Co esponding au ho : Oko o C.N
Copy igh © 2025 Au ho (s) e ain he copy igh o his a icle. This a icle is published unde he e ms o he C ea i e Commons A ibu ion License 4.0.
In ege p og amming o op imized nu se scheduling: A model o e icien wo k o ce
alloca ion in heal hca e sys ems
Oko o, C.N. *, O i C. G. and Mmahi, E.C.
Depa men o Indus ial Ma hema ics and Applied S a is ics, Ebonyi S a e Uni e si y, Abakaliki Nige ia.
Wo ld Jou nal o Ad anced Resea ch and Re iews, 2025, 27(01), 1430-1439
Publica ion his o y: Recei ed on 03 May 2025; e ised on 08 July 2025; accep ed on 11 July 2025
A icle DOI: h ps://doi.o g/10.30574/wja .2025.27.1.2464
Abs ac
E ec i e nu se scheduling is essen ial o op imizing heal hca e wo k o ce managemen . This s udy p esen s an in ege
p og amming model ha ensu es op imal shi alloca ion while balancing ope a ional e iciency, legal cons ain s, and
s a p e e ences. By minimizing scheduling ine iciencies and imp o ing wo kload dis ibu ion, he model enhances
bo h cos -e ec i eness and nu se sa is ac ion. The esul o implemen a ion a mile ou hospi al Abakaliki, Nige ia
p oduces op imal alloca ion o nu ses o days o and also op imal numbe o nu se equi emen o each wa d o uni .
Resul s demons a e supe io pe o mance o e adi ional scheduling me hods by minimizing 0.09% o he hospi al’s
o al nu sing s a cos . This amewo k o e s a scalable solu ion o imp o ing wo k o ce alloca ion in heal hca e
sys ems.
Keywo ds: Nu se Scheduling; In ege P og amming; Wo k o ce Op imiza ion; Heal hca e Ope a ions
1. In oduc ion
A ma hema ical p og amming app oach has been shown o e ec i ely minimize nu sing sho ages and sa is y s a ing
cons ain s (Wa ne and P awda, 1972).
Abdalka eem e al. (2021) p o ided a comp ehensi e su ey o heal hca e scheduling p oblems, highligh ing key a eas
such as nu se scheduling, pa ien admissions, and ope a ing oom planning, and analyzing 190 a icles ac oss a ious
op imiza ion app oaches.
E icien nu se scheduling is undamen al o he smoo h ope a ion o heal hca e acili ies, di ec ly impac ing pa ien
ca e, hospi al e iciency, and s a well-being. Gi en he c i ical ole ha nu ses play in pa ien managemen , ensu ing
adequa e s a ing le els while balancing nu se wo kload, egula o y equi emen s, and ins i u ional cons ain s emains
a complex challenge. The Nu se Scheduling P oblem (NSP) is a well-documen ed issue in heal hca e ope a ions,
equi ing hospi als o de elop shi alloca ions ha minimize ine iciencies, p e en s a bu nou , and op imize
esou ce u iliza ion. An e ec i e scheduling sys em ensu es unin e up ed pa ien ca e, adhe ence o labo laws, and
ai wo kload dis ibu ion among nu ses.
T adi ionally, hospi als ha e elied on manual scheduling me hods, which a e o en ime-consuming, e o -p one, and
in lexible. These adi ional app oaches s uggle o accommoda e luc ua ing pa ien demands, s a p e e ences, and
legal cons ain s, leading o unde s a ing o o e s a ing, inc eased o e ime cos s, and nu se dissa is ac ion. S udies
ha e shown ha poo scheduling con ibu es o low job sa is ac ion and high u no e a es, which in u n a ec he
quali y o heal hca e deli e y (Cheang e al., 2003; Bu ke e al., 2004). Addi ionally, ine ec i e scheduling can esul in
highe ope a ional cos s, as hospi als o en ely on expensi e empo a y s a ing solu ions o ill gaps in co e age (Smi h
Wo ld Jou nal o Ad anced Resea ch and Re iews, 2025, 27(01), 1430-1439
1431
and Wiggins, 1997). These challenges necessi a e he adop ion o mo e sys ema ic and da a-d i en app oaches o
wo k o ce scheduling.
