Asignacion Mix Camas

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Document de travail du LEM

2009-03

HOSPITAL BED CAPACITY AND MIX PROBLEM

FOR STATE SUPPORTED PUBLIC AND FEE PAYING

PRIVATE WARDS

Yazgi Ttnc, David Newlands

ISEG School of Management, CNRS-LEM (UMR 8179)


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Hospital Bed Capacity and Mix Problem for State Supported

Public and Fee Paying Private Wards

Yazgi Ttnc and David Newlands

IESEG School of Management

in collaboration with Izmir University of Economics and Ege University Hospital

Oncology Department, Izmir, Turkey

1. Abstract

Hospital bed capacity planning accuracy is important to satisfy patients needs, organize

departments and improve the quality and amount of service provided. Health care management

thus should aim to optimise activities within the imposed. This study applied the branch-and-

bound method, M/M/s algorithm, genetic algorithm and Dijkstra algorithm. This article

presents techniques and solutions used to solve the bed capacity planning problem. Results

obtained via these techniques enabled the efficiencies to be compared. Comparison to actual

data suggest specific techniques generate higher accuracy predictions that we recommend by

applied by hospital managers to commit resources that more closely match demand.

In this study, Bed Capacity Planning Problems have been modeled using two time buckets:

annual bed capacity planning and monthly bed capacity planning. The annual capacity problem

is investigated using three techniques:

1) as an Integer Programming Model. The objective is to minimize total waiting cost

and total cost,

2) as an M/M/s Queue Model in which the beds are considered as servers, and

3) as a Genetic Algorithm.

Monthly time buckets were defined as Shortest Path Problems (SPP) in which the vertices

meaning have been considered as possible monthly bed capacity decisions for the department

and solved using the Dijkstra Algorithm. The variances between results and observations were

analysed to identify accurate and repeatable methods for the application.


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2. Introduction

To satisfy patients needs and organize departments, hospital managers periodically review

tactical and strategic plans for anticipated bed capacity. The objective is to define requirements

for months and years ahead. In most of the cases, patients arrival times appear to vary

randomly and their length of stay are functions of various internal and external influences and

constraints. Hence, the factors that influence bed capacity planning can be modeled

stochastically. A stochastic variable construction of the health care process increases the model

complexity. Estimating ranges, probabilities and limiting values are required. Underestimating,

planned resourcing slow response times and random events. This in turn makes the

determination of the optimum bed capacity, measured via utilisation for a time period, more

problematic. In order to overcome these difficulties some hospitals keep their bed capacity

sufficiently large to overcome this periodic fluctuations. This approach reduces inbound

registration delays, turning away walk-in patients, and transfers to other health care sites are

kept minimum. Typically, the occupancy level decreases as the bed capacity increases. Keeping

more beds and staff active on a just-in-case basis (uncontrolled bed capacity) incurs extra

hospital costs as a result of constantly available additional service provision that is may be in

emergency situations be required but for normal operations are not required. Management,

funding bodies and society then can perceive that the cost to benefit ratio is unjustified and

prevision should be trimmed to suit reliable and forecastable demand profiles. Hence, hospital

administration perceive it as vital to estimate the optimum bed capacity for any period throughthe year.

Literature [1] suggests various methods including ratios, discrete event analysis, queuing

dynamics and stochastic simulation have been to solve the bed planning problem. Ratio-based

methods using average treatment durations have been applied extensively to determine required

bed capacities [2-3]. The average required bed capacity is derived from a function of the length

of stay (LOS) ratio [3]. Long and short term capacity decisions tend to be made based on target

occupancy levels. These two methods tend not to take into account day to day and hour by hour

admissions request fluctuations. This type of variation in part is due to the unpredictable nature

of hospital admission rates and patients length of stays. Stochastic simulation models also have

been used to determine the size of the required bed capacity based on the number of patients in

the hospital [4-5]. A combination of simulation, queuing theory, statistical analysis and

mathematical modeling are used to determine the bed requirements and to predict the

probability that patients needing admission will be turned away or transferred [6-7-8].

Stochastic methodologies that combine queuing network analysis with integer programming

models and discrete event simulation are used to balance bed occupancy rates against reduceddelays, reduced patient blocking and reduced poor bed assignment [9-10]. Network flow


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models that incorporate facility performance and budget constraints also have been developed

to determine the optimum size of bed capacity [11]. The studies cited to a certain extent are

theoretical and are arduous to apply by hospital management in a real life.

This article proposes a decision analysis method that easily can be used by decision makerswithout specific technical modeling knowledge. The proposed approach enables hospital

administration to choose the most appropriate method and remains simple to understand and

apply. The project objective was to enable them to include their knowledge and insight in the

solution process.

This article defines the analysed problem in section 3. Annual bed capacity planning models,

specifically an Integer Programming Model, M/M/s Queue Model and Heuristic Methods are

reviewed in section 4. Monthly time bucket bed capacity planning models based on the

Network Model and Modified Network Model are reviewed in section 5.

