A Cyclical Nurse Schedule Using Goal Programming

Transcription

1 ITB J. Sci., Vol. 43 A, No. 3, 2011, A Cyclical Nurse Schedule Usig Goal Prograig Ruzzaiah Jeal 1,*, Wa Rosaira Isail 2, Liog Choog Yeu 3 & Ahed Oughalie 4 1 School of Iforatio Techology, Faculty of Sciece ad Iforatio Techology, Uiversiti Kebagsaa Malaysia, Selagor, Malaysia 2,3,4 School of Matheatical Scieces, Faculty of Sciece ad Techology, Uiversiti Kebagsaa Malaysia, Selagor, Malaysia *Correspodig author: [email protected] Abstract. Schedulig is a very tedious tas i orgaizatios where duty is aroud the cloc. Costructig tietable for urses i hospital is oe of the challegig jobs for the head urse or urse aager. It requires a lot of tie to sped for geeratig a good ad fair tietable. Thus, i this study, we propose a cyclical urse schedulig odel usig a 0-1 goal prograig that would help the head urse or urse aager to have less effort o buildig ew schedules periodically. The proposed odel satisfies the stated hospital s policies ad the urses prefereces. The result obtaied fro this odel gives a optial solutio where all goals are achieved. The odel also provides a ubiased way of schedulig the urses ad thus leads to a overall higher satisfactio ad fairess to the urses ad the hospital aageet. Keywords: cyclical schedulig; goal prograig; urse schedulig. 1 Itroductio Schedulig probles are foud i ay differet types of orgaizatios ad idustries icludig trasportatio, call cetres, health care, protectio ad eergecy services, civic services ad utilities, veue aageet, fiacial services, hospitality ad touris, retail ad aufacturig. This paper will focus o the health care idustry ad i particular o urse schedulig. Besides urse schedulig [1,2]; other schedulig i the health care idustry icludes the physicias [3]; the eergecy edicie residets [4]; ad the aster surgery schedulig [5]. The urse schedulig proble ivolves geeratig a schedule of worig days ad days off for each urse. The urse schedulig is the ost highly costraied ad difficult persoel schedulig probles. Schedulig is the process of costructig wor tietables for the staff so that a orgaisatio ca satisfy the deads for their services. The people ivolved i developig schedules eed decisio support tools to help provide the right eployees at the right tie ad at the right cost while achievig a high level of eployee job satisfactio. Received April 14 th, 2010, Revised March 11 th, 2011, Accepted for publicatio April 13 th, Copyright 2011 Published by LPPM ITB, ISSN: , DOI: /itbj.sci

2 152 Ruzzaiah Jeal, et.al. The study o urse schedulig has bee revised over 30 years ago. There are a few approaches itroduced by the past researchers i order to solve the urse schedulig proble such as usig the atheatical prograig, goal prograig, costraits prograig, artificial itelligece, heuristics ad eta heuristics [6-8]. A goal prograig is oe of the techiques that has bee studied ad used widely because of its capability to solve ad see the optiu of ulti objectives proble that occurred i urse schedulig [3,4,8-14]. It defies a target level for each objective or goal ad relative priorities to achieve these goals. This techique is flexible eough to cope with relative raig assiged to various goals. The research o odellig urse schedulig usig goal prograig has bee studied by Arthur & Ravidra [9] which focused o two phases. Phase 1 is to assig the worig days ad days off for each urse while phase 2 is to assig the shift types of their worig days. Arthur & Ravidra [9] liits the scope with sall set of costraits ad restricts the proble diesios with the size of urses is 4. Musa & Saxea [15] have used a 0-1 goal prograig that applied to oe uit of a hospital with the cosideratios of the hospital policies ad urses prefereces. However, the coplexity of the proble is rather low with a two wee plaig period ad just oe sigle shift. I Ozaraha & Bailey [13], the urse schedulig odellig showed the flexibility of goal prograig i hadlig various goals which fulfilled the hospital s objectives ad the urses prefereces. The proble defies three basic goals ad divides the wor ito two phases. The 0-1 goal prograig odel i urse schedulig has bee applied i Berrada, et al. [11] with adiistrative ad uio cotract specificatios has bee cosidered as hard costraits while wor patters ad urses prefereces has bee forulated as soft costraits. The outcoe shows satisfactory result by cobiig goal prograig approach ad tabu search techique. Azaiez & Al Sharif [10] ad Wa Rosaira, et al. [14] also used the 0-1 goal prograig approach with the cosideratios of hospital s objectives as hard costraits ad the urses prefereces as soft costraits to develop the schedules. Both odels solved by Azaiez & Al Sharif [10] ad Wa Rosaira, et al. [14] are easured to execute reasoably well. Nevertheless, the odels liit to oe off schedule where they have to build ew schedule for each plaig period. The odels are ot the cyclical schedulig. Hece, i this paper, the authors deal with the cyclical urse schedulig proble with a 0-1 goal prograig approach that icludes several objectives or goals to achieve subject to both several hard ad soft costraits.

