GOAL PROGRAMMING BASED MASTER PLAN FOR CYCLICAL NURSE SCHEDULING

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1 Joural of Theoretical ad Applied Iforatio Techology 5 th Deceber 202. Vol. 46 No JATIT & LLS. All rights reserved. ISSN: E-ISSN: GOAL PROGRAMMING BASED MASTER PLAN FOR CYCLICAL NURSE SCHEDULING WAN ROSMANIRA ISMAIL 2 RUZZAKIAH JENAL 3 NUR ASYIKIN HAMDAN 3 School of Matheatical Scieces Faculty of Scieces ad Techology Uiversiti Kebagsaa Malaysia Bagi Selagor. 2 School of Iforatio Techology Faculty of Techology ad Iforatio Scieces Uiversiti Kebagsaa Malaysia Bagi Selagor. E-ail: wrisail@uk.y 2 ruzza@fts.uk.y ABSTRACT Costructig tietable for urses i hospital is a challegig job for the head urse or urse aager as the task is doe repeatedly for every specified period. It requires a lot of tie ad effort for geeratig a good ad fair tietable. I order to overcoe the proble a cyclical urse schedule usig a 0- goal prograig is proposed. The tietable will be the aster pla that would help the head urse or urse aager to schedule the urses i a yearly basis. The proposed odel has also aaged to satisfy the hospital s policies ad the urses prefereces. The odel has provided a ubiased schedule ad thus leads to a overall satisfactio ad fairess to the urses. The developed odel has bee solved usig LINGO software ad the result is foud to be better copared to the preset ethod used. Keywords: Matheatical Model; Goal Prograig; Cyclical Schedule; Nurse Schedulig. INTRODUCTION The study o urse schedulig is ot ew i the world of research related to hua resource aageet i health care istitutios []. Nurse schedulig is however a dyaic ad challegig tasks to be addressed fro tie to tie. At a local hospital i Malaysia the urse tietable is geerated aually based o the experiece ad kowledge of head urse ad approval of adiistratio. Sice there is o writte policies ad procedures as a guidace i the preparatio of the tietable the schedule becoes iflexible ad biased to soe urses at certai ties. I this study a urse schedulig odel will be developed to solve ufairess i workload distributio aog the urses. A 0- goal prograig (0GP) techique will be used to build the odel. The 0GP is a variatio fro liear prograig techique that cosiders ore tha oe objective. The 0GP prefers to iiize slack variables for each objective fuctios accordig to their priorities rather tha to iiize the objective fuctios itself [2][3]. There are a few research applied the 0GP techique to solve their proble such i acquisitio allocatio proble [4] recyclig syste [5] operatio waitig list proble [6] facility locatio selectio [7] ad staff schedulig proble [8][9][0][]. Musa ad Saxea [8] have applied GP techique to oe uit of a hospital that tackled the hospital s policies ad urses requests. But the developed odel is quite low coplexity with a two weeks plaig period ad oly oe shiftwork. Isail et. al. [9] also used GP techique to build a periodic urse rosterig odel. However the odel liited to a 4 days of plaig period with three types of shift works. Azaiez ad Al-Sharif [0] have developed a urse schedulig odel with four weeks plaig period ad two shift works i Riyadh Al-Kharj Hospital. The odel has cosidered the hospital s objectives ad urses request. The result fro the odel has icreased the hospital perforace while reducig about 4% of their overtie cost. Nevertheless the odel has to be reewig for each four weeks. Jeal et. al. [] has solved a cyclical urse schedulig proble usig GP techique. The odel has cosidered ad achieved the hospital s objective ad urses request for three weeks plaig period ad three shift works. The developed odel has produced a schedule patter for each of the 8 urses i 2 days (three weeks). For the ext 2 days i order to preserve the fairess ad cotiuity of the schedule urse 2 will take urse s patter urse 3 will take urse 2 s patter ad fially urse will take urse 8 s 499