3. Problem Definition

This article summarises analysis of a bed capacity planning problem at Ege University Hospital

Oncology Department in zmir Turkey. At the beginning of the study, the department manager

acknowledged the Ege University Hospital Oncology Department needed to increase bed

capacity in order to satisfy their patients needs and to improve the departments service

prevision key performance metrics. The departments manager defined space and budget

limitations (53000 TL) to increase the bed capacity. At the beginning of this study, there were

22 beds in the department, of which 10 were reserved for private practice and the remaining 12

served both state assisted and private health care requirements. The limited floor place limited

the potential department capacity increase to a maximum of 30 beds. It was widely accepted by

administration and medical team representatives alike, that should the enlargement occur, there

were enough doctors to provide sufficient health care attention for the anticipated maximum

capacity. Nurses and auxillliaries would need to be recruited to fully service the increased

capacity. Initially there were 11 nurses and 9 nurse-aids.

Another issue that the department was dealing with was that since most of the patients who

come to this department must be treated urgently, the use of private beds as common beds.

This was attributed to there being insufficient common beds in the original prevision mix.

Initial meetings with the hospital manager lead to defining the problem as determining the

number and the allocation of the beds needed in total in order to decrease the occupancy level

(waiting time per patient) without exceeding the departments budget. The randomness in the

number of patients arriving and their length of stay typically presupposes a stochastic analysis.


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Such processes make determining the optimum capacity more arduous. The key parameters that

affect the optimum size of the required bed capacity are: occupancy levels, the number of

admissions per day, service level expectations and the incurred staff cost to service the demand.

Some of these parameters are known. Most, however, have to be estimated. This study

combined deterministic, stochastic and heuristic approaches to solve the problem.

4. Annual Bed Capacity Planning Models

4.1. A Generic Mathematical Model for a Hospital Bed Capacity Planning Problem

4.1.1. Decision Variables

The decision variables are defined as

iCPB

iCB

iPB

iCBI

iPBI

ijNA

ijN

ijD

ijNAI

ijNI

ijDI

i

i

i

i

i

ij

ij

ij

ij

ij

ij

departmentinbedscommonasusedbedsprivateofNumber:

departmenttoaddedbedscommonofNumber:

departmenttoaddedbedsprivateofNumber:

departmentinbedscommonofnumberInitial:

departmentinbedsprivateofnumberInitial:

departmenttoaddedaids-nurseType:

departmenttoaddednursesType:

departmenttoaddeddoctorsType:

departmentinaids-nursetypeofnumberInitial:

departmentinnursestypeofnumberInitial:

departmentindoctorstypeofnumberInitial:


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4.1.4. Explanation of the Constraints

departmenteachforconstraintBudget(4.6)

departmenteachtoaddedbecanbedsofnumberMaximum(4.5)

departmenteachinbedcommonasusedbedsPrivate(4.4)

tenlargementhetoduedepartmenteachtoneededaids-nurseofnumberandType(4.3)

tenlargementhetoduedepartmenteachtoneedednursesofnumberandType(4.2)

tenlargementhetoduedepartmenteachtoneededdoctorsofnumberandType(4.1)

4.2. Integer Programming Model

The objective function and constraints stated in section 3 were introduced into the problem.

This outputs required were minimized total waiting cost and total cost of the oncology

department. Since existing staff costs including doctors salaries and the cost of operating beds

already were financed by the government, the only augmented cost anticipated is limited to the

salaries of the nurses and nursing assistants. As a consequence, reducing the non-occupancy

level became a primary objective.

4.2.1. Decision Variables

The decision variables are defined as

addedaids-nurseofNumber:

addednursesofNumber:

leveloccupancytheofbecausebedcommonaasusedbedsprivateofNumber:

addedbedsprivateofNumber:

addedbedscommonofNumber:

NA

N

PCB

PB

CB

4.2.2. Objective Function

The objective function is

)122700(16200)10450(16200

)10(16200)9(8760)11(10980MinimizeCBPB

PCBPBNAN

++++

++++

Here, the first two coefficients are the yearly costs of nurses and nursing assistants respectively.

The third statement is the annual contribution to profit generated by operating private beds. The

fourth is the opportunity cost occured when a potential private patient that had been waiting for

a private bed goes elsewhere due to semi-temporary full capacity situation exists. The

remaining expression is a fairness coefficient that balances the opportunity cost of loosing a

patient that waited for a common bed. This is required because the department does not incur


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treatment costs nor is there a lost profit opportunity from operating common beds. The

coefficient statements in this model were set using the data that is generated empirically at the

Ege University Hospital Oncology Department.

4.2.3. Constraints

The constraints that are used to solve the problem are given as

(4.12)integersare,,,

)11.4(8

)10.4(8.220))12(2700(

)9.4()(409.0

)8.4()(5.0

)7.4(53000)10(16200)9(8760)11(10980

NANPBCB

PBCB

PCBCB

NAPBCB

NPBCB

PCBPBNAN

+

+

+

+

++++

4.2.4. Explanation of the Constraints

addedbecanbedsofnumberMaximum(4.11)

leveloccupancytheofbecausebedcommonasusedbedsPrivate(4.10)

occurstenlargementhecaseinneededaids-Nurse(4.9)

occurstenlargementhecaseinneededNurses(4.8)

constraintBudget(4.7)

4.2.5. Solution and Interpretation

Most integer problems in practice are solved by searches using Branch-and-Bound techniques.

