reference place where it has

to be done, if possible.

The system is composed of several places. There

is a known distance between these places. In each

place, there are one to several resources. Every re-

117

Gourgand M., Grangeon N. and Klement N..

Activities Planning and Resource Assignment on Multi-place Hospital System - Exact and Approach Methods Adapted from the Bin Packing Problem.

DOI: 10.5220/0004803901170124

In Proceedings of the International Conference on Health Informatics (HEALTHINF-2014), pages 117-124

ISBN: 978-989-758-010-9

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

sources cannot treat all the tasks, there is a list of com-

patibilities between tasks and resources. Resources

are from one type. Resources have a planning deﬁn-

ing their available time, we call it resources opentime.

The considered horizon planning is known. This

horizon is divided into periods. Each task starts in one

period and ﬁnishes in the same period.

The aim is to assign each task to one resource and

one period, respecting the compatibilities and the dif-

ferent times. Tasks have to be done as soon as pos-

sible, so the assigned periods have to be the smallest

possible.

The model has to respect some constraints which

are the following:

• Resources have to be able to process their as-

signed tasks (resource compatibility).

• Tasks have to be assigned to a resource during the

opentime of this resource.

• A task has to be assigned to one resource and one

period.

2.2 Objective

The aim of this model is not to make a precise sched-

ule of the tasks but to attribute a period to each task.

The criteria that can be studied are the following:

• The completion time of the last planned task. It

is the biggest period assigned to the tasks. All the

tasks are done before or during this period. By

misnomer, we call it in the following makespan.

• The sum of the completion times of all the tasks

planned in the system, that is to say the sum of

each period assigned to all the tasks.

• The number of tasks done before their due date.

• The number of tasks done at their reference place,

or the distance between the reference place and

the effective one for each task.

The criteria which are the most relevant are about

the economic aspect of the problem. The earlier the

tasks will be planned, the earlier they will be com-

pleted. So resources will be available earlier to prac-

tice the next tasks. With the two ﬁrst criteria, we can

assume that the tasks will be done as soon as possible.

2.3 Medical Imaging Example

For the hospital system, a task is an exam. The

considered system is the HCT, composed of several

places. In each place, there are one to several material

resources. A material resource is from one type: MRI,

x-ray, or scanner. Each material resource has a plan-

ning with its available time per period. For example,

we can assume that an MRI engine is only available

ninety minutes on Monday morning because it needs

a maintenance operation or because an external physi-

cian reserved it. Its opentime will span over ninety

minutes during this period. The considered horizon

planning is divided into periods. Each period can rep-

resent one half-day, one day or one week, the horizon

planning can represent one week, one month or one

year.

Depending on the used horizon planing, the de-

scribed model can be used at three levels:

• Strategic: to determine the system dimensional,

the number of places needed in the system, how

many hospitals are part of the HCT.

• Tactical: to determine the number of resources

needed in each place, how many material re-

sources of each type (MRI, x-ray, scanner, etc.)

are needed in each hospital.

• Operational: to plan exams and assign them one

resource and one period.

In this paper, we are dealing with instances in

which the horizon planning is not greater than one

week, we are focusing on the operational level.

In the following, we will used the word exam and

not task.

3 STATE OF THE ART: BIN

PACKING

The bin packing problem considers N objects and

some bins. The aim is to put all the objects in bins

by minimizing the number of bins, with respect of the

size of objects and bins (Garey and Johnson, 1979).

It is a NP-hard problem and has been widely studied

(Beigel and Fu, 2012).

Our problem consists in the assignment of exams

to a material resource and a period. If we consider

that the horizon planning is composed of couples (l, s)

with material resource l ∈ (1, L) and period s ∈ (1, T ),

the aim of the problem is to assign exams to these cou-

ples (l, s). Exams have to be planed as soon as pos-

sible, the aim is to minimize the number of different

couples (l, s), that is to say to minimize the number

of bins. Table 1 summarizes the links between the bin

packing problem and the problem described in this

paper.

Several heuristics have been developed to solve

this problem (Coffman Jr et al., 1996).

