research article modelling a nurse shift schedule with...

Research ArticleModelling a Nurse Shift Schedule with Multiple PreferenceRanks for Shifts and Days-Off

Chun-Cheng Lin1 Jia-Rong Kang1 Wan-Yu Liu2 and Der-Jiunn Deng3

1 Department of Industrial Engineering and Management National Chiao Tung University Hsinchu 300 Taiwan2Department of Tourism Information Aletheia University New Taipei City 251 Taiwan3Department of Computer Science and Information Engineering National Changhua University of Education Changhua 500 Taiwan

Correspondence should be addressed to Der-Jiunn Deng djdengccncueedutw

Received 4 October 2013 Revised 22 February 2014 Accepted 22 February 2014 Published 27 March 2014

Academic Editor Ching-Ter Chang

Copyright copy 2014 Chun-Cheng Lin et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

When it comes to nurse shift schedules it is found that the nursing staff have diverse preferences about shift rotations and days-off The previous studies only focused on the most preferred work shift and the number of satisfactory days-off of the schedule atthe current schedule period but had few discussions on the previous schedule periods and other preference levels for shifts anddays-off which may affect fairness of shift schedules As a result this paper proposes a nurse scheduling model based upon integerprogramming that takes into account constraints of the schedule different preference ranks towards each shift and the historicaldata of previous schedule periods tomaximize the satisfaction of all the nursing staff rsquos preferences about the shift scheduleThemaincontribution of the proposed model is that we consider that the nursing staff rsquos satisfaction level is affected by multiple preferenceranks and their priority ordering to be scheduled so that the quality of the generated shift schedule is more reasonable Numericalresults show that the planned shifts and days-off are fair and successfully meet the preferences of all the nursing staff

1 Introduction

Nurse-scheduling problem is a classical combinatorial opti-mization problem and has been shown to be NP-hard [12] The objective of the nurse-scheduling problem is todetermine the rotating shifts of the nursing staff over aschedule period (weekly or monthly) [3] A nurse scheduleincludes the work shifts and days-off of the nursing staffensuring that all the combinations of shifts and days-off meetthe manpower requirements of each shift (including totalnumber of staff members daily minimum number of staffmembers and number of senior staffmembers required) andat the same time the number of basic days-off of each staffmember should be fulfilled [3]

In general there are a lot of types of work shifts anddays-off in shift schedules The most common ones includethe 2-shift rotation (ie 12-hour day shift and 12-hour nightshift) and the 3-shift rotation (ie 8-hour day shift 8-hour evening shift and 8-hour night shift) Regular days-off allow the nursing staff to rest and each staff member is

entitled to the same number of days-off [4ndash7] Due to diversepersonal lifestyles and different degrees of physical tolerancefor continuous working days the nursing staff usually havedifferent preferences for work shifts and days-off

The satisfaction of the nursing staff rsquos preference for workshifts and days-off enables them to take the proper rest toincrease the quality of medical service and reduce medicalcost of the hospital as well as risks of occupational hazard[8 9] Therefore it has been an interesting problem in therecent works to take into consideration the nursing staff rsquospreferences in planning the schedule of work shifts and days-off and adopt maximization of satisfaction and minimizationof penalty cost to evaluate the quality of the shift schedulewith preferences [2 10ndash13]

For example as far as the hard constraints and softconstraints of the nurse shift schedule (including the nursingstaff rsquos preferences and the demands of hospitals) are con-cerned Hadwan et al [2] aimed to minimize the penalty costof a nurse schedule Aickelin and Dowsland [11] applied agenetic algorithm to solve the nurse shift scheduling problem

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 937842 10 pageshttpdxdoiorg1011552014937842

2 Mathematical Problems in Engineering

with the objective of minimizing the penalty cost for notfulfilling the preferences of the nursing staff Maenhout andVanhoucke [12] investigated the penalty costs with multipleconstraints (including the nursing staff rsquos preferences andsome specific combinations of work shifts and days-off)Topaloglu and Selim [13] considered a variety of uncertainfactors in nurse shift scheduling to propose a fuzzy mul-tiobjective integer programming model which takes intoconsideration the fuzziness of the objective and the nursingstaff rsquos preferences