The approach determines an optimal solution to an integer problem by quantifying enumerating

points within sub-problem variables credible ranges. Applying the branch-and-bound method

requires a lower limit to the optimization problem to be identified and the feasible region must

be divided to obtain smaller graduations that are treated as separate subproblems. There also

must be a way to compute a feasible solution at the other extreme - an upper bound. It would be

a rare event to expect every variable to be the best case or worst case extreme. However, to

generate a viable model these extremes must be considered in order to identify more likely

combinations of scenarios. For practical purposes, it should be possible to compute upper

bounds for a set of non-trivial, yet feasible, variables.

Initially the problem is considered with the complete feasible region, which is called the root

problem. Lower-bounding and upper-bounding procedures are applied to the root problem. If

the bounds match, then an optimal solution has been found and the procedure terminates.

Otherwise, the feasible region is divided into two or more regions, each are equal divisions of

the original. All together they cover the whole feasible region; ideally, these subproblems


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partition the feasible region at a high enough resolution to identify the solution, and with as few

partitions as possible in order to reduce redundant mathematical complexity. The sub-problems

replace the single root of the search node. The algorithm approach is applied repeatedly to the

sub-problems in order to generate a complete set or tree of sub-problems. If an optimal

solution is found to a subproblem, this may be a feasible solution to the problem even though it

may not optimum solution for all the conditions and permutations. Since it is feasible, it can be

used to prune the rest of the tree: if the lower bound for a node exceeds the best known feasible

solution, no globally optimal solution can exist in the subspace of the feasible region

represented by the node. Therefore, the node can be removed from consideration. The search

proceeds until all nodes have been solved or pruned, or until some specified threshold is met

between the best solution found and the lower bounds on all unsolved subproblems. The solved

the problem using the LINGO 11.0 [12] on a standard Windows formatted computer using the

branch-and-bound method. In total 960 iterations were computed and a local optimum solution

has been found.

The analysis suggested Ege University Hospital Oncology Department required 3 common and

5 private extra beds and in addition to these, 4 nurses and 4 nursing assistants should be

recruited or assigned. A minimum additional budget of 51780 TL has been identified to fund

this enlarged service provision. By applying these changes, it was anticipated that the

occupancy level would likely decrease. The trade off anticipated was to be able to treate more

patients without exceeding the departments gross annual budget.

4.3. M/M/s Queue Model

Queuing theory can be used in various areas of real life, for example in parking places, banks,

supermarkets, post offices, traffic lights, toll booths and in data analysis, telecommunication,

and production systems.

In the hospital setting, the group of individuals from which arrivals come is referred to as the

call-in population. Variations occur in this population's size. Total customer demand requiring

service from time to time constitute the size of it. The cluster of customers waiting for

admission and subsequent service is known as the queue. The basic queueing process can be

considered as the following. The size of groups of customers waiting for service varies over

time. The hospital perceives such customers as a call-in population composed of unannounced

immediate requests for urgent service and those that have reserved low urgency diagnoses or

interventions. These customers enter the queuing system and join an admission or treatment

queue.


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A member of the queue is selected for service by some rule known as the queue discipline at

specific times. Queue discipline refers to the order in which customers are selected for the

service. Once selected, customer then transfers from the admissions queuing system to the

service mechanism [13] in order to receive or undergo the initiated service. Queue participants

requests can be assessed in batches or one by one. Batches are groups of requests that have

accumulated over a time interval. Convention usually dictates they are processed on a first-

come-first-served (FCFS) basis. Exceptionally, last-come-first-served (LCFS) when clearly

urgent requirements are identified. Part of the admissions process service policy criteria is to

explicitly state at the admissions stage that such cases shall be prioritized. Further prioritization

criteria can be enforced, such as premium service contracts versus basic provision cover.

Random sequencing can be identified in four respects:

Demand from individual from a given population, Specific demand from extra-populous traveling or visiting populations, Unscheduled and random arrival at times. A key variable is the frequency of individual

and group arrivals. Given sufficient data sets these can be modeled as an arrivals

distribution.

Demand for specific health care services that may be clustered into a limited number ofgeneric families. The number of families may be increased or reduced to facilitate themodeling procedure, as well as create clear service routes through the health care

process. Increasing the number of clusters may reduce overall capacity in each process

and reduce the systems resources flexibility. The capacity of the system elements

becomes more apparent as more divisions are created. Each cluster then will be limited

to a specific capacity. Overall, the number of beds will be fixed in the short term,

although there can be specialized beds for certain conditions and client requirements,

such as day care, short and long term sequences, isolation wards, critical care and post

intervention recovery.

Most of the random criteria may be assessed retrospectively using marketing and client survey

techniques in order to better generate and validate quantitative demand forecasting models. The

health care service mechanism consists of one or more service facilities called server hubs.

Periods of saturation can be experienced. Hands may be put up to say no more for now, or

we cant take any more at the moment. At this level of activity, the admissions system shall

start diverting non-essential and even urgent cases to other health care centres.


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The number of random drop-in events can be reduced when ambulance and other organized

patient transport services co-ordinate with central regional demand and capacity management

decision support centres. The patients in such cases can be routed directly from pick up to an

assigned health care centre in the region, such as when demand exceeds bed availability and

day-care capacity are over stretched, or in exceptional circumstances transferred outside the

region to neighbouring regions or further afield, including across national frontiers. A certain

amount of medicare tourism exists for patients that have sufficient funds or insurance to be able

to afford travel, subsistence and treatment abroad. In the United States, some 36 million

inhabitants do not have health care insurance. State aid is available for individuals with servere

medical conditions that also have had their financial resources assessed. Many others attend

day care centres. With little choice, they have to be prepared to wait many hours for dental and

other interventions.