3 A Cyclical Nurse Schedule Usig Goal Prograig 153 There are few studies doe i cyclical schedulig proble. A cyclic schedule cosists of a set of wor patters which is rotated aog a group of worers over a set of schedulig horizo. At the ed of the schedulig horizo each worer would have copleted each patter exactly oce. Harvey ad Kiragu [16] preseted a atheatical odel for cyclic ad o-cyclic schedulig of 12 hours shift urses. The odel is quite flexible ad ca accoodate a variety of costraits. I spite of this, the odel deals with sall requireets which are ot appropriate to ebed i real situatios. Cha ad Weil [12] studied the cotext ad the use of wor cycles with various costraits to produce tietables of up to 150 people. Therefore, this study is carried out to highlight the ew odel of the urse schedulig proble. A cyclical urse schedulig with ultiple objectives ad subject to various costraits is developed. The schedule will rely o fairess aog urses ad will cosider urses prefereces to axiize their satisfactio. This will help the urses to provide adequate quality of service. 2 Proble Descriptios This wor is catered i oe ward that has 18 urses with the uber of urses required for orig shift is at least 4 urses, eveig shift is at least 4 urses ad ight shift is exactly 3 urses. The plaig period for the proble is 4- wee or 28 days with 3 differet shifts. There are orig shift startig fro 7 a.. till 2 p.. for 7 hours, eveig shift startig fro 2 p.. till 9 p.. for 7 hours, ad ight shift startig fro 9 p.. till 7 a.. for 10 hours. O the other had, this wor was focused o solvig the cyclical urse schedulig proble. Thus, there is a adjustet to the plaig period for the proble. The legth of the cyclical schedules for this proble is 21 days. As the staff requireet per shift is hoogeous, the iiu required uber of days to assig to each urse is calculated as follows: Miiu Required Nuberof Days MRND uber of urses uber of cosecutive ight shift staff requireet per ight shift I this case, the uber of urses is 18 while the staff requireet per ight shift is 3 ad the uber of cosecutive ight shift is 3. Thus, MRND = 18/3 3= 18. For ergooic purposes, the schedule legth would be easured i wees. Therefore, the schedule legth chose is 21 days (3 wees) where a urse would have to wor 3 cosecutive ight shifts twice i 3 wees. The cyclical schedules would however have the sae set of costraits but with sall chages due to the ew legth of the schedule.