From the literature we discover that in the penalty cost(or satisfaction) due to the nursing staff rsquos preferences in theobjective function all the previous works only focused onthe most preferred work shift and the number of satisfactorydays-off of the shift schedule at the current schedule periodand further investigated neither different preference ranks(such as three ranks good normal or bad) of the nursingstaff toward different work shifts or planned days-off nor thenumber of times in which their preferences are satisfied in theprevious shift schedules and days-off Since the constraintsof the schedule lead to the fact that the most preferred workshifts and days-off of each staff member cannot be fulfilledcompletely we believe that if the penalty cost (or satisfactionlevel) due to preferences of the nursing staff does not considerdifferent preference ranks and the historical data of previousschedules the long-term fairness and justice of the nurseshift schedules is affected For example if each nursing staffmember has three different preference ranks (good normalor bad) towards a 3-shift rotation those scheduled to a badwork shift would have a larger preference penalty cost (orlower preference satisfaction level) as compared with thosescheduled to work a normal shift and should be assigneda higher priority to their higher preference rank in theirschedule at the next scheduling period

In light of the above this paper proposes a binary integerlinear programming model for the nurse shift schedule withdifferent levels of satisfaction preference and a differentpriority ordering of the nursing staff for planning their shiftschedule The objective of this model is to maximize theoverall satisfaction level of the nursing staff towards theirwork shifts and days-off schedule In addition some commonnurse-scheduling constraints are considered including thelimited number of nursing staff members work shift lim-itation and day-off limitation Our mathematical model iscapable of effectively helping schedule planners to design anurse shift schedule that attempts to within all schedulingconstraints satisfy the shift and day-off preferences of mostof the nursing staff members in a fair manner and to achievethe highest overall satisfaction level

Themain contributions of this paper are stated as followsIn the proposed mathematical model for nurse schedule thework shift and day-off preferences of the nursing staff arecategorized into different levels and are then integrated to besolved In addition in the case where the preferred number ofshifts or days-off exceeds the actual shifts or days-off available(due to schedule constraints) a priority ordering mechanismof the nursing staff when planning their shift schedule isapplied to solve the contradictory situations among theirpreferences

The rest of this paper is organized as follows Section 2gives the literature review of our work Section 3 firstdescribes our concerned problem and then constructsits mathematical programming model Section 4 gives anumeric example to analyze the performance of the shiftschedule constructed by our mathematical model Finally aconclusion is made in Section 5

2 Literature Review

Thenurse-scheduling problems have been solved by a varietyof methods which are mainly introduced by mathematicalprogramming heuristics and others in this section

First the mathematical programming approaches for thenurse-scheduling problem are introduced Maenhout andVanhoucke [12] proposed an integrated analysis method tosolve human resource planning and shift scheduling problemof nurses on the long run Azaiez and Al Sharif [14] solvedthe nurse-scheduling problemwith a 0-1 linear programmingmodel which takes into consideration the ratio of nursesworking night shifts or having days-off on weekends andtries to avoid unnecessary overtime so that hospital costscan be reduced Topaloglu [15] proposed a multiobjectiveprogrammingmodel to tackle the nurse-scheduling problemsof house physicians in the emergency room Based on theAHP method soft constraints in the model are weighted inaccordance with their relative importance and this becomesthe basis for weighting objective functions Topaloglu [16]proposed a multiobjective scheduling model for planningthe shifts of resident physicians in which seniority of aresident physician is used for the weight setting Empiricalanalysis showed that the model is far superior to the manualscheduling method in terms of efficiency and time-saving

Belien and Demeulemeester [17] proposed an integratedscheduling method for nurses and surgeons They appliedthe branch-and-price method to perform integration whichis one of the most common methods to generate explicitsolutions among the column generation techniques GlassandKnight [18] identified four categories of nurse-schedulingproblems and solved them with mixed-integer linear pro-gramming Results showed that the optimal solutions wereproduced in all benchmarking examples within 30 minutesThe characteristic of this model is that it reduces the collec-tion space based on the structure of the problem so that theproblem-solving efficiency can be enhanced Valouxis et al[19] proposed a 2-stage solution to solve the nurse-schedulingproblem where the workload and days-off of each nurse arefirst determined before the shifts are planned