Extreme conditions are identifiable in various locations. The hospital studied had a multi-server

queuing system represented by Figure 1.

By convention queuing models are labeled:

ZmKsBA ///// , where

queuein thecustomersordertousedDiscipline

stopsprocessarrivalthebeforearrivetoallowedcustomersofNumber

systemin theallowedcustomersofnumberMaximum

servicesofNumber

timesserviceofonDistributi

timesalinterarrivofonDistributi

Z

m

K

s

B

A

A will be filled with one ofM ,D , where is Ek? kE , GIand B will be filled with M ,D , kE , G ,

where

allowed)ondistributiarbitrary(anyondistributitindependenGeneral

allowed)ondistributiarbitrary(anyondistributiGeneral

times)(constantondistributiDegenerate)(MarkovianondistributilExponentia

GI

G

DM


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Figure 1. A Multi-server Queuing System Representation

Source? Replace with ppt version

4.3.1. Terminology and Notation

State of system Number of customers in the queueing system, that is, number of customers waiting in

the queue + number of customers that are being served at this time.

Queue length Number of customers waiting in the queue.

)(tN Number of customers in the queueing system at time t, ( 0t )

)(tpn Probability of exactly n customers in the queueing system at time t, ( 0t ), giventhe number at time 0.

s Number of servers in the queueing system. Expected number of arrivals per unit time of new customers.

Mean long term arrival rate.

1 Expected inter-arrival time.

Expected number of customers completing the service per unit time.

)(1 ALOS= Expected service time = average length of stay.

The utilization factor for the service facility = s np Probability of exactly n customers in the queuing system.

)(LE Expected number of customers in the queuing system.

)( qLE Expected queue length.

)(WE Expected waiting time in the queuing system for each individual customer.

)( qWE Expected waiting time in the queue for each individual customer.

There are some relationships between )(LE , )( qLE , )(WE and )( qWE such as

)()( WELE = , )()( qq WELE = and 1)()( += qWEWE [14].

4.3.2. Birth-and-Death Process

Births and deaths are significant variable that influence the current size of a population. Special

continuous-time Markov processes can be used to anticipate long-term macro-level population

fluctuations of regions and states. Expatriation, immigration, repatriation and illegal or

unregistered residents are other variables that can be configured. However, to simplify Markov


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models these typically only focus on population changes of the resident majority, hence the key

variables represented are limited to births and deaths. Birth-and-death processes have many

applications such as macro and regional population demographics, queuing models for local,

regional and national facilities and genetic pool associated biological susceptibilities that can be

measured via reported cases and treatment frequencies.

When a birth occurs, the population changes state, from n to 1+n . When a death is

registered, the process goes from current state n to state 1n . The process is specified by

birth rates )0(, = Kii and death rates )1(, = Kii .

Figure 2. The Rate Diagram for the Birth-and-Death Process Reference

A fundamental flaw in the birth-and-death process structure is a reliance on an equilibrium

between birth and death rates. This assumes the overall population shall remain constant,

particularly in the short term. There are significant examples of exceptions to this principle: the

government of The United Kingdom of Great Britain and Northern Ireland anticipate a

population increase from just over 60 million to a desired maximum 70 million during the first

half of the 21st

century. Countries such as Italy have negative population growth. Both Italy

and the UK now have more of retired people than under sixteen years of aging.

The balanced population model models queuing system dynamics on a Rate in = Rate out

basis. This principle imposes that for any state of the system can be expressed by an equation

which is called the balance equation for state n .

,,2,1,0 K=n

mean entering rate = mean leaving rate.

Constructing balanced equations for all the states in terms of an unknown population np , we

can find probabilities for given levels by solving the the equation 10

=

=n

np . Balance equations

are given in Table1.

Table 1. Population Equilibrium Equations


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State Rate In = Rate Out

0

1

2

M

1n

n

M

0011 pp =

1112200 )( ppp +=+

2223311 )( ppp +=+

M

11122 )( +=+ nnnnnnn ppp

nnnnnnn ppp )(1111 +=+ ++

M

Given11

021

K

K

=

nn

nnnK for K,2,1=n where 10 =K and 0=n

Hence, probabilities to achieve a given steady-state are expressed as:

0pKp nn = for K,2,1,0=n

This requires that

1

0

0

=

=

n

nKp .

Once np is calculated, )(LE , )( qLE , )(WE and )( qWE can be obtained using the

following:

=

=0

)(n

nnpLE ,

=

=sn

nq psnLE )()( ,

)()(

LEWE = and

)()(

q

q

LEWE = , where

=

=0n

nn p .

These steady-state expressions require both n and n to have realistic steady-state values.

Results are always valid when 0=n for values of n greater than the initial state. Hence only

a finite number of states are possible. The results also are always valid when and are


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defined by expression 4.3.1 and 1s , we have

=n for K,2,1,0=n

+=

==

K

K

,1,

,,2,1

ssnfors

snfornn

The rate diagram for the multiple-server case is given in Figure 3.