2 Joural of Theoretical ad Applied Iforatio Techology 5 th Deceber 202. Vol. 46 No JATIT & LLS. All rights reserved. ISSN: E-ISSN: patter. The patter for each urse will chage i a cascadig aer util week 55 where the patter will agai retur to the origial schedule. Thus i this study a aster pla for cyclical urse schedulig odel will be developed to solve the urse schedulig proble. The proposed odel would be cyclic ad have a shorter tie to recycle agai. 2. A MASTER PLAN FOR NURSE SCHEDULING MODEL 2. A Case Study A case study cosiders a 2 days plaig period ad three shift works i oe ward of Tuaku Miza Ary Hospital Kuala Lupur. The shift works divides ito orig shift (7 a 2 p) eveig shift (2 p 9 p) ad ight shift (9 p 7 a). There are 2 urses available i this ward. The odel cosiders both hospital s objectives ad urses request. The hospital s objectives act as hard costraits that ust be satisfied while the urses requests act as soft costraits that ca be violated. Table. Notatios Notatios Defiitio Nuber of days work per schedule (=2). Nuber of urses available i a ward (=2). i Idex for days k Idex for urses k=2.... P i Nuber of urses required for orig shift of day i. T i Nuber of urses required for eveig shift of day i M i Nuber of urses required for ight shift of day i. Decisio Variables A ik C ik E ik F ik Table 2. Decisio Variables Defiitio = if urse k is assiged orig shift for day i = if urse k is assiged eveig shift for day i = if urse k is assiged ight shift for day i = if urse k is assiged a day off for day i 2.2 Notatios ad Decisio Variables Table shows the otatios while Table 2 shows the decisio variables used i this study. 2.3 Hard Costraits The hospital s objectives or hard costraits cosiders i this study are as follow: to esure the iiu uber of urses are fulfilled for each shift each day; k = A P; i = 2... ik i () Aik Ti; i = 2... (2) k = Aik Mi; i = 2... (3) k = to esure each urse has oe ad oly oe shift work i a day; A C E F i = 2... k = 2... ik ik ik ik = ; (4) 500

3 Joural of Theoretical ad Applied Iforatio Techology 5 th Deceber 202. Vol. 46 No JATIT & LLS. All rights reserved. ISSN: E-ISSN: to esure each urse has a three cosecutive days of ight shifts the followed by two cosecutive days of days off; E E E F F = i = k = 2 3 i k i k i i 3 k i 4 k 5; E E E F F = i = 4 k = 456 Ei k Ei k Ei k Fi k Fi k = i = 7 k = 78 9 Ei k Ei k Ei = 3 ; i = 0 k = 02 i k i k i i 3 k i 4 k 5; ; (5) (6) (7) (8) to esure each urse has the uber of workload betwee eight to te days per schedule; ( Aik Cik Eik ) 8; (9) k = 2... ( Aik Cik Eik ) 0; (0) k = 2... to esure each urse has o ore tha six cosecutive days o; ad Fi k Fi k Fi Fi 3 k Fi 4 k Fi 5 k Fi 6 k ; () i = k = 2... to esure each urse works accordig to the uber of required shift works per schedule. A 3; k = 2... ik (2) Fi k Ai k Ci k Ei k Fi 2; i = k = 2... (5) F A C E F ηk ρ k= 2; i = k = 2... i k i k i k i k i (6) to esure each urse has the sae uber of workload per schedule; ( Aik Cik Eik ) = 9; (7) k = 2.. ( A C E ) ik ik ik η2k ρ2k = 9; k = 2.. (8) to avoid a eveig shift followed by orig or ight shift; Ci k Ai k Ei k ; i = 2... k = 2... Ci k Ai k Ei k η3k ρ3k = ; i = 2... k = 2... (9) (20) to avoid a orig shift followed by eveig or ight shift; ad A C E i = 2... k = 2... i k i k i k ; Ai k Ci k Ei k η4 k ρ4 k = ; i = 2... k = 2... (2) (22) Cik = 3; k = 2... (3) Eik = 3; k = 2... (4) 2.4 Soft Costraits The urses requests or soft costraits cosider i this study are as follow: to avoid a isolated days o; to avoid a isolated days off. A C E F A C ik ik ik i k i i Ei 2; i = k = 2... A C E F A ik ik ik i k i Ci Ei η5k ρ5k = 2; i = k = 2... (23) (24) 50