In esponse o hese ine iciencies, In ege P og amming (IP) has eme ged as a powe ul op imiza ion ool o sol ing
complex scheduling p oblems, including NSP. Unlike heu is ic o manual me hods, IP ensu es ma hema ically op imal
solu ions by sys ema ically assigning nu ses o shi s while sa is ying mul iple cons ain s. This app oach enhances
ai ness, e iciency, and cos -e ec i eness, ensu ing compliance wi h ins i u ional policies and labo laws. P io
esea ch has demons a ed ha IP-based scheduling models imp o e nu se alloca ion, enhance wo k o ce e iciency,
and educe scheduling con lic s in hospi al se ings (Al a es, 2002; El-Quli i and Al-Da ab, 2009). By le e aging
ma hema ical op imiza ion, hospi als can achie e a balance be ween wo k o ce demands and s a well-being, leading
o imp o ed pa ien ca e ou comes.
This s udy applies an In ege P og amming model o op imize nu se scheduling a Mile Fou Hospi al, Abakaliki, which
ope a es 21 wa ds wi h a o al o 64 ull- ime nu ses. Gi en ha each nu se is equi ed o wo k i e days pe week wi h
wo consecu i e days o , scheduling becomes a highly cons ained p oblem ha equi es ca e ul op imiza ion. The
s udy aims o de elop a s uc u ed, da a-d i en model ha e icien ly assigns nu ses o shi s while ensu ing compliance
wi h hospi al policies and s a ing equi emen s. The key objec i es o his esea ch a e o: De elop an op imized nu se
scheduling model ha ensu es p ope shi co e age while adhe ing o s a ing equi emen s. Minimize ine iciencies
such as unde s a ing, excessi e o e ime, and wo kload imbalances. Enhance nu se sa is ac ion by p omo ing ai
wo kload dis ibu ion. Ensu e egula o y compliance, pa icula ly in e ms o maximum wo king hou s and manda ed
es pe iods.
By applying In ege P og amming, his esea ch con ibu es o he ield o heal hca e ope a ions managemen , o e ing
a scalable and decision-suppo amewo k o hospi al adminis a o s seeking o imp o e wo k o ce planning. The
indings will no only bene i Mile Fou Hospi al bu can also se e as a benchma k o op imizing nu se scheduling in
o he heal hca e ins i u ions. Th ough his s udy, hospi als can achie e a balance be ween cos -e ec i eness, s a well-
being, and pa ien ca e quali y, ein o cing he signi icance o ma hema ical op imiza ion in mode n heal hca e
managemen .
2. Me hodology
2.1. P oblem Fo mula ion
The Nu se Scheduling P oblem (NSP) is a highly cons ained combina o ial op imiza ion p oblem, whe e he goal is o
c ea e an e icien os e ha assigns nu ses o shi s while sa is ying mul iple cons ain s. The p ima y challenge in
nu se scheduling is balancing hospi al s a ing equi emen s, employee p e e ences, legal egula ions, and ope a ional
e iciency.
To add ess his, he p oblem is o mula ed as an In ege P og amming (IP) model, which ensu es ma hema ically
op imal solu ions while adhe ing o de ined cons ain s. The scheduling p oblem in ol es alloca ing shi s o nu ses
while ensu ing adequa e shi co e age (mee ing daily nu se demand pe wa d), ai wo kload dis ibu ion (a oiding
nu se bu nou and excessi e o e ime), egula o y compliance (adhe ing o labo laws and hospi al policies). days-o
scheduling cons ain s (ensu ing e e y nu se ge s wo consecu i e days o pe week).
This s udy ocuses on days-o scheduling, whe e he decision a iables ep esen he assignmen o nu ses o speci ic
o -day pa e ns. The IP model minimizes scheduling ine iciencies while sa is ying all hospi al cons ain s.