Figure 3. Rate Diagram for the Multiple-Server Case

The queuing model proposed as a result of this project is an M/M/s )1( >s model that is based

on the rate in = rate out principle of the birth-and-death process. The queue model assumes in

the short term a given steady-state condition in which customers are patients and the servers are

beds. Results here are derived when )1( >s . Hence, when 1>s , the nK factors become

( )

( )

+=

=

=

K

K

,1,!

,,2,1!

ssnforss

snfornK

sn

n

n

n

Consequently, if 1


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( )

( )20

1!)(

=

s

pLE

s

q ,( ) ( )

( )2

2

0

1!

1!)(

+=

s

spLE

s

,

( )

( ),

1!)(

2

0

=

s

pWE

s

q ( ) ( )

( )2

2

0

1!

1!)(

+=

s

spWE

s

.

The bed optimization project proposed an M/M/s algorithm shown in Algorithm 1 below, and

tested with variables from the parameters defined previously. When examinging this type of

system dynamic, analysts may develop and select existing verified algorithms embedded in

optimization software. Variable fields then are filled (see Figure 4) and the solution is posted in

a results box.

Algorithm 1. Pseudo-code of the M/M/s algorithmStep 1 1.1 Set

current next

Step 2 2.1

while current next current=Calculate expected queue length for #common bed next= Calculate expected queue length for

#common bed+1 #common bed= #common bed + 1

2.2 Set current next

Step 3 3.1

while current nextcurrent= Calculate expected queue length for #private bed next= Calculate expected queue

length for #private bed+1 #private bed= #private bed + 1

Step 4 4.1

Display #common bed and #private bed

Figure 4 shows the Ege University Hospital Oncology Department needs 8 common beds and

one private bed. Historic data and trends can be analysed to determine expected short term and

long term demand for common and private health care treatment. The mix of preventative,

requested non-essential and urgent survival treatments must be assessed. The registrationprocess scheduled non-essential private treatment for defined weekly time buckets where

immobilization and monitoring where essential. Demand for non-urgent beds used defined

scheduled. Given annual current demand for this, intuitively only one extra bed was expected to

the solution. Since common beds that also can be used privately for urgent survival monitoring,

interventions and recoveries, the extra common beds shall be equipped on a just-in-case basis to

cope with these differing treatment phases. Demand traditionally had been higher for common

beds, though these did not contribute to margins. Space also was limited in the building. Taking

into account the equipment and patient access space required limited an increase in capacity up

to 8 beds, the best fit between physical and budgetary constraints and the theoretical optimum


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(8 common and 1 private) the researchers recommended 7 common beds and 1 private bed.

Figure 4. A display window for an M/M/s algorithm

4.4. Genetic Algorithm Overview

Genetic algorithms are a class of non-deterministic algorithms that derive from Darwinian

evolutionary principles [15]. Genetic algorithms have two principal applications: to model

systems such as political systems that evolve overtime [16] and to model provision of goods.

The latter may not be optimal solutions for combinatorial optimization problems [17]. Jones,

Robertson and Willett asserted that for both uses, exploring all possible solutions using a

deterministic algorithm almost would be infeasible within an acceptable time frame [15].

With colleagues and students, John Holland [21] developed genetic algorithms (GAs). Goldberg

[18] categorises aims of their research in two groups: to abstract and rigorously explain the

adaptive processes of natural systems and to design artificial systems software that retains the

important mechanisms of natural systems. He summises these views guided subsequent

important innovations in natural and artificial systems science.

GAs are methaphorically similar to the evolutionary processes since both accept and model the

survival of the fittest concept. GAs have populations that are demographically diversified as aresult of randomly generated combinations of individual elements that have different attributes


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and variables. In effect, mutations are permitted that build on and create permutations as a

result of selection criteria and inherited trait principles [19]. Box [20] first proposed to develop

solutions to problems using selection and variation. This work did not include the use a

computer. Some the studies conducted in the 1960s, most notably Holland [21] and Rechenberg

[22], were based on evolution principles were analysed using mainframes.

Rechenberg [22] highlighted that selection and mutation are important mechanisms used to

solve difficult numeric optimization problems. By contrast Holland [21] emphasized adaptive

properties of populations and the importance of recombination or crossover mechanisms.

GAs start with a population constituting randomly created solutions. Better fitting solutioning

have a higher probability of being recombined into new solutions via an information transfer to

subsequent generations of solutions. Furthermore, new solutions may occur through mutation

or crossover operators. Some of the fittest solutions of each generation are retained, while

others are replaced by the newly constituted solutions. The process is repeated until goals are

achieved.

In a given population, each individual is formed by a string of binary digits (bits). There are

numerous permutations possible though self selection via mutual compatibility and acceptance

criteria effectively limit a solutions fit within the acceptable limits tolerated by a population.

Binary cases are argued to be the most rudimentary repeating form [18]. In a given biological

lifeform population, each individual can be classified into a single genotype (string of bits) and

has specific or unique genetic material. Forrest [19] asserted the term bit string refers to both

genotype and the individuals that they define and proposed the metaphor that each bit position

represents one gene.

4.4.1. Crossover and Mutation Operators

The crossover proposes enables building blocks to contributions that combine into new strings.Various crossover methods exist. This study used two different crossover methods: order-based

crossover and position-based crossover. In order-based crossover, three chromosomal elements

are combine: parent 1, parent 2 and temp_crom (temporary chromosomal abnormality/variant).