• The typical one is NextFit. Objects are put in the

current bin. If the current bin is full, objects are

put in the next one. We cannot come back to the

current bin anymore.

HEALTHINF2014-InternationalConferenceonHealthInformatics

118

Table 1: Analogies between the bin packing and the consid-

ered problem.

Bin packing problem Considered problem

Object Exam

Bin Couple (l, s)

Size of an object Process time

∅ Due date

∅ Reference place

To assign exams to

To assign objects in bins a period and a

material resource

Constraint of the Constraint of material

capacity of the bins resources opentime

∅ Compatibility constraint

To minimize the number To minimize the number

of used bins of used couples (l, s)

∅ To minimize the last

assigned period

∅ To minimize the sum

of the number of bins

• FirstFit is different from NextFit because objects

are put in the ﬁrst available bin. A new bin is taken

into account if and only if all the previous bins do

not have enough space to accept the current ob-

ject. Bins are not deﬁnitively closed. If a smaller

object ﬁts in a previous bin, it will be put in this

one. In the worst case, the quadratic complexity

in time of this algorithm is O(N log(N)), with N

the number of objects (Corcoran and Wainwright,

1995).

• BestFit consists in putting objects in the best bin

which has some available space. The best bin is

the one which will have the least available space

after this object will be put in it.

FirstFit heuristic is adapted to our problem. This

will be explained in Section 5.2.

4 MATHEMATICAL MODEL

The used data in this problem are the following. Let

be:

• N the number of exams to plan.

• K the number of distinct places.

• L the number of available material resources.

• T the horizon planning. The planning will be done

for s(s = 1, T ) periods.

• t

(i = 1, N) the process time of exam i.

• d

(i = 1, N) the due date of exam i.

• r

i,k

(i = 1, N and k = 1, K) = 1 if place k is the

reference place of exam i, 0 otherwise.

• c

i,l

(i = 1, N and l = 1, L) = 1 if exam i can be

handled by material resource l, 0 otherwise.

• s

l,k

(l = 1, L and k = 1, K) = 1 if material resource

l is located on place k, 0 otherwise.

• p

l,s

(l = 1, L and s = 1, T ) the opentime of material

resource l during period s.

The considered variables are the following:

• x

i,l,s

(i = 1, N; l = 1, L and s = 1, T ) = 1 if exam i

is assigned to material resource l during period s,

0 otherwise.

• C

is the period assigned to exam i.

The constraints are written as follows:

• Material resources opentimes have to be respected

(1).

∑

i=1

i,l,s

≤ p

l,s

, ∀l ∈ {1, L}, ∀s ∈ {1, T } (1)

• An exam is assigned to one material resource and

one period (2).

∑

l=1

∑

s=1

i,l,s

= 1, ∀i ∈ {1, N} (2)

• An exam is assigned to a compatible material re-

source (3).

∑

l=1

(1 − c

i,l

)

∑

s=1

i,l,s

= 0, ∀i ∈ {1, N} (3)

• (4) deﬁnes C

, which is the period assigned to

exam i.

∑

s=1

∑

l=1

s.x

i,l,s

, ∀i ∈ {1, N} (4)

• (5) and (6) integrity and non negativity con-

straints.

i,l,s

∈ {0, 1}, ∀i ∈ {1, N}, ∀l ∈ {1, L}, ∀s ∈ {1, T } (5)

≥ 0, ∀i ∈ {1, N} (6)

Four criteria can be studied:

• The sum of the completion times, i.e. of the as-

signed periods to the planned exams (7).

∑

i=1

(7)

• The makespan, i.e. the period of the last planned

exam (8).

= max

(8)

ActivitiesPlanningandResourceAssignmentonMulti-placeHospitalSystem-ExactandApproachMethodsAdapted

fromtheBinPackingProblem

119

• The number of exams done after their due date

(9).

= N −

∑

i=1

∑

l=1

∑

s=1

i,l,s

(9)

• The number of exams performed out of their ref-

erence place (10).

= N −

∑

i=1

∑

k=1

∑

l=1

∑

s=1

i,l,s

l,k

i,k

(10)

In the following, we are taking into consideration only

the economics aspects, that is to say to assign the

maximum number of exams during the smaller num-

ber of periods and material resources. The studied

objective function is the minimization of the ﬁrst two

criteria.

