EFFECTIVE GENETIC OPERATORS OF COOPERATIVE

GENETIC ALGORITHM FOR NURSE SCHEDULING

Makoto Ohki, Shin-ya Uneme, Shigeto Hayashi and Masaaki Ohkita

Department of Electric and Electronic Engineering, Faculty of Engineering, Tottori University

Keywords: Genetic Algorithm, Cooperative GA, Mutation

Operator, Virus Operator, Nurse Scheduling, Scheduling

Problem.

Abstract: This paper proposes effective genetic operators for cooperative genetic algorithm (GA) to solve a nurse

scheduling problem. A clinical director of a medical department makes a duty schedule of all nurses of the

department every month. Such the scheduling is very complex task. It takes one or two weeks to create the

nurse schedule even by a veteran director. In conventional ways using the cooperative GA, a crossover

operator is only employed for the optimization, because it does not lose consistency between chromosomes.

We propose a mutation operator and a virus operator for the cooperative GA, which does not lose

consistency of the nurse schedule. The cooperative GA with these new operators has brought a surprisingly

good result, it has never been brought by the conventional algorithm.

1 INTRODUCTION

In hospitals, 15-30 nurses are working in a medical

section such as the internal medicine department or

the pediatrics department. A clinical director of the

department constitutes a schedule of all the nurses

every month. Constituting the nurse schedule (Goto,

1993, Ikegami 2001), or the nurse scheduling, is

very complex task. In our investigation, even a

veteran director needs one or two weeks for the

nurse scheduling. To reduce such the problem, a

computer software creating the nurse schedule is

developed, and it is recentlly used at many hospitals.

Those software must find the better schedule in the

shorter time.

In conventional ways (Goto, 1993, Kawanaka

002, Inoue, 2002, Itoga, 2003) using the

cooperative genetic algorithm (CGA), a crossover

operator is only employed for the optimization,

because it does not lose consistency between

chromosomes. Theirfore, the conventional technique

optimizes the nurse schedule at slow speed, and it is

difficult to acquire a good solution.

We propose mutation and virus operators for the

CGA, wh

ich do not lose consistency of the nurse

schedule. The CGA with these new operators has

brought a surprisingly good result, it has never been

brought by the conventional algorithm.

2 NURSE SCHEDULING BY CGA

WITH CROSSOVER

In the nurse scheduling, we have to consider many

requirements, for example, duty load of each nurse,

fairness of assignment of day time and night duty,

intensiveness of night duty and so on. These

requirements are implemented by twelve penalty

functions in this paper. These twelve penalty

functions, F

-F

, are classified into four partial

penalty groups, G

-G

. Finally, we define an total

penalty function, E, and the partial penalty

functions, G

-G

, as follows,

∑

)α(

, (1)

(

)

∑

iii

FhFhFhG

5134121111

, (2)

(

)

∑

iii

FhFhFhG

6233222212

, (3)

(

)

∑

++=

jjj

FhFhFhG

9338327313

, (4)

and

(

)

∑

++=

jjj

FhFhFhG

1243114210414

, (5)

where h

(p=1,2,3 or 4, q=1,2 or 3) denotes energy

coefficients as follows,

=0.2, h

=0.4, h

=0.2, h

=0.4,

347

Ohki M., Uneme S., Hayashi S. and Ohkita M. (2007).

EFFECTIVE GENETIC OPERATORS OF COOPERATIVE GENETIC ALGORITHM FOR NURSE SCHEDULING.

In Proceedings of the Fourth International Conference on Informatics in Control, Automation and Robotics, pages 347-350

 SciTePress

=0.1, h

=0.7, h

=0.7.

Each nurse schedule of one month is evaluated by

the following penalty function,

iiiiiii

FhFhFhFhFhFhH

623322221513412111

++++=

(6)

In the cooperative GA, the population represents the

whole nurse schedule of one month. Each individual

is coded in a chromosome which shows one-month

schedule of one nurse. The individual consists of the

sequence of the duty symbols as shown in Fig.1. The

duty sequence consists of thirty fields, since one

month includes thirty days in this practical example.

Each individual expresses one-month schedule of

the nurse n

. There are not two or more individuals

including the same nurse’s schedule in the

population. An example of the population is shown

in Fig.2.