4 154 Ruzzaiah Jeal, et.al. The odel cosiders the hospital s objectives as the hard costraits which ust be satisfied ad the urses prefereces as the soft costraits. The schedules would satisfy the followig objectives set by the aageet of the hospital: 1. Each uit is covered by 3 shifts for 24 hours a day ad 7 days a wee. 2. Miiu staff level requireet ust be satisfied. 3. Each urse wors at ost oe shift a day. 4. Avoid ay isolated days patters of off-o-off. 5. Each urse ust have three days off after havig three cosecutive ight shifts. 6. Each urse wors betwee 12 to 14 days per schedule. 7. Each urse wors ot ore tha 6 cosecutive days. 8. Eveig shift costitutes at least 25% of total worload. 9. Morig shift costitutes at least 30% of total worload. Meawhile the urses prefereces are as follows: 1. Avoid worig i a eveig shift followed by a orig shift or a ight shift the ext day. 2. Avoid worig i a orig shift followed by a eveig shift or a ight shift the ext day. 3. Each urse has at least oe day off i oe weeed. 4. All urses have the sae aout of total worload. 3 Forulatios The developet of urse schedulig odel is based o the hospital s objectives ad the urses prefereces. The hospital s objectives are a set of hard costraits that ust be satisfied while the urses prefereces are a set of soft costraits that ay be violated. The odel will attept to iiize the violatios of the soft costraits. The schedulig proble cotais 13 sets of costraits. It is ot expected however that a feasible solutio ay be obtaied while satisfyig all sets of costraits. Therefore, these sets are divided ito two groups: oe group cosists of sets of hard costraits that ust be satisfied. The other group cosists of reaiig sets of soft costraits, which if could ot be fulfilled, the odel will reduce to at least the violatios of these costraits. The hospital s policies will be cosidered as hard costraits while the urses prefereces will be cosidered as soft costraits.

5 A Cyclical Nurse Schedule Usig Goal Prograig Notatios The followig otatios are used to specify the odel: = uber of days i the schedule ( = 21) = uber of urses available for the uit of iterest ( = 18) i = idex for days, i = 1 = idex for urses, = 1 P i = staff requireet for orig shift of day i, i = 1 T i = staff requireet for eveig shift of day i, i = 1 M i = staff requireet for ight shift of day i, i = Decisio Variables The decisio variables are defied as follows: X i, 1 if urse is assiged a orig shift for day i 0 otherwise Y i, Z i, 1 if urse is assiged a eveig shift for day i 0 otherwise 1 if urse is assiged a ight shift for day i 0 otherwise C i, 1 if urse is assiged a day off for day i 0 otherwise 3.3 Hard Costraits The hard costraits of the forulatio are give below. Set 1: Miiu staff level requireet ust be satisfied: Xi, Pi, i 1,2,..., (1) 1 Yi, Ti, i 1,2,..., (2) 1 Zi, Mi, i 1,2,..., (3) 1 Set 2: Each urse wors oly oe shift a day: X Y Z C 1, 1,2,..., ad 1,2,..., (4) i, i, i, i, i

6 156 Ruzzaiah Jeal, et.al. Set 3: Avoid ay isolated days patters of off-o-off : C X Y Z C 2, 1,2,..., 2 ad 1,2,..., i, i1, i1, i1, i2, i Set 4: Each urse wors 3 cosecutive days of ight shift ad followed by 3 days off. Each urse will be assiged to their ight shifts ad off days as follow: Z Z Z C C C Z Z Z 9, 1,7,13 (6) 1, 2, 3, 4, 5, 6, 19, 20, 21, C C C Z Z Z C C C 9, 2,8,14 (7) 1, 2, 3, 16, 17, 18, 19, 20, 21, Z Z Z C C C 6, 3,9,15 (8) 13, 14, 15, 16, 17, 18, Z Z Z C C C 6, 4,10,16 (9) 10, 11, 12, 13, 14, 15, Z Z Z C C C 6, 5,11,17 (10) 7, 8, 9, 10, 11, 12, Z Z Z C C C 6, 6,12,18 (11) 4, 5, 6, 7, 8, 9, Set 5: Each urse wors betwee 12 to 14 days per schedule: Xi, Yi, Zi, 12, 1,2,..., (12) i1 Xi, Yi, Zi, 14, 1,2,..., (13) i1 Set 6: Each urse wors ot ore tha 6 cosecutive days: C C C C C C C 1, i, i1, i2, i3, i4, i5, i6, i 1,2,..., 4 ad 1,2,..., 6v i, i, 1 i1 (5) (14) C C 1, v 0,1,...,5; 1,2,..., 1 (15) iv 6v C C 1, v 0,1,...,5 (16) iv i, i,1 i1 Set 7: Eveig shift costitutes at least 25% of total worload: Yi, 3, 1,2,..., (17) i1 Set 8: Morig shift costitutes at least 30% of total worload: Xi, 4, 1,2,..., (18) i1