5 APPROACH METHOD

5.1 Coding of the Set of Solutions

A solution X is the assignment of exam i to mate-

rial resource l during period s. It is coded by a three-

dimensional matrix, each dimension deﬁnes each set

of subscript. The dimension of the matrix is N ∗ L ∗ T .

X = (x

i,l,s

), ∀i ∈ {1, N}, l ∈ {1, L}, s ∈ {1, T }

Let be Ω the set of all the solutions. Let be C ∈ Ω

the set of admissible solutions.

Size of Ω is given in (11).

card(Ω) = L

(11)

For each solution X , all the exams are assigned

to one period and one material resource (2). If the

material resources opentimes (1) or the compatibili-

ties between exams and material resources (3) are not

respected, the solution is called non admissible solu-

tion.

5.2 Initial Solution

Two ways are used to compute an initial solution.

An admissible initial solution can be built by a

heuristic based on bin packing: FirstFit heuristic. It

is described in Algorithm 1. The maximum number

of exams is assigned to the ﬁrst material resource dur-

ing the ﬁrst period. The exam is assigned to the ma-

terial resource if they are compatible. When there is

no free time anymore on this material resource dur-

ing this period, exams are assigned during the same

period to the second material resource, then the third

one, etc. When there is no free time anymore on all

the material resources during the ﬁrst period, exams

are assigned to the second period in the same way.

Algorithm 1: Heuristic for an initial solution.

Require:

temp

l,s

= 0, ∀l ∈ {1, L}, ∀s ∈ {1, T }

begin

for each exam i = 1, N do

OK = f alse, l = 1, s = 1

while s ≤ T.and.OK = f alse do

while l ≤ L.and.OK = f alse do

if exam i is compatible with resource l then

if p

l,s

−temp

l,s

≥ t

then

i,l,s

= 1

temp

l,s

= temp

l,s

OK = true

end if

l = l + 1

end while

s = s + 1

end while

end for

end

We can also randomly build an initial solution. It

consists in assigning a random material resource to

each exam during a random period. In this case, the

initial solution is admissible or not, compatibilities

between exams and material resources and resources

opentimes are not taken into account.

5.3 Metaheuristics

The used metaheuristics are single solution based.

Two stochastic algorithms are detailed and used: sim-

ulated annealing and kangaroo algorithm (iterated lo-

cal search). Under some hypotheses, inhomogeneous

algorithm for simulated annealing and kangaroo algo-

rithm converge in probability to an optimal solution

(Aarts and van Laarhoven, 1987) and (Fleury, 1993).

Principle algorithms are given, using the following

writing:

• X current solution,

• Y candidate solution (or neighbour),

• V,W two neighbourhood systems,

• H cost function.

HEALTHINF2014-InternationalConferenceonHealthInformatics

120

5.3.1 Simulated Annealing

Originally, the inhomogeneous simulated annealing

was used by Metropolis (Metropolis et al., 1953) to

simulate the physical annealing in metallurgy. Sim-

ulated annealing converges in probability to the set

of optimal solutions if neighbourhood system V sat-

isﬁes the accessibility and reversibility property. The

algorithm of simulated annealing is described in Al-

gorithm 2.

Algorithm 2: Simulated Annealing.

Require: temperature T

, initial solution X, record

solution RX := X , decreasing factor α, maximum

number of iterations IterMax

begin

iter := 0, T := T

while iter < IterMax do

Choose randomly and uniformly Y ∈ V (X )

if H(Y ) < H(RX) then

RX := Y

end if

if H(Y ) ≤ H(X) then

X := Y

else

X := Y with probability e

−

H(Y)−H(X )

end if

iter = iter + 1

Generate a new temperature T = α ∗ T

end while

end

Two parameters have to be chosen: the initial tem-

perature and the decreasing factor. The initial temper-

ature T

is chosen so as all the transitions are autho-

rized at the beginning i.e. e

−

H(Y)−H(X )

' 1, ∀(X,Y ).

(Aarts and van Laarhoven, 1987) proposes an algo-

rithm to compute the initial temperature.