Second the heuristics for nurse-scheduling problems areintroduced Hadwan et al [2] proposed a harmony searchalgorithm for the nurse-scheduling problem which wastested in hospitals in Malaysia and was shown to be betterthan the genetic algorithm approach as well as most heuristicapproaches Aickelin andDowsland [11] proposed an indirectgenetic algorithm for the nurse-scheduling problem inwhicha chromosome encoding is performed recombination isconducted with a heuristic approach evolution is conductedvia mixed-crossover to locate better solutions The method

Mathematical Problems in Engineering 3

has been tested and was found to be superior to some of thealready published Tabu search methods

Tsai and Li [20] proposed a 2-stage programming modelto analyze and solve the nurse-scheduling problem in whichdays-off are planned in the first stage and then shifts aredetermined in the second stage The two stages were thenanalyzedwith a genetic algorithm Sadjadi et al [21] proposeda mixed-integer nonlinear programming model to randomlyplan shift schedules in which the demand for humanresource is considered a variable and is based on a certainprobability distribution Then the authors applied the GAand Taguchi method to solve the nurse-scheduling problemGutjahr andRauner [22] proposed ant colony optimization tosolve dynamic regional nurse-scheduling problem in a publichospital in Vienna Austria Upon verification via simulationexperiments the model was shown to be superior to a greedyassignment algorithm

Finally except for the above two methods some othersolutions for the nurse-scheduling problem are introducedLu and Hao [23] proposed adaptive neighborhood search(ANS) for the nurse-scheduling problem which performs aneighborhood search and changes based on three differentlevels of intensity and change The approach was tested on60 examples and the results were quite impressive M VChiaramonte and L M Chiaramonte [24] proposed to use acompetitive agent-based negotiation algorithm for the nurse-scheduling problem which aims to maximize preferences ofnurses and minimize the costs Vanhoucke and Maenhout[25] proposed a set of complexity indicators for the nurse-scheduling problem which can indicate the complexity ofthe problem and automatically come up with a simulationsolution that meets the level of complexity of the problemThey can be used as the baseline analysis to compare theperformance of different approaches

From the above literature review it can be found thatmany works did not explore the preference ranks of the nurs-ing staff towards each shift rotation or day-off In additionthe preferred shift and day-off are not given any priorityordering according to the previous scheduling periods Basedon this this paper proposes a mathematical programmingmodel for the nurse-scheduling problem in order to producea preliminary shift schedule that can fulfill the needs ofpractical work and at the same time satisfy most nursing staffmembers

3 Methodology

In this sectionwe construct amathematicalmodel to plan theshift rotation and day-off based on the preference ranks of thenursing staff The construction process of the mathematicalmodel includes description of the problem identification ofthe satisfaction level of the nursing staff and the developmentof the mathematical model

31 Problem Description In this subsection we explain theactual work scenario inside the hospital and then the prob-lems encountered when planning for a shift schedule

We first describe the work scenario which includes thestructure and constraints of shift schedule combination ofthe nursing staff and the preference ranks of the nursingstaff towards each shift rotation and days-off Consider a shiftschedule for a 2-week work in which the shifts of a day startat 000 AM and the hospital runs on a 3-shift rotation a dayshift (800 AMsim400 PM) an evening shift (400 PMsim000AM) and a night shift (000 AMsim800 AM) Note that onlyregular days-off are planned in the schedule

The planned schedule has some constraints on shiftrotations each nursing staff member is only assigned to afixed type of shift within each schedule period the numberof nursing staff members required for each shift is fixed(after deducting the number of nursing staff members onregular days-off) each person should have at least an 8-hourrest before continuing on to the next shift Note that theconstraint of a fixed shift for each staff member is reasonablesince in practice in order to ensure that the nursing staffenjoy the health with fixed work and rest some hospitalsassign each nursing staff member a fixed work shift type forall working days of the scheduling period In addition theplanned schedule has some constraints on days-off the totalnumber of days-off of each nursing staff member within theschedule period is the same and each nursing staff memberis entitled to at least one day-off each week The maximumnumber of the nursing staffmembers is allowed to be on day-off and the number of senior staff members working in eachshift each day are known and flexible

As for composition of the nursing staff the qualifiednursing staff members are categorized into junior and seniorstaff members where the staff members with at least 2 yearsof nursing working experience are considered senior andthose below 2 years junior Also all the nursing staff are fulltime When the total number of staff members is insufficientto cover all the shifts the concerned department has tohire new staff members to fill this shortage in manpowerOutsourcing nursing staff members from other departmentsis not allowed