7 A Cyclical Nurse Schedule Usig Goal Prograig Soft Costraits The soft costraits of the forulatio are give below. Set 1: Avoid worig i a eveig shift followed by a orig shift or a ight shift the ext day: Y X Z 1, 1,2,..., 1 ad 1,2,..., (19) i, i1, i1, i Y X Z 1, 1,2,..., 1 (20), 1, 1 1, 1 Y, X1,1 Z1,1 1 (21) Set 2: Avoid worig i a orig shift followed by a eveig shift or a ight shift the ext day: X Y Z 1, 1,2,..., 1 ad 1,2,..., (22) i, i1, i1, i X Y Z 1, 1,2,..., 1 (23), 1, 1 1, 1 X, Y1,1 Z1,1 1 (24) Set 3: Each urse has at least oe weeed off: C7, C14, C21, 1, 1,2,..., (25) Set 4: All urses have the sae aout of total worload: 3.5 Goals Xi, Yi, Zi, 13, 1,2,..., (26) i1 The soft costraits are icorporated i the odel as the goals ad forulated as follows: Goal 1: It avoids assigig a urse to have a eveig shift followed by a orig shift or a ight shift the ext day. Here η1 (respectively ρ1 ) is the aout of egative (positive) deviatio fro goal 1 for urse. Oly positive deviatios are pealized. Y X Z 1 1 1, 1,2,..., 1 ad 1,2,..., (27) i, i1, i1, i, i, i Y X Z 1 1 1, 1,2,..., 1 (28), 1, 1 1, 1,, Y X Z (29), 1,1 1,1,,

8 158 Ruzzaiah Jeal, et.al. Goal 2: It avoids assigig a urse to have a orig shift followed by a eveig shift or a ight shift the ext day. Here η2 (respectively ρ2 ) is the aout of egative (positive) deviatio fro goal 2 for urse. Oly positive deviatios are pealized. X Y Z 2 2 1, 1,2,..., 1 ad 1,2,..., (30) i, i1, i1, i, i, i X Y Z 2 2 1, 1,2,..., 1 (31), 1, 1 1, 1,, X Y Z (32), 1,1 1,1,, Goal 3: It esures that each urse has at least oe day off o weeed i the 3- wee schedule. Here η3 (respectively ρ3 ) is the aout of egative (positive) deviatio fro goal 3 for urse. Oly egative deviatios are pealized. C7, C14, C21, 3 3 1, 1,2,..., (33) Goal 4: It esures that all urses are scheduled to have 13 days as possible i the 3-wee schedule. Here η4 (respectively ρ4 ) is the aout of egative (positive) deviatio fro goal 4 for urse. Both egative ad positive deviatios are pealized. Xi, Yi, Zi, , 1,2,..., (34) i1 Thus, the preeptive goal prograig for this odel is Subject to Miiize 1,, 2,, 3, 4 4 Equatios (1)-(18); Equatios (27)-(34); i i i1 1 i X 0 or 1; Y 0 or 1; Z 0 or 1; C 0 or 1; 1, 1, 2, 2, 3, 3, 4, Results ad Discussio The 0-1 goal prograig odel was ipleeted i oe ward that has 18 urses with the uber of urses required for orig shift is at least 4 urses, eveig shift is at least 4 urses ad ight shift is exactly 3 urses. The odel