4 Joural of Theoretical ad Applied Iforatio Techology 5 th Deceber 202. Vol. 46 No JATIT & LLS. All rights reserved. ISSN: E-ISSN: For each soft costrait it will be forulated as a goal by addig up the slack variables. Equatios (6) (8) (20) (22) ad (24) show the goal goal 2 goal 3 goal 4 ad goal 5 for each soft costrait respectively. 2.5 Objective Fuctios Each goal will be iiized accordig to their priority. The first priority is to iiize goal followed by goal 2 goal 3 goal 4 ad goal 5 i successio. Goal is aied to iiize the positive slack variable. Goal 2 is aied to iiize both positive ad egative slack variables. Goal 3 goal 4 ad goal 5 are aied to iiize the positive slack variables respectively. Thus the objective fuctio for cyclical urse schedulig odel accordig to their priority is as follow: Miiize k= k = ρ ik η2k ρ2 k ρ3 ik ρ4 ik ρ5ik k= k= k= 3. RESULTS AND DISCUSSIONS The cyclical urse schedulig odel solves usig LINGO 8.0 software. The result shows that the developed odel has fulfilled goal goal 2 ad goal 3 while goal 4 ad goal 5 are violated. For goal the positive slack variable gives zero value which eas the objective of avoidig the isolated days o is achieved. There is o urse have the isolated days o i their schedule s patter. For goal 2 both the positive ad egative slack variables give zero value which eas the objective of havig the sae uber of workload per schedule for each urse is achieved. All urses have the 9 days of workload per schedule. For goal 3 the positive slack variable gives zero value which eas the objective of avoidig eveig shift followed by orig shift or ight shift is achieved. There is o urse have the schedule s patter of havig eveig shift followed by orig shift or ight shift. For goal 4 there is a value for the positive slack variable. The objective of avoidig orig shift followed by eveig shift or ight shift is ot achieved. For goal 5 there is a value for the positive slack variable. The objective of avoidig the isolated days off is ot achieved. Table 3 shows the duty roster doe aually while Table 4 shows the duty roster developed by usig the 0GP techique. The schedule produced aually has show that there is a icosistecy i the total uber of workloads for the urses ad also the uber of urses o duty each day. The workloads vary fro days to 6 days with oe urse ot havig ay shifts. The distributios of shifts are also ubalaced. There is oe urse who has ot bee assiged the ight shifts at all i the 4 days period. These icosistecies were the reasos for the urse satisfactio ad quality of work. The developed odel usig the 0GP produced a very cosistet ad fair schedule for urses. I the 2 day period the total uber of workloads ad the uber of urses o duty daily are the sae ad also the equal distributios of shift works. The reaso why 2 day schedule is chose is because the sae patter for each urse ca be repeated for the ext 2 days without violatig both hard ad soft costraits. This will produce a aster pla that would be cyclical 30 ties i a year for 360 days. With the aster pla the head urse ca have a good plaig o the urses activities such as attedig the courses or seiars. While for the urses theselves they ca have a good plaig of their ow activities such as havig log holidays with their faily. The aster pla also deostrates the fairess of schedule. The uber of shift work ad workload per schedule distributes equally aog the urses ad this leads to the urses satisfactio. 502