Initially, the population is randomly generated as

satisfying the necessary number of nurses on every

duty at every date. In this paper, the necessary

number of nurses is specified as ten, six and five for

day time duty on a week day, Saturday and Sunday

respectively, three for a semi night duty and three

for night duty respectively. An overview of the

crossover operator is shown by Fig.3. First, two

hundreds pairs of the individual are selected as a

parent for the crossover. One of the pair is selected

from the population under the roulette selection

manner. The roulette selection gives an individual

with higher energy, H

. This manner tends to select

the worse individual. Another one of the pair is

randomly selected from the population. Exchanging

parts of the individuals divided at two crossover

points, two new pairs of individual are generated as

children. This exchanging process does not

exchange the fixed duties, such as the meeting, the

training and the requested holiday. These child pairs

are temporally returned to the original nurse position

respectively. The temporal population with the

children is performed by the energy function, E.

After all the pairs of individual have been

performed, one pair giving the smallest energy is

selected for the next generation.

3 NEW GENETIC OPERATORS

In this paper, two new genetic operators are

proposed to accelerate the nurse scheduling by

CGA. One of them is a mutation operator. In the

conventional way, it is considered that the mutation

loses the consistency of the population and does not

work effectively. However, the mutation is a very

strong genetic operator to widely search in the

solution space. In this paper, new effective mutation

operator is proposed for the cooperative GA which

does not lose the consistency of the schedule. The

aim of mutation is to bring small change into the

population. However if one of the duty fields is

randomly changed, the whole schedule become

meaningless thing, which is very hard to recover by

a genetic operation. Therefore, the mutation operator

must preserve the number of nurses in every duty at

all date. The basic operation of the mutation is

shown in Fig.4. One of dates is randomly selected.

Two nurses at the same date are decided. Finally, the

duties of these two positions are exchanged. If one

of the selected duties is the fixed duty, another nurse

duty schedule for one month

・・・・・

MRTSHHH HMRS

nurse n

DD DDD

duty schedule for one month

・・・・・

MRTSHHH HMRS

nurse n

DD DDD

Figure 1: An individual coded in chromosome giving

duty schedule of one nurse for one month.

・

individual set

duty schedule of nurse n

for one month

duty schedule of nurse n

for one month

duty schedule of nurse n

for one month

duty schedule of nurse n

for one month

duty schedule of nurse n

for one month

・

individual set

duty schedule of nurse n

for one month

duty schedule of nurse n

for one month

duty schedule of nurse n

for one month

duty schedule of nurse n

for one month

duty schedule of nurse n

for one month

・

Figure 2: Population including one-month schedules of

each nurse.

select parents

crossover

・

※

denotes fixed duty, for example,

meetin

dut

and re

uested holida

crossover

・

individual set

pareto

selection

return

select parents

crossover

・

※

denotes fixed duty, for example,

meetin

dut

and re

uested holida

crossover

・

individual set

pareto

selection

return

Figure 3: Optimization cycle of the CGA with the

crossover operator.

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

348

is randomly selected again. The basic mutation

operation is applied to CGA as shown in Fig.5.

We also propose a virus operator as another new

technique. When a virus in immunology is infected

within a cell, it overwrites in a part of gene forcibly.

The virus copies own gene pattern in a genetic part

of the cell in many cases. Our virus operator

simulates this work. An aim of the virus operator is

to replace some individuals with a good thing

forcibly when the optimization is stagnant.

In normal optimization cycle, the CGA searches

by using the crossover and the mutation operators

and preserves the best performing individuals after

the crossover. When the mutation operator is

executed G

times, the virus operator is applied

instead of the mutation operator. One of individual

who gives a bad penalty is selected by using a

roulette selection manner. The virus operator

overwrites the best performed individual onto the

selected individual as shown in Fig.6.

4 COMPARISON OF GENETIC

OPERATORS

We have tried three types of the CGA in this paper,

the CGA only with the crossover operator, the CGA

with the crossover and the mutation operators and

the CGA with those three operators. An optimization

of each algorithm is performed for One hundred

thousands generations. It takes about ten minutes for

one optimization. Ten times of trials are carried out

under each condition.

Firstly, a difference of the mutation period is

examined. As shown by Fig.7, the mutation period,

, is tried to change from fifty to two thousands

generations. We have decided that the mutation

period should be set at 150. Average value of the

energy function for ten times trials is shown in Fig.8.

The mutation with a period at less than 200

generation effectively works.

We have examined the virus operator. Different

viral infection frequencies are examined as shown

by Fig.9. The viral infection frequency is able to be

set in a considerably wide range.

11 12 13 14 15

・・・・・・

・

・・・

(1) randomly select a date.

(2) randomly select two nurses.

(3) exchange the duties

of them at the date.

(2) randomly select two nurses.

11 12 13 14 15

・・・・・・

・

・・・

(1) randomly select a date.

(2) randomly select two nurses.

(3) exchange the duties

of them at the date.

(2) randomly select two nurses.