9 A Cyclical Nurse Schedule Usig Goal Prograig 159 was solved usig preeptive ethod where the priority orderig used is G1 G2 G3 G4. The odel is optiized usig oe goal at a tie such that the optiu value of a higher priority goal is ever degraded by a lower priority goal. Before ruig the odel usig LINGO software, a coputer code has bee developed. A few odels has bee developed ad adjusted i order to get a good solutio. Nevertheless, the odel preseted here is the best odel for the urse schedulig. Table 1, Table 2 ad Table 3 suaries the result of the odel usig the Ligo software. Table 1 The schedule s patter usig 0-1 goal prograig techique. Day Schedule's Patter S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 S16 S17 S18 1 E A M A M E A M E A M 2 E A M M E A E A M A M 3 E A M E M M A E A M A 4 A M E M M M A E A A E 5 A M M E M M A E A A A E 6 A M M E M M E A A A E 7 A A M E M M M E A A E 8 A A E M M M E M A M A E 9 A M A E M A E M A M E 10 M E M M A A E A M E A 11 M E M A A E A M M E A 12 M M E M A E A A M E A 13 M M E M A E A A M E A 14 M E M M A E A A M E A 15 M E M M A E A A E M A 16 M E A M A E A A M E M 17 E A A A E A M M E M M 18 E A A A E A M M E M M M 19 E E A A A E A M M E M 20 E E A E A M M E A A M 21 E E A E A M M E A A M (M=Morig, E=Eveig, A=Afteroo) Table 1 shows the patters of the shift of the worig day ad the day off for the 3-wees (21 days) plaig period that resulted fro the odel. The schedule satisfied all the hard costraits ad soft costraits where all goals are achieved. Table 1 shows that both goals 1 ad 2 are fulfilled. Thus, there is o eveig shift followed by orig shift or ight shift the ext day ad also, there is o orig shift followed by eveig shift or ight shift the ext day is assiged to each schedule s patter.

10 160 Ruzzaiah Jeal, et.al. Table 2 patter. Suary of the uber of shifts ad weeed off for each schedule s Nuber Schedule's patter of S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 S16 S17 S18 Morig Afteroo Eveig Total Weeed Table 2 shows the suary for the uber of shifts ad weeed off for each schedule s patter. Here, goal 3 where all urses i each schedule s patter ust have at least oe weeed off i 21 days is satisfied. Goal 4 is also satisfied where all schedule s patter have the sae 13 days of total uber of shifts i 21 days plaig period. Table 3 shows the suary for the uber of shifts for each day. The distributios of shifts for each day is also see i balace for each day i the 21 days of plaig period. The total urses o duty for the 21 days varied betwee 11 to 12 urses per day. Table 3 Suary of the uber of shifts for each day. Day Morig Afteroo Eveig Total