Lastly the nursing staff are asked to rank their preferencesfor each shift and day- off which are called preference ranksThe preference ranks of each shift are classified into threetypes ldquogoodrdquo ldquonormalrdquo and ldquobadrdquo shifts The preferenceranks of days-off are classified into ldquogoodrdquo and ldquobadrdquo days-off based on the ldquopreferredrdquo and ldquonot preferredrdquo days-offrespectively Note that each staff member has a fixed numberof days-offwithin each schedule periodTherefore we assumethat there may be more than one ldquogoodrdquo day-off and nofurther rank is made among all the ldquogoodrdquo days-off

As the nursing staff have rather diverse preferences theyare asked to fill out a preference form before the scheduleis formulated so that the schedule planner has adequateinformation about the staff rsquos preferences for shifts and days-off and the total number of the nursing staff preferring eachshift or day-off In the preference form the preference ranksof shifts are expressed in a numerical ordering 1 indicatesldquogoodrdquo (most preferred) 2 indicates ldquonormalrdquo and 3 indicatesldquobadrdquo (least preferred) The preference ranks of days-off areexpressed as follows 1 indicates ldquogoodrdquo (preferred) while 3indicates ldquobadrdquo (not preferred) Note that the numbers of


the preference ranks for shifts and days-off are three (ie1 2 and 3 for good normal and bad resp) and two (iegood and bad) respectively We assume that the good andbad preference ranks for shifts and days-off are correspondedwith each otherTherefore we let the good and bad preferenceranks for days-off be 1 and 3 respectively

In the actual work scenario each nursing staff mem-ber has different preference ranks for shifts and days-offmarkedly increasing the computing time and difficulty toformulate a shift schedule Also a lot of constraints forschedule make it impossible for all the staff members towork their preferred shifts and have their preferred days-off Therefore it is important to formulate a preliminaryshift schedule recommendation so that schedule plannerscan perform necessary and flexible adjustments on therecommended schedule reducing the difficulty andworkloadof manpower planning

32 Preference Satisfaction of Shifts and Days-Off This sub-section discusses the satisfaction of the preference ranks ofshifts and days-off which is called preference satisfactionThis paper aims to maximize the overall shifts and days-offpreference satisfaction of the nursing staffThere are differentpreference ranks of shifts and days-off and the preferencesatisfaction increases with the preference ranksThemore thenumber of preferred shifts and days-off is satisfied the higherthe overall satisfaction level towards the shift schedule will be

However due to the constraints on shifts and days-offnot all the preferred shifts and days-off could be satisfied Forthe staff members not scheduled to their preferred shifts ordays-off for consecutive schedule periods if their preferencesin the next schedule are satisfied with a higher prioritythen their preference satisfaction must be higher Thereforein designing the preference satisfaction the past schedulesare reviewed first (to count the number of times in whicha preferred shift of an individual is satisfied in the pastfew schedule periods) and the weights of preferred shiftsand days-off of each nursing staff member are calculatedbefore the current schedule is formulated Note that the staffmember with a larger preference weight must be scheduledwith a higher priority order Then based on these twoweighted values the preference satisfaction of work shifts anddays-off of the current schedule can be calculated

33 Mathematical Model In this subsection a binary integerlinear programming model is constructed which aims tomaximize the overall preference satisfaction of the nursingstaff towards the shift schedule by taking into considerationthe preference ranks of the nursing staff for different workshifts and days-off despite the constraints of manpowershifts and days-off

331 Symbols

Subscript119894 Index of a nursing staff member119895 Index of a shift type119896 Index of date

Parameters

119868 Set of the nursing staff (ie 119894 isin 119868)119869 Set of shift types (ie 119895 isin 119869 note that J = 1 (dayshift) 2 (evening shift) 3 (night shift) in this paper)119870 Set of days-off (ie 119896 isin 119870 note that |119870| = 14 in thispaper)119872 Set of preference ranks of shifts M = 1 (good) 2(normal) 3 (bad)119873 Set of preference ranks of days-off N = 1 (good)3 (bad)119875 Number of the past schedule periods considered120572 Coefficient of the most preferred shift 120572 gt 1120573 Total number of days-off of each staff memberwithin the schedule period 120573 gt 1119877

119894 The variable to identify whether staff