The used formula to compute the decreasing fac-

tor α is deﬁned in Equation (12). T

is the latest tem-

perature computed in the simulated annealing, close

to 0.

α =

IterMax

(

) (12)

5.3.2 Kangaroo Algorithm

Kangaroo algorithm consists in a stochastic descent,

but if there is no improvement of the current solution

after a number A of iterations, a jump is made. To

make this jump, a solution is chosen in a neighbour-

hood system W different from V . Kangaroo algorithm

converges in probability to the set of optimal solutions

if neighbourhood system W satisﬁes the accessibility

property. Algorithm 3 describes kangaroo algorithm.

Algorithm 3: Kangaroo Algorithm.

Require: number of iterations A before a jump, ini-

tial solution X, record solution RX := X , maximum

number of iterations IterMax

begin

iter := 0, c := 0

while iter < IterMax do

if c < A then

Choose randomly and uniformly Y ∈ V (X )

if H(Y ) ≤ H(X) then

if H(Y ) < H(X) then

c := 0

if H(Y ) < H(RX) then

RX := Y

end if

X := Y

end if

else

Choose randomly and uniformly Y ∈ W (X )

if H(Y ) 6= H(X) then

if H(Y ) < H(RX) then

RX := Y

end if

c := 0

end if

X := Y

end if

c := c + 1

iter = iter + 1

end while

end

The used formula to compute A is given in (13)

(Fleury, 1993).

A ≥ card(V ).ln(2) (13)

5.3.3 Neighbourhood System

The used neighbourhood system V is deﬁned in Algo-

rithm 4. An exam i is randomly chosen. Its current as-

signment is deleted. A material resource is randomly

chosen among the compatible ones: l ∈ {1, L} such

as c

i,l

= 1. A period is randomly chosen: s ∈ {1, T }.

Exam i is assigned to material resource l and period

s. Equation (14) gives the size of V .

card(V ) ≤ N.L.T (14)

ActivitiesPlanningandResourceAssignmentonMulti-placeHospitalSystem-ExactandApproachMethodsAdapted

fromtheBinPackingProblem

121

Algorithm 4: Neighbourhood System.

Choose randomly and uniformly i ∈ {1, N}

begin

for l and s such as x

i,l,s

= 1 do

i,l,s

= 0

end for

Choose randomly and uniformly l ∈ {1, L} such as

l is compatible with i

Choose randomly and uniformly s ∈ {1, T }

i,l,s

= 1

end

In the kangaroo algorithm, a second neighbour-

hood system W is used. Its principle is to choose ran-

domly and uniformly an exam and to assign it to an-

other period and another material resource randomly

and uniformly chosen. W is used eight times each

time it is called. W satisﬁes the accessibility and re-

versibility property.

5.3.4 Studied Criteria

According to the mathematical model, two constraints

have to be respected:

• Material resources opentimes (1),

• Compatibility between assigned material resource

and the considered exam (3).

To check these constraints, overruns constraints

are computed. So a new criterion Cont is deﬁned. It

is equal to the sum of differences between the sum

of the process times of the assigned exams to a mate-

rial resource during period and the opentime resource

over this period, plus the number of exams which are

assigned to non compatible material resources. It is

computed thanks to the constraints (1) and (3). If

Cont is equal to 0, all the constraints are respected.

Cont(X) =

∑

l,s

max(0,

∑

∗ x

i,l,s

− p

l,s

∑

i,l,s

∗ (1 − c

i,l

) (15)

A weight sum of the three criteria, H

, H

and

Cont can be used. The general form of the objective

function is given in Equation (16), with α

, α

the

weight coefﬁcient.

H(X) = α

(X) + α

.Cont(X) (16)

6 EXPERIMENTS AND RESULTS

The HCT is composed of three places. There are four

or eight material resources, each one located on one

place. The horizon planning is composed of several

periods, each period represents one half-day. The aim

is to plan N exams in the HCT, that is to say to assign

one period and one compatible material resource to

each exam. Exams have to be planned during material

resources opentimes. The objective is to plan all the

exams at the beginning of the horizon planning, the

last exam must be done over the smallest period.