Figure 4: Primitive operation of the mutation operator.

mutation operator

preserve the best

individual set

basic operation

best individual setα

temp

mutated

individual set

individual

setα

optimization only with

crossover operator

for G

generations

generation g

optimization only with

crossover operator

for G

generations

preserve the best

individual set

basic operation

best individual setα

temp

mutated

individual set

individual

setα

optimization only with

crossover operator

for G

generations

mutation operator

preserve the best

individual set

basic operation

best individual setα

temp

mutated

individual set

individual

setα

optimization only with

crossover operator

for G

generations

generation g

optimization only with

crossover operator

for G

generations

preserve the best

individual set

basic operation

best individual setα

temp

mutated

individual set

individual

setα

optimization only with

crossover operator

for G

generations

Figure 5: Mutation operator executed by a period for G

generations.

We have investigated the optimization with the

virus operator in detail. After twenty or thirty

thousands generations, penalty function, F

, has big

value, and the value of all other penalty functions is

comparatively small. Then we have modified a part

of the virus operator: when the best individual is

preserved after the crossover operation, the

following penalty function is applied,

iiiiii

FhFhFhFhFhH

623322221513412

. (7)

As shown by Fig.10, the modified virus operator

gives good results with any virus frequency.

The modified virus operator does not effectively

work after thirty thousands generation as shown in

one month schedule

of each nurse

individual set

best performed

individuals

overwrite

one month schedule

of each nurse

individual set

best performed

individuals

overwrite

Figure 6: The virus operator forcibly overwrites an one-

month schedule of one of nurses selected in the roulette

selection manner, when the optimization is stagnant.

EFFECTIVE GENETIC OPERATORS OF COOPERATIVE GENETIC ALGORITHM FOR NURSE SCHEDULING

349

Fig.11. In contrast, the mutation operator slowly

searches. Then we apply the modified virus operator

until thirty thousands generations. After that, the

mutation operator is only applied to the CGA with

the crossover. As shown in Fig.11, the rearranged

technique gives the best result.

390

410

430

450

470

490

510

12345678910

generat ion

energy funct ion

50 100

150 200

250 500

1000 1500

2000 crossover

Figure 8: Average value of the energy function for ten

times trials. The unit of the horizontal line is ten thousand.

390

400

410

420

430

440

450

460

470

10 20 40 80 120

fr equency of t he virus oper at or

energy funct ion

maxi mum

minimum

average

Figure 9: Comparison of the frequency of the virus

operator.

390

400

410

420

430

440

450

460

470

10 20 40 80 120

fr equency of t he vir us oper at or

energy funct ion

maxi mum

minimum

average

Figure 10: Comparison of the frequency of the modified

virus operator.

390

410

430

450

470

490

510

530

02468

generat ion

energy funct ion

5 CONCLUSION

We have proposed two new genetic operators for the

CGA applied to the nurse scheduling. The only one

search technique of the conventional CGA is the

crossover operator, because it dose not lose the

consistency of the nurse schedule. In contrast, the

new techniques which we proposed in this paper

give good results without losing the consistency. By

means of these new techniques, the optimization of

the nurse schedule has been accelerated.

REFERENCES

Goto, T., Aze, H., Yamagishi, M., Hirota, M. and Fujii, S.

1993, Application of GA, Neural Network and AI to

Planning Problems, NHK Technical report, No.144.

Ikegami, A. 2001, Algorithms for Nurse Scheduling, Proc.

of 11th Intelligent System Symposium.

Kawanaka, S., Yamamoto Y., Yoshikawa D., Shinogi T.

and Tsuruoka N., 2002, Automatic Generation of

Nurse Scheduling Table Using Genetic Algorithm,

Trans. on IEE Japan, Vol.122-C.

Inoue, T., Furuhashi, T., Maeda H. and Takabane, M.,

2002, A Study on Interactive Nurse Scheduling

Support System Using Bacterial Evolutionary

Algorithm Enegine, Trans. on IEE Japan, Vol.122-C.

Itoga, T., Taniguchi N., Hoshino Y. and Kamei K., 2003,

An Improvement on Search Efficiency of Cooperative

GA and Application on Nurse Scheduling Problem,

Proc. of 12th Intelligent System Symposium.

crossover

mutation

virus

virus unt il 3000

Figure 11: Comparison of the optimization process among

CGA only with the crossover operator, CGA with the

mutation operator, CGA with the modified virus operator

and CGA applied with the modified virus operator until

30000 generations. The unit of the horizontal line is ten

thousand.

390

400

410

420

430

440

450

460

470

50 100 150 200 250 500 1000 1500 2000

mutation interval GM

energy function

maximu m

min imu m

ave r a

Figure 7: Comparison of the mutation period.

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