2.2. Ma hema ical Model
The NSP is o mula ed as an In ege Linea P og amming (ILP) model, which ensu es ha he numbe o nu ses
assigned o each shi is an in ege alue. The key componen s o he model a e:
2.3. Decision Va iables
Le Xi=Numbe o nu ses assigned o a speci ic days-o pa e n (i=1,2, 7)
• X1: Sa u day-Sunday o
• X2: Sunday-Monday o
• X3: Monday-Tuesday o
Wo ld Jou nal o Ad anced Resea ch and Re iews, 2025, 27(01), 1430-1439
1432
• X4: Tuesday-Wednesday o
• X5: Wednesday-Thu sday o
• X6: Thu sday-F iday o
• X7: F iday-Sa u day o
2.4. Objec i e Func ion
The goal is o minimize he o al numbe o days-o alloca ions while ensu ing ha each wa d mee s i s daily s a ing
equi emen s.
Min Z=X1+X2+X3+X4+X5+X6+X7
2.5. Cons ain s
Each day mus mee a minimum s a ing equi emen bj (nu ses needed pe shi pe wa d). The model ensu es ha
enough nu ses a e scheduled on du y each day:
• X1+X2+X3+X4+X5 ≥ b1 (Sa u day)
• X2+X3+X4+X5+X6≥b2 (Sunday)
• X3+X4+X5+X6+X7≥b3 (Monday)
• X1+X4+X5+X6+X7≥b4 (Tuesday)
• X1+X2+X5+X6+X7≥b5 (Wednesday)
• X1+X2+X3+X6+X7≥b6 (Thu sday)
• X1+X2+X3+X4+X7≥b7 (F iday)
Addi ionally, he numbe o assigned nu ses mus be non-nega i e in ege s
Xi≥0, XS is an in ege o all i.
2.6. Solu ion App oach (B anch and Bound Algo i hm)
2.6.1. S ep 1: Sol e he Linea Relaxa ion (LP Relaxa ion)
• Igno e in ege cons ain s and sol e he linea p og amming (LP) e sion o he p oblem using he Simplex
Me hod.
• The solu ion ob ained will ha e ac ional alues o some a iables, which a e no easible since nu ses canno
be assigned in ac ional numbe s.
• I he LP solu ion is al eady an in ege , i is he op imal solu ion. O he wise, p oceed o b anching
2.6.2. S ep 2: B anching (Di ide he P oblem in o Sub p oblems)
• Iden i y he i s non-in ege a iable in he LP solu ion.
• C ea e wo new sub p oblems (b anches) by o cing he non- in ege a iable o ake in ege alues.
• These wo cons ain s c ea e wo new LP p oblems, each wi h a smalle easible egion.
2.6.3. S ep 3: Bounding (Elimina ing Non-Op imal Solu ions)
• Sol e bo h sub p oblems using he LP elaxa ion.
• I a b anch leads o an in easible solu ion (i.e., iola es nu se s a ing cons ain s), i is p uned (disca ded).
• I a b anch gi es an in ege solu ion, eco d i as a candida e op imal solu ion.
• I he objec i e unc ion alue o a new in ege solu ion is wo se han an al eady known easible solu ion,
disca d i
2.6.4. S ep 4: Node explo a ion
• Selec he nex non-in ege a iable and epea he b anching and bounding p ocess.
• I a b anch leads o a be e easible in ege solu ion, upda e he bes -known solu ion.
• S op when all b anches ha e ei he been explo ed o p uned.
• The bes easible in ege solu ion ound is he op imal nu se schedule.
Wo ld Jou nal o Ad anced Resea ch and Re iews, 2025, 27(01), 1430-1439
1433
2.7. Da a Collec ion and Rep esen a ion
This s udy is based on eal hospi al da a collec ed om Mile Fou Hospi al, Abakaliki, Ebonyi S a e, Nige ia. The hospi al
has 21 ya ds and 64 ull- ime nu ses, each equi ed o wo k i e days pe week wi h wo consecu i e days o .