Parent 1 and parent 2 are selected from the population while temp_crom is created randomly

from binary integers. This method enables parent genes to be copied to direct decendents (child

1 and child 2). The individual trait exhibited from either parent are selected in accordance with

the temp_crom gene selection template. Remaining decendent elements are defined by forcing

gene elements into the mix from the parent deselected by temp_crom.


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In position-based crossovers, parents genes are copied to children based on temp_crom genes

states 0 and 1 that indicate parent 1 and parent 2. Decendents missing genes are filled using the

order of elements in the childs own. Order-based and position-based crossovers are illustrated

for comparison in Figure 5.

Parent1 : 2 4 6 8 10

Parent2 : 1 3 5 7 9

Temp_crom: 0 1 1 0 0

Child1 : - 4 6 -Child2 : 1 - - 7 9

Child1 : 1 4 6 3 5

Child2 : 1 2 4 7 9

Order-based crossover

Parent1 : 2 4 6 8 10

Parent2 : 1 3 5 7 9

Temp_crom: 0 1 1 0 0

Child1 : - 3 5 - -Child2 : - 4 6 - -

Child1 : 2 3 5 8 10

Child2 : 1 4 6 7 9

Position-based crossover

Figure 5. Order-based and Position-based Crossover Procedures

There are also several mutation methods exist. This project used two: scrumble-sublist and two

point mutation methods. Scramble-sublist mutation selects one parent a from the population and

two positions are selected from this parent randomly. After that all elements before and

including the smallest one and also after and including the other elements are copied to

decendent chromosomes in the corresponding positions. Mutations are created when a

decendents missing chromosome elements are filled randomly. Two point mutation selects two

genes from the parent chromosomes. These are reproduced in these positions. Other elements

are copied to decendents chromosomes. Examples of scramble-sublist mutation and two-point

mutation methods are illustrated in Figure 6.

Parent1 : 2 4 6 8 10 12

Chosen points: 2 and 5

Child1 : 2 4 7 9 10 12

Scramble-sublist Mutation

Parent1 : 2 4 6 8 10 12

Chosen points: 2 and 5

Child1 : 2 7 6 8 10 12

Two-point Mutation


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Figure 6. Scramble-sublist and two point mutations

To ensure the long term viability of sequential of chromosomal element transfers without

creating a form of sterility due to inbreading within a relatively small population, there is a 2%

probability that a mutation may occur. This then creates the possibility of enabling a population

to adapt organically over generations to external influences such as climate and other

environmental factors. In a business context various factors include: financial crisis and credit

crunches, regional and global recession and depressions, war efforts, currency value

fluctuations, political imperatives, customer requirements, supply shortages, demographic

changes, level of general and specialized employee education, use of common language and

other communication difficulties, culture clash, industrial and employee relations, technical

advances in materials used, product and process design, ability to learn and adapt, resistance to

change, entry investments, exist and diversification strategies, actions undertaken by

competitors, and un-anticipated or low probability high impact events.

4.4.2. Parent Selection

The parent selection method assigns a probability that genes shall be propogated to each

individual in a population. This is governed by a proportionality rule that evaluates individuals

fitness. In function of this viability, attractiveness or other comparable performance, the

algorithm assigns a number of opportunities. Several methods for the parent selection are widely

applied. This project used the roulette-wheel method to analyse the bed capacity problem. Davis

[23] proposed the roulette-wheel method works as follows:

1) Sum the fitness of all the individuals in the current population. This is the sum

total_fitness,

2) Generate a random number k between 0 and total_fitness,

3) Select the first fitness value for T such that:

=

T

i

kifitness1

)(

This procedure forces theth

T chromosome to be a parent.

Although the selection process is random, the chance of selection of a parent is directly

proportional to that individuals fitness. Since selection is random, a population member that

has a poor fitness has a lower probability, though still in with a chance, of being selected as a

parent.


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The starting conditions for the model used in this study had randomly created a population, with

a range of both feasible and unfeasible solutions. The procedure enabled each individual, that

had a fitness value and a individual code, to be kept in the init_pool. The init_pool already has a

maximum capacity and after this capacity is reached, one of the two operators (mutation or

crossover), is selected randomly. Operators are chosen using a roulette-wheel method. One or

both parents can be replaced by the selected operator. In most cases only one parent is replaced.

Both parents can only be replaced when the crossover operator is used. In this context, a new

generation can only be created using selection and operator methods. While the new generation

is being created, the 2% of the best solution is kept and copied to the next generation base upon

the survival of the fittest principle. This process is iterated until evolutionary process suspension

criteria are activated. Algorithm 2 enumerates a GA pseudo code based on boulean logic and

BBC BASIC language.

Algorithm 2. Pseudo-code of the genetic algorithm

Step 11.1. Randomly generate an initial population composed of distinct individuals and combine

them into the init_pool (initial populations pool)

Step 2

2.1. Assign a fitness value to each individual

2.2. Retain and duplicate the best 2% of population to the prod_pool (producing pool) thatsubsequently shall form part of the new generation.