6.1 Dataset

The data are randomly generated but the size of the

data represents real instances. Two kinds of data are

created:

• For instances 50A to 500A, each process time is

between 5 and 45, with 5 minutes steps. Each ma-

terial resource opentime is equal to 300 minutes.

• For instances 50B to 500B, each process time is

between 1 and 100, and each material resources

opentime is equal to 100.

Table 2 summarizes the data.

6.2 Exact Method

To solve this model, we use the version 12.4.0 of

CPLEX. The host machine is powered by an Intel

Xeon X5687 quad-core CPU running at 3.6 GHz. For

each size of problem, results about makespans and

sums of the different periods assigned to each exam

are summarized in Table 3. Computation times are

given in seconds.

The computations which are stopped after thirty

minutes are written with a *. In this case, results about

lower and upper bounds are given. The bigger the

problem is, the longer the computation times are.

is the sum of the completion times, of the as-

signed periods to all the planned exams in the HCT.

is the makespan, the period of the last planned

exam. The used objective function is given in Equa-

tion (17). α

equals 10.000 because in all the in-

stances, H

is always smaller than 10.000. Thus, each

criterion is easily readable.

H(X) = H

(X) + 10.000.H

(X) (17)

For the smallest size of problem, ﬁfty exams have

to be planned. The makespan is equal to two, that

means that at least two periods are used to plan all

of them. The available time in one period is equal to

HEALTHINF2014-InternationalConferenceonHealthInformatics

122

Table 2: HCT Data.

Exams N 50A 50B 100A 100B 200A 200B 300A 400A 500A 500B

Resources L 4 4 4 4 8 8 8 8 8 8

Periods T 8 8 8 15 10 20 15 20 25 40

Table 3: Results with the exact method.

Exams 50A 50B* 100A 100B* 200A 200B* 300A 400A* 500A* 500B*

51 - 128 - 262 - 486 - - -

2 - 2 - 2 - 3 - - -

Time 0.22 - 0.16 - 0.98 - 4.41 - - -

Lower bound - 70135 - 120483.1 - 91056.3 - 40774.2 51114.8 162329.9

Upper bound - 70137 - 120520 - 141074 - 40776 51118 327231

material resources opentime during this period, three

hundreds minutes in this case. So one period is more

or less equal to half-a-day. At least, one day is used

to plan ﬁfty exams over the four material resources in

the HCT.

6.3 Approach Methods

Results of the approach methods are compared to the

exact method’s ones. Two cases are possible:

• [X ∈ Ω]: we can accept non admissible solutions.

Initial solution is randomly and uniformly gener-

ated. The criteria Cont is used to compute over-

runs constraints. In the objective function, α

set to 10

. Thus Cont is the ﬁrst criterion to min-

imize. At the beginning of the computation, it is

greater than zero but at the end, it is always equal

to zero. So the best solution will be admissible.

• [X ∈ C]: we only accept admissible solutions. Ini-

tial solution is created by the proposed heuristic.

Results about initial solution are given in table

4. The generated neighbour has to be admissible,

that is to say Cont = 0.

Table 4 summarizes results about the makespan

) and the sum of the assigned periods to each exam

For each replication, the number of iterations is

equal to 1.000.000. The gold standard are the results

of the exact method given by CPLEX in the previous

section. CPLEX optimal solution is written if found.

If not, upper bounds are written because they are the

best found integer solutions (*). Mostly, methaheuris-

tics ﬁnd the optimal makespan.

For big instances (**), the makespan is not the

same in approach and exact methods. In these cases,

in the approach way can be better than in the exact

one.

Moreover, the heuristic can ﬁnd a better makespan

than the metaheuristics for instances B. In these in-

stances, size of a ”bin” is equal to 100 and some ex-

ams have got a process time which is equal to 100.

With the heuristic, exams are sorted by decreasing

processing times. Thus, the biggest exams are pro-

cessed in the ﬁrst periods. With the metaheuristics,

once a big exam is put in one of the last period, it can-

not be moved to another ”bin” otherwise it will not

respect resources opentimes. But every time, meta-

heuristics give better results than the heuristic about

. Most of the times, simulated annealing gives the

best results.