2.7.1. Wa d-wise S a ing Da a
The able below p esen s he o al nu se a ailabili y and he daily s a ing equi emen pe wa d
Table 1 Nu ses dis ibu ion pe wa d
Wa d/Uni
To al Nu ses A ailable
Daily Requi emen
Pos na al (1) and (2)
9
4
Pos na al (3)
6
2
Nu se y
6
4
Labou Wa d
12
6
An ena al Clinic (ANC)
6
6
Anes hesia Uni (ART)
2
2
Mul iple D ug Resis an (MDR)
2
2
Ou pa ien Dep . (OPD)
3
2
Child en's Wa d
8
4
Ope a ing Thea e
2
1
Admin/Counseling Uni
2
2
Tube culosis Wa d
1
1
2.8. Cons ain Rep esen a ion o Each Wa d
Fo each wa d/uni , we o mula e sepa a e cons ain s using hei daily s a ing equi emen s.
2.8.1. Fo mula ion o Pos na al (1) and (2) Wa d
This wa d equi es 4 nu ses pe day hus, he cons ain s speci ic o his wa d will be:
• X1+X2+X3+X4+X5≥4 (Sa u day)
• X2+X3+X4+X5+X6≥4 (Sunday)
• X3+X4+X5+X6+X7≥4 (Monday)
• X1+X4+X5+X6+X7≥4 (Tuesday)
• X1+X2+X5+X6+X7≥4 (Wednesday)
• X1+X2+X3+X6+X7≥4 (Thu sday)
• X1+X2+X3+X4+X7≥4 (F iday)
Simila cons ain s apply o all o he wa ds, adjus ing bj o each day's speci ic s a ing needs.
The abo e is ep esen ed in he ollowing ma ix o m
Wo ld Jou nal o Ad anced Resea ch and Re iews, 2025, 27(01), 1430-1439
1434
Figu e 1 In ege linea p og amming model o nu ses dis ibu ion pe wa d
Simila ly, in ege p og amming o mula ion o F2 ep esen ing Anes hesia uni , Counseling uni and Mul iple d ug-
esis an uni s wi h a daily equi emen o 2 nu ses, F3 ep esen ing Labo wa d and An ena al clinic wi h a daily
equi emen o 6 nu ses, and F4 ep esen ing Ope a ing hea e and Tube culosis wa d wi h daily equi emen o 1
nu se we e o mula ed wi h alues o bj’s on he igh -hand side o he cons ain s as 2, 6 and 1 espec i ely.
3. Resul analysis
The excel sol e was used o execu e he b anch and bound algo i hm and he esul s a e shown in he ables and igu es
below.
Table 2 LP Solu ion o F1
X1
X2
X3
X4
X5
X6
X7
Objec i e
1
1
1
1
1
1
1
6
Cons ain 1
1
1
1
1
1
0
0
4
4
Cons ain 2
0
1
1
1
1
1
0
6
4
Cons ain 3
0
0
1
1
1
1
1
4
4
Cons ain 4
1
0
0
1
1
1
1
4
4
Cons ain 5
1
1
0
0
1
1
1
4
4
Cons ain 6
1
1
1
0
0
1
1
4
4
Cons ain 7
1
1
1
1
0
0
1
4
4
Bound
0
0
0
0
0
0
0
Decisions
0
2
0
2
0
2
0
Wo ld Jou nal o Ad anced Resea ch and Re iews, 2025, 27(01), 1430-1439
1435
Figu e 2 B anch and Bound Enume a ion T ee o solu ion o F1
Table 3 LP solu ion o F2
X1
X2
X3
X4
X5
X6
X7
Objec i e
1
1
1
1
1
1
1
3
cons ain 1
1
1
1
1
1
0
0
2
2
cons ain 2
0
1
1
1
1
1
0
3
2
cons ain 3
0
0
1
1
1
1
1
2
2
cons ain 4
1
0
0
1
1
1
1
2
2
cons ain 5
1
1
0
0
1
1
1
2
2
cons ain 6
1
1
1
0
0
1
1
2
2
cons ain 7
1
1
1
1
0
0
1
2
2
Bound
0
0
0
0
0
0
0
Decisions
0
1
0
1
0
1
0
Wo ld Jou nal o Ad anced Resea ch and Re iews, 2025, 27(01), 1430-1439
1436
Simila ly, he objec i e alue(minimum) o F3 consis s o 7 nodes wi h Z =9, while he objec i e alue o F4 consis s Bes o 3 nodes wi h Z= 1.5.