2.3. Select operators randomly

IF Mutation operator is selected, THENchoose the mutation type randomly

(Scramble-sublist or two-point mutation)

ELSEChoose crossover type randomly

(Order-based or position-based crossover)

2.4. Choose parent by the roulette-wheel method

IF crossover operator is selected, THENchoose another parent by the same

way

2.5. Apply chosen operator to the chosen parent or parentsIFcrossover operator is selected THEN2 children are produced

ELSEone child is produced

2.6. Take all children into prod_pool

Take also feasible solutions (children) to the feas_pool (feasible solutions pool). Keep

the best feasible solution as a best_of_generation.

Step 3

3.1 IF the prod_pool is full and there is no improvement in the best_of_generation for 30

generations or there is improvement in 3000 generations produced, THENSTOP;ELSE

go to Step 2


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Figure 7 shows the use of genetic algorithm and results for Ege University Hospitals

Oncology Department; 3 common and 5 private extra beds are recommended plus 4 nurses and

4 nurse-aids. The minimum total extra budget is 51780 TL.

Figure 7. Client needs analysis using a Genetic Algorithm

5. Monthly Bed Capacity Planning Models

5.1. Network Model

Network model structers are defined below. In order to find the optimum solution for the mix

of the beds required, let the edges i to j denote the type of the beds that are used during the

thk month. kijc is the incurred cost of using beds i to j at thethk month. The variable kijx is

denoted as follows:

=

otherwise,0

monthat theusedistofromedgetheif,1 th

kij

kjix


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An optimising mathematical model is formulate as follows:

)3.5(12,,2,1,1

)2.5(12

Subject to

)1.5(Minimize

9

1

9

1

12

1

9

1

9

1

12

1

9

1

9

1

K==

=

= =

= = =

= = =

kx

x

xc

i j

kij

k i j

kij

k i j

kijkij

The objective function )1.5( minimizes costs that occur if the edge from i to j is used at the

thk month. Constraints )2.5( ensure the network model will be complemented MEANING?

by passing 12 edges. Constraints )3.5( ensure that for each month only 1 edge can be used.

Such mathematical models require significant computation time in order analytically to solve

the model. Figure 8 shows all possible routes from starting point to end point. Due to the fact

that it is nearly impossible to arrive at a single optimum solution, the network model is

modified as a shortest path problems (SPP). The Dijkstra algorithm is viable to solve the

proposed SPP model.

Figure 8. Network flow repsentation


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5.2. Shortest Path Problems

Shortest path problems (SPP) focus on finding a path between two vertices in order to minimize

the sum of the weights of its constituent edges. Finding the shortest route to get from one

location to another on a road map can be given as an metaphor. In this problem, vertices show

locations and edges show segments of a route. Time needed to travel a specified distance is

used to weight these two variables. There are many types of algorithms available to solve SPPs

including: Dijkstra algorithm, Bellman-Ford algorithm, A* search algorithm, Floyd-Warshall

algorithm, Johnson's algorithm and Perturbation theory. This study used the Dijkstra algorithm

to solve the SPP.

5.2.1. The Dijkstra Algorithm

Dijkstras algorithm takes its name from the Dutch computer scientist Edsger Dijkstra. In his

1959 paper, he explained a graph search algorithm that solves the single-source shortest path

problem using only positive edge path costs. The result is a shortest path tree.

The model assumes that a weighted, connected simple graph is viable and all weights are

positive. This graph, called G , has vertices K,,, 210 vvv where 00 =v for the starting

vertex and nvn = is the terminating vertex for the path. Weights are stored in a 2-dimensional

array where =ji vvw , EQUALS WHAT? an integer when there is an edge between iv and

jv , else infinity . The proposed Dijkstra Algorithm's pseudo code is given in Algorithm 3.

During processing, an array [ ]nL K1 records path lengths, called vertex labels. This array is set

to infinity for all entries except the starting vertex that initially is set to 0. Set S contains all

vertices processed so far. According to Dijkstras algorithm, when the terminating vertex

equates to Sn , the main loop ends and the length of the shortest path becomes equal to value

[ ]nL . To find the shortest path from vertex 1 to vertex n requires analysis backward from

vertex n . The aim is to find vertex labels differing by exactly the length of the connecting

value in the weighted matrix.


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Algorithm 3. Pseudo-code of the Dijkstra algorithm

Step 1

1.1 for i = 1 to nL[vi] = L[0] = 0

S =

Step 2

2.1 while n S u=vertex notin S with L value minimal S = S {u}

2.2 for all vertices v S do if L[u]+ w[u,v] < L[v] then L[v] = L[u] +

w[u,v]

Step 3 3.1

while n 1for i=1 to n

if L[n]- L[n-i] = w[n,n-i] then record n-In = n-i

5.3. Changing Bed Classification using a Modified Network Model

Let kijc and kijx be defined as 5.1, then the mathematical model can be given as follows:

)3.6(12,,2,1,1

)2.6(12

Subject to

)1.6(Minimize

8

2

1

1

9

9

11

1

1

12

1

9

1

9

1

12

1

9

1

9

1

K==++

=

=

+

= =

==

+

=

= = =

= = =

kxxx

x

xc

i

i

ij i

i

ij

kijkij

i

i

ij

kij

k i j

kij

k i j

kijkij

In this problem, we have derived a cost function that represents the cost of changing type of the

total 8 beds. The cost function is as follows:

== =

+++++

12

1

12

1

12

1

)1212(1350)1010(135010(1350m

m

m m

m cbWCpbWPpcbpb ,

MISSING ) IN FIRST EXPRESSION where pb and cb represent an initial

number of private and common beds respectively. New variables waiting

cost for a common bed WC, waiting costs a for private bed WP and average

stay for patients APS are required to define marginal costs incurred when

beds are reallocated. Values ofWC, WP and APS are given in Table 2.