11 A Cyclical Nurse Schedule Usig Goal Prograig 161 Table 4 The cyclical schedule s patter for each urse. Schedule (wee) Nurse J1 J2 J3 J4 J5 J6 J7 J8 J9 J10 J11 J12 J13 J14 J15 J16 J17 J18 1 (1-3) S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 S16 S17 S18 2 (4-6) S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 S16 S17 S18 S1 3 (7-9) S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 S16 S17 S18 S1 S2 4 (10-12) S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 S16 S17 S18 S1 S2 S3 5 (13-15) S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 S16 S17 S18 S1 S2 S3 S4 6 (16-18) S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 S16 S17 S18 S1 S2 S3 S4 S5 7 (19-21) S7 S8 S9 S10 S11 S12 S13 S14 S15 S16 S17 S18 S1 S2 S3 S4 S5 S6 8 (22-24) S8 S9 S10 S11 S12 S13 S14 S15 S16 S17 S18 S1 S2 S3 S4 S5 S6 S7 9 (25-27) S9 S10 S11 S12 S13 S14 S15 S16 S17 S18 S1 S2 S3 S4 S5 S6 S7 S8 10 (28-30) S10 S11 S12 S13 S14 S15 S16 S17 S18 S1 S2 S3 S4 S5 S6 S7 S8 S9 11 (31-33) S11 S12 S13 S14 S15 S16 S17 S18 S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 12 (34-36) S12 S13 S14 S15 S16 S17 S18 S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 13 (37-39) S13 S14 S15 S16 S17 S18 S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 14 (40-42) S14 S15 S16 S17 S18 S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 15 (43-45) S15 S16 S17 S18 S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 16 (46-48) S16 S17 S18 S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 17 (49-51) S17 S18 S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 S16 18 (52-54) S18 S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 S16 S17 The head urse will allocate each schedule s patter to each urse as show i Table 4. For schedule i 1,...,18, urse j wors accordig to patter j i 1 od18 if ji1 18, ad 18 if S ; where j j ji1 18. The cyclical schedulig for each urse is as show i Table 4. The patter will be rotated aog the urses ad each urse will be worig accordig to each schedule s patter at the ed of wee 54, or 18 schedules. After copletig 18 schedules, the each urse will revisit the startig schedule. I Table 4, it shows that the cyclical urse schedulig rotates equally through the desirable ad udesirable wor stretches aog the urses ad requires relatively less schedulig effort of the head urse. The schedule satisfies the factors of copleteess ad cotiuity. While the fairess factor is dealt with sice the schedule s patter is goig to rotate aog the urses. All urses will have the opportuity to wor with the satisfactory ad usatisfactory schedule s patters. The urse will have the least satisfactory schedule s patter whe they have the patter S6 with total uber of orig shift is 7 as show i Table 2. While the ost satisfactory schedule s patter is whe they have the patters S6, S12 ad S18 with the total uber of weeed off is 2 days (see Table 2). Each urse will also have a log days off (3 days off after 3 cosecutive eveig shifts followed by 2 days off) whe they have the j

12 162 Ruzzaiah Jeal, et.al. schedule s patter S8 followed by schedule s patter S9, the schedule s patter S11 ad S17. The ore weeed off ad a log days off are good for urses to pla ay activities with their failies ad social life. This iplies ubiased of the schedule to all urses. Furtherore, with this cyclical schedulig, it gives urses ore cotrol over their wor life because they ow the type of shift schedule i the future which should have a positive effect o their job satisfactio. 5 Coclusios Modelig urse schedulig usig a 0-1 goal prograig has show its capability of geeratig schedules cosiderig all the hard ad soft costraits i the schedulig eviroet. The developed schedulig has bee foud ot oly to satisfy hospital s objectives but also urses prefereces. Both parties obtaied higher satisfactio whe all goals are achieved. All the urse s prefereces or goals o havig a eveig shift ot followed by a orig shift or a ight shift the ext day; havig a orig shift ot followed by a eveig shift or a ight shift the ext day, havig at least oe day off o weeed i 21 days of plaig period; ad havig the sae total uber of shifts are fulfilled with the optiu solutio. The 0-1 goal prograig techique has bee proved to solve the ultiple objectives proble effectively ad aided the decisio aer o aig a wise ad appropriate decisio of the schedule. The developed odel with various costraits ad goals usig the 0-1 goal prograig techique gives the optiu solutio that showed both the hard costraits ad soft costraits are satisfied. The optiu solutio gathers whe all the goals are achieved with the objective fuctio value is equal to zero ad thus there is o pealty. The cyclical schedulig for the developed odel help the head urse to have less effort o buildig the ew schedules. All urses have 18 patters of schedule i periodicity of 378 days (54 wees) or approxiately 12 oths or a year. The they eet the first schedule s patter agai. The urses will go through the satisfactory ad usatisfactory schedule s patter without feelig biased aog the ad thus lead to a overall higher satisfactio of the urses. New schedule will oly eed to be produced whe chages occur i its average daily staff requireets. For further research, oe of possible wor is to ebed the odel ito userfriedly software that would be easy to use ad reliable. The cyclical urse schedulig should be cosidered to iprove the ways of developig the schedule ad save ore tie. The odel also should be exteded to accout for