The able below summa izes he op imal solu ion o he nu se scheduling p oblem o each g ouping.
Figu e 3 B anch and Bound T ee o Enume a ion solu ion o F2
Table 4 Summa y o he op imal solu ion o he Nu se Scheduling p oblem
G oup
X1
X2
X3
X4
X5
X6
X7
Objec i e alue (z)
F1
0
2
0
2
0
2
0
6
F2
0
1
0
1
0
1
0
3
F3
1
1
1
1
2
1
2
9
F4
1
0
0
0
1
0
0
2
4. Discussion o Resul s
Based on he analysis we ob ained he ollowing key indings
In he pos -na al (1) and (2) wa d, Nu se y wa d and Child en wa d (F1) which has o al a ailabili y o 9,6 and 8
employed nu ses espec i ely; (see Table 1), an op imal o al o 6 nu ses o each o he wa ds in his g oup (F1) is
needed in o de o sa is y hei daily equi emen o 4 nu ses. Also, x2 =2, x4 =2, x6 =2 indica es ha wo nu ses should
be assigned o Sunday – Monday o , Tuesday – Wednesday o , Thu sday – F iday o espec i ely; (see able 7), x1=0,
x3=0, x5=0, x7=0 implies ha no nu se should be assigned o Sa u day –Sunday o , Monday –Tuesday o , Wednesday
–Thu sday o and F iday –Sa u day o .
In pos -na al (3) wa d, Anes hesia uni (A R T), Mul iple d ug- esis an uni s and ou –pa ien s depa men (F2) which
has o al o a ailabili y o 6,6,2 and 3 employed nu ses espec i ely; (see able 1), an op imal o al o 3 nu ses o each
o he wa ds/uni in his g oup (F2) is needed in o de o sa is y hei daily equi emen o 2 nu ses. Also, x2=1, x4=1,
x6=1 indica es ha one nu se should be assigned o Sunday –Monday o
Tuesday –Wednesday o , and Thu sday –F iday o espec i ely. (see able 8) Then, x1=0
Wo ld Jou nal o Ad anced Resea ch and Re iews, 2025, 27(01), 1430-1439
1437
x3=0 x5=0 and x7=0 implies ha no nu se should be assigned o Sa u day – Sunday o , Monday –Tuesday o ,
Wednesday-Thu sday o and F iday-Sa u day o .
In labou wa d and An ena al clinic (F3) which has o al o a ailabili y o 12 and 6 employed nu ses espec i ely; (see
able 1), an op imal o al o 9 nu ses o each o he wa ds in his g oup (F3) is needed in o de o sa is y hei daily
equi emen o 2 nu ses. Also, x1=1, x2=1, x3=1, x4=1, x6=1 indica es ha one nu se should be assigned o Sa u day –
Sunday o , Sunday –Monday o , Monday –Tuesday o Tuesday –Wednesday o , and Thu sday –F iday o espec i ely.
(see able 9) Then, x5=2 and x7=2 implies ha wo nu ses should be assigned o Wednesday-Thu sday o and F iday-
Sa u day o .