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Table 2. Waiting cost for a common bed and private bed (WP) with average

length of stay per patient (APS) data

Months WC WP APS

1 19.115 13.793 1.56

2 16.59 13.793 1.8

3 15.165 2.297 2.13

4 17.593 3.448 1.8

5 16.282 1.148 2.2

6 25.131 1.253 1.53

7 26.653 1.148 1.5

8 19.546 1.060 1.93

9 16.282 2.297 2

10 17.241 1.253 2.06

11 16.282 1.493 2.1

12 19.115 1.919 1.76

Figure 9 provides an example of a network representation for the shortest path problem. This

model has 110 nodes that including a source and a terminal node. Arcs connect the nodes.

These represent the reclassification of the type of the bed. Algorithm 3 is proposed in a user

interface shown in Figure 10. The user must enter the number of total nodes in the data box,

shown in the top of the left corner, and click the OK button. An empty matrix appears. This

matrix is filled by the user. After the matrix is filled, the user should click the OK button that

to set the Show Route box. The shortest route is shown in this window. The shortest route

result appears in the Additional Needed Beds Per Month data field.

Figure


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9. A network flow repsentation for a shortest path problem

5.3.1 Interpretation of the result

Figure 10 shows the shortest path from the starting to end points for this Dijkstra algorithm

problem as:

1091059788796959504234261880

This path represents reallocations of bed type required for each month.

The numbers of needed additional private and common beds are given in Table 3.

Since the demand of the private beds is larger than the common beds during the first month of

the year; Table 3 shows that Ege University Hospital Oncology Department needs an

additional 7 private beds and a common bed. Some beds must be reclassificed during the

summer, for example in June they require 4 additional private beds and 4 common beds. The

result for June shows that demand for additional common and private beds can be similar.


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Figure 10. A display of the Dijkstra algorithm

Table 3. Additional type of beds per month

Table 3. Optimum bed types calculated in monthly time buckets

6. Conclusion and Further Work

This study identified annual demand and monthly bed classifications for Ege University

Hospital Oncology Department. An integer programming model, a M/M/s queue model and a

genetic algorithm approaches were used to solve annual bed capacity requirements. A network

model and a modified network model were used to solve monthly bed allocations.

For the annual bed capacity problem, results are found from the integer programming model

differed from results that are obtained from the M/M/s queue model. These inaccuracies

principally stem from the M/M/s queue model since the budget, costs of additional nurses and

nurse-aids were not taken into account. The M/M/s queue model does not consider constraints

and objective functions. Data used by the model are limited to expected number of patients, and

waiting times for private or common beds. When budget limits and the costs of the employees

are not known or estimated, the proposed M/M/s queue model can be used effectively. Given

the information available, the genetic algorithm approach yielded the same results as those

obtained using integer programming. These results show the robustness and effectiveness of the

proposed genetic algorithm since the results obtained from the integer programming model are

short time bucket optimums found by applying the branch-and-bound method. For more

Months The number of needed

additional private bed

The number of needed

additional common bed

1 7 1

2 8 0

3 7 1

4 6 2

5 5 3

6 4 4

7 4 4

8 5 39 6 2

10 6 2

11 6 2

12 5 3


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complex problems, when branch-and-bound methods are unable to cope, can be solved within

an acceptable computation time by the proposed genetic algorithm.

For the monthly bed capacity planning problem, results obtained from the modified network

model enable the client to use the additional beds more efficiently by defining the optimumcombination of bed types for each month. Weekly time buckets may also be used in response to

demand.

All of these results demonstrate that the client can increase bed capacity and avoid exceeding

their budget. This leads to an increase in the service quality and a decrease in expected waiting

time of the patients; both are vital since patients that come to this department typically have to

be treated urgently. Comparison between analysis techniques illustrate the contribution to

understanding made by each method. This enables analysts to propose an appropriate solution.

Research into different kinds of bed capacity planning problems is still in its infancy. This

research project contributes to the debate by focusing on identifying the value and accuracy of

various analytical techniques. Further work should be undertaken with other departments and

hospitals in order to validate the proposed algorithms.

7. References

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[2] Jung, A.L., Streeter, N.S., (1985) Total population estimate of newborn special-care bed needs,

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method to optimize hospital bed capacity, Int. J. Med. Inform 74, 39-49.

[4] El-Darzi, E., Vasilakis, C., Chaussalet, T., Millard, P.H., (1998) A simulation modeling approach to

evaluating length of stay, occupancy, emptiness and bed blocking in a hospital geriatric department,

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[5] Bagust, A., Place, M., Posnett, J.W., (1999) Dynamics of bed use in accommodating emergency

admissions: Stochastic simulation model, Br. Med. J. 319.

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application to hospital facilities. Oper. Res. 29, 122.

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[9] Cochran, J.K., Bharti, A., (2006) A multi-stage stochastic methodology for whole hospital bed

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[18] Goldberg, D.E., (1989) Genetic Algorithms in Search, Optimization and Machine Learning,

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