Results about H

are optimal if the number of ex-

ams is small. For large instances, there is no opti-

mal solution. The exact method gives an approxi-

mation after thirty minutes of computation. With the

approach methods, optimal makespan, if existing, is

found in some seconds and a good result of H

some minutes. Almost all the exams are assigned to

the optimal period, only some of them are assigned

during the next period of the optimal one.

7 CONCLUSIONS

In the current economic context, solutions have to be

found to improve hospital efﬁciency. By dividing up

exams over an Hospital Community of Territory, ex-

ams could be done more quickly. In a same horizon

planning, more exams could be planned.

In this paper, a formulation of the problem is

given. Exact and approach methods are compared.

For large instances, exact method does not return the

optimal solution while approach method ﬁnd a good

solution in some minutes. Optimal makespan, if ex-

isting, is found in a few seconds and a good result

for all assignments is found in a few minutes. Other

heuristics and other neighbourhood systems have to

be tested to improve the best solution.

At the operational level, in this paper, assignment

are made over seven days. In real life, planner needs

to anticipate the planning one month in advance. By

ActivitiesPlanningandResourceAssignmentonMulti-placeHospitalSystem-ExactandApproachMethodsAdapted

fromtheBinPackingProblem

123

Table 4: Results with the metaheuristics.

Exams 50A 50B* 100A 100B** 200A 200B* 300A 400A 500A 500B**

CPLEX (upper bound)

2 7 2 12 2 14 3 4 5 32

51 137 128 520 262 1074 486 776 1118 7231

Solution generated by the heuristic

2 7 3 12 3 14 4 4 6 31

54 203 142 684 295 1435 579 957 1408 8228

Kangaroo algorithm, x ∈ Ω

2 7 2 14 2 16 3 4 5 36

51 154 129 554 266 1220 504 815 1185 6520

Kangaroo algorithm, x ∈ C

2 7 2 13 2 15 3 4 5 33

51 140 128 508 264 1126 496 800 1161 6465

Simulated annealing, x ∈ Ω

2 7 2 13 2 15 3 4 5 35

51 138 128 503 263 1154 491 796 1164 6326

Simulated annealing, x ∈ C

2 7 2 13 2 15 3 4 5 34

51 139 128 488 262 1127 494 799 1172 6228

increasing the number of periods, or by increasing

the size of the periods, we could apply this study to

the tactical level or the strategic one. This could be

used to determine the number of resources or places

needed in a HCT.

The made hypothesis here is that human resources

are always available. In further research, we could in-

clude them to our model. Human resources can work

or not overtime and they can be allowed to move over

the HCT. So the load between the places can be bal-

anced.

REFERENCES

Aarts, E. and van Laarhoven, P. (1987). Simulated Anneal-

ing: Theory and Applications. Kluwer Academic Pub-

lishers.

Beigel, R. and Fu, B. (2012). A dense hierarchy of sublinear

time approximation schemes for bin packing. In Fron-

tiers in Algorithmics and Algorithmic Aspects in Infor-

mation and Management, pages 172–181. Springer.

Coffman Jr, E. G., Garey, M. R., and Johnson, D. S. (1996).

Approximation algorithms for bin packing: A survey.

In Approximation algorithms for NP-hard problems,

pages 46–93. PWS Publishing Co.

Corcoran, A. L. and Wainwright, R. L. (1995). Using libga

to develop genetic algorithms for solving combinato-

rial optimization problems. Practical Handbook of

Genetic Algorithms, 1:143–172.

Fleury, G. (1993). M

ethodes stochastiques et d

eterministes

pour les probl

emes NP-difﬁciles. Ph.D., Universit

Blaise Pascal, Clermont-Ferrand II.

Garey, M. R. and Johnson, D. S. (1979). Computers

and Intractability: A Guide to the Theory of NP-

completeness. WH Freeman and Company, New

York.

Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N.,

Teller, A. H., and Teller, E. (1953). Equation of state

calculations by fast computing machines. The Journal

of Chemical Physics, 21:1087–1092.

HEALTHINF2014-InternationalConferenceonHealthInformatics

124