In ope a ing hea e and ube culosis wa d (F4) which has o al a ailabili y o 2 and 1 employed nu ses espec i ely;
(see able 1), an op imal o al o 2 nu ses o each o he wa ds/uni in his g oup (F4) is needed in o de o sa is y hei
daily equi emen o 1 nu se. Also, x1=1 x5=1 indica es ha one nu se should be assigned o Sa u day –Sunday o , and
Wednesday –Thu sday espec i ely, (see able 9) Then, x2=0, x3=0 x4=0, x6=0 and x7=0 implies ha no nu se should be
assigned o Sunday – Monday o , Monday –Tuesday o , Tuesday –Wednesday o , Thu sday –F iday o and F iday-
Sa u day o .
These esul s a e summa ized in able 4 below
Table 5 Op imal solu ion o each wa d in mile ou hospi al Abakaliki, Ebonyi S a e
Wa d/uni s
Op imal alloca ion o Nu se o days o
Op imal numbe o Nu se equi ed
X1
X2
X3
X4
X5
X6
x7
Pos na al (1) and (2) wa d
0
2
0
2
0
2
0
6
Pos na al (3) wa d
0
1
0
1
0
1
0
3
Nu se y
0
2
0
2
0
2
0
6
Labou wa d
1
1
1
1
2
1
2
9
An ena al clinic (A N C)
1
1
1
1
2
1
2
9
Anaes hesia uni (A R T)
0
1
0
1
0
1
0
3
Mul iple d ug esis an (MDR)
0
1
0
1
0
1
0
3
Ou -pa ien depa men (OPD)
0
1
0
1
0
1
0
3
Child en wa d
0
2
0
2
0
2
0
6
Ope a ing hea e
1
0
0
0
1
0
0
2
Admin / Counselling uni
0
1
0
1
0
1
0
3
Tube culosis wa d
1
0
0
0
1
0
0
2
Table 6 Sample o one week Ros e o nu se in F1 wa ds/uni s
Nu se ID numbe
Sunday
Monday
Tuesday
Wednesday
Thu sday
F iday
Sa u day
1
O
On
On
On
On
On
O
2
O
On
On
On
On
On
O
3
On
O
O
On
On
On
On
4
On
O
O
On
On
On
On
5
On
On
On
O
O
On
On
6
On
On
On
O
O
On
On
Requi ed
4
4
4
4
4
4
4
Wo ld Jou nal o Ad anced Resea ch and Re iews, 2025, 27(01), 1430-1439
1438
Assigned
4
4
4
4
4
6
4
Excess
0
0
0
0
0
0
0
Table 7 Sample o one week Ros e o nu se in F2 wa ds/uni s
Nu se ID numbe
Sunday
Monday
Tuesday
Wednesday
Thu sday
F iday
Sa u day
1
O
On
On
On
On
On
O
2
On
O
O
On
On
On
On
3
On
On
On
O
O
On
On
Requi ed
2
2
2
2
2
2
2
Assigned
2
2
2
2
2
3
2
Excess
0
0
0
0
0
1
0
Table 8 Sample o one week Ros e o nu se in F3 wa ds/uni s
Nu se ID numbe
Sunday
Monday
Tuesday
Wednesday
Thu sday
F iday
Sa u day
1
On
On
On
On
On
O
O
2
O
On
On
On
On
On
O
3
O
O
On
On
On
On
On
4
On
O
O
On
On
On
On
5
On
On
O
O
On
On
On
6
On
On
On
O
On
On
On
7
On
On
On
O
O
On
On
8
On
On
On
On
O
O
On
9
On
On
On
On
O
O
On
Requi ed
6
6
6
6
6
6
6
Assigned
7
7
6
6
6
6
7
Excess
1
1
0
0
0
0
1
Table 9 Sample o one week Ros e o nu se in F4 wa ds/uni s
Nu se ID numbe
Sunday
Monday
Tuesday
Wednesday
Thu sday
F iday
Sa u day
1
On
On
On
On
On
O
O
2
On
On
O
O
On
On
On
Requi ed
1
1
1
1
1
1
1
Assigned
2
2
1
1
2
1
1
Excess
1
1
0
0
1
0
0