5 Joural of Theoretical ad Applied Iforatio Techology 5 th Deceber 202. Vol. 46 No JATIT & LLS. All rights reserved. ISSN: E-ISSN: Table 3. The Duty Roster Doe Maually D a P ca T da M ea fa J b S S M T W T F S S M T W T F J Office-hour duty J2 T T T P P T T P M M J3 P T P P P P M M M J4 P P T P P M M M T P J5 M M M T T P P T T T J6 M M M T T T P P T T J7 T P M M M P J8 T P P P P M M M J9 T T T T T P P M M J0 P T P T T P P P P T P J T T M M M seiar P P J2 T P P T T M M M T T P cb T db M eb fb a.days; b. Nurses; c. Morig shifts; d. Eveig shifts; e. Night shifts; f. Total Table 4. The Duty Roster Developed By Usig The 0GP Techique D a P ca T da M ea fa J b S S M T W T F S S M T W J M M M T T T P P P J2 M M M T T T P P P J3 M M M T T T P P P J4 P P P M M M T T T J5 P P P M M M T T T J6 P P P M M M T T T J7 T T P P P M M M T J8 T T P P P M M M T J9 T T P P P M M M T J0 T T T P P P M M M J T T T P P P M M M J2 T T T P P P M M M P cb T db M eb fb a.days; b. Nurses; c. Morig shifts; d. Eveig shifts; e. Night shifts; f. Total 4. CONCLUSION Schedulig the urse to their duty roster is a iportat ad repetitive task to the head urse. The developet of the aster pla for cyclical urse schedulig odel helps the head urse solvig the urse schedulig proble ad gives a better result of the duty roster copared to the aual duty roster. Although the process of gettig the result takes few ties to ru but the output is reasoable ad beig accepted by both the hospital ad the urses. The 0GP techique has bee proved to solve ore tha oe objective effectively. The cyclical schedulig aides the head urse to have less effort durig the schedulig process. The head urse also ca prepare the log ter plaig of activity for each urse i advace. For further research the odel ca be iproved by addig up differet sizes of urses to cater differet types of ward. 503

6 Joural of Theoretical ad Applied Iforatio Techology 5 th Deceber 202. Vol. 46 No JATIT & LLS. All rights reserved. ISSN: E-ISSN: REFERENCES: [] E. Burke P. De Causaecker G. Berhge ad H. Va Ladeghe The state of the art of urse rosterig Joural of Schedulig 2004 pp [2] S.H. Zaakis ad S.K. Gupta A categorized bibliographic survey of goal prograig Oega Vol. 3 No pp [3] B. Aoui ad O. Kettai Goal prograig odel: a glorious history ad a proisig future Europea Joural of Operatioal Research Vol pp [4] K. Wise ad D.E. Perushek Goal prograig as a solutio techique for the acquisitios allocatio proble Library & Iforatio Sciece Research Vol. 22 No pp [5] R.K. Pati P. Vrat ad P. Kuar A goal prograig odel for paper recyclig syste Oega Vol pp [6] M. Areas A. Bilbao R. Caballero T. Goez M.V. Rodriguez ad F. Ruiz Aalysis via goal prograig of the iiu achievable stay i surgical waitig lists Joural of the Operatioal Research society Vol pp [7]O.T. Arogudade A.T. Akiwale A.F. Adekoya ad G. Awe Oludare A 0- odel for fire ad eergecy service facility locatio selectio: a case study i Nigeria Joural of Theoretical ad Applied Iforatio Techology Vol. 9 No pp [8] A.A. Musa ad U. Saxea Schedulig urses usig goal-prograig techiques IIE Trasactios Vol. 6 No pp [9] W.R. Isail R. Jeal C.Y. Liog ad M.K. Muda Pejaduala kerja berkala jururawat egguaka kaedah pegaturcaraa gol 0- (periodic rosterig for urses usig 0- goal prograig ethod) Vol. 38 No pp [0] M.N. Azaiez ad S.S. Al-Sharif A 0- goal prograig odel for urse schedulig Coputers & Operatios Research Vol pp [] R. Jeal W.R. Isail C.Y. Liog ad A. Oughalie A cyclical urse schedule usig goal prograig ITB Joural of Sciece Vol. 43A No pp