13 A Cyclical Nurse Schedule Usig Goal Prograig 163 other iportat schedulig aspects such as requested day off i order to beig acceptable to all parties. Refereces [1] Bard, J.F., & Puroo, H.W., Preferece Schedulig for Nurses Usig Colu Geeratio, Europea Joural of Operatioal Research, 164, pp , [2] Bester, M.J., Nieuwoudt, I. & va Vuure, J.H., Fidig Good Nurse Duty Schedules: A Case Study, Joural of Schedulig, 10, pp , [3] Beaulieu, H., Ferlad, J.A., Gedro, B. & Michelo, P., A Matheatical Prograig Approach for Schedulig Physicias i The Eergecy Roo, Health Care Maageet Sciece, 3, pp , [4] Topaloglu, S., A Multi-Objective Prograig Model for Schedulig Eergecy Medicie Residets, Coputers ad Idustrial Egieerig, 51, pp , [5] Belie, J., Deeuleeester, E. & Cardeo, B., A Decisio Support Syste for Cyclic Master Surgery Schedulig with Multiple Objectives, Joural of Schedulig, , (2008). [6] Bure, E.K., de Causaecer, P., va de Berghe, G. & va Ladeghe, H., The State of The Art of Nurse Schedulig, Joural of Schedulig, 7, pp , [7] Cheag, B., Li, H., Li, A. & Rodrigues, B., Nurse Schedulig Probles A Bibliographic Survey, Europea Joural of Operatioal Research, 151, pp , [8] Erst, A.T., Jiag, H., Krishaoorthy, M. & Sier, D., Staff Schedulig Ad Schedulig: A Review of Applicatios, Methods ad Models, Europea Joural of Operatioal Research, 153(1), pp. 3-27, [9] Arthur, J. L., & Ravidra, A., A Multiple Objective Nurse Schedulig Model, IIE Trasactios, 13(1), pp , [10] Azaiez, M.N., & Al Sharif, S.S., A 0-1 Goal Prograig Model for Nurse Schedulig Proble, Coputers & Operatios Research, 32, pp , [11] Berrada, I., Ferlad, J. A., & Michelo, P., A Multi-objective Approach to Nurse Schedulig with Both Hard ad Soft Costraits, Socio-Ecooic Plaig Scieces, 30(3), pp , [12] Cha, P. & Weil, G., Cyclical Staff Schedulig Usig Costrait Logic Prograig, Lecture Notes o Coputer Scieces 2079, pp , 2001.

14 164 Ruzzaiah Jeal, et.al. [13] Ozaraha, I. & Bailey, J.E., Goal Prograig Model Subsyste of A Flexible Nurse Schedulig Support Syste, IIE Trasactios, 20(3), pp , [14] Wa Rosaira Isail, Ruzzaiah Jeal, Liog Choog Yeu & Mohd Khairi Muda, Pejaduala Kerja Berala Jururawat Megguaa Kaedah Pegaturcaraa Gol 0-1(Periodic Rosterig for Nurses Usig 0-1goal Prograig Method), Sais Malaysiaa, 38(2), pp , [15] Musa, A. A., & Saxea, U., Schedulig Nurses Usig Goal-Prograig Techiques, IIE Trasactios, 16(3), pp , [16] Harvey, H.M., & Kiragu, M., Cyclic ad No-cyclic Schedulig of 12 h Shift Nurses by Networ Prograig, Europea Joural of Operatioal Research, 104, pp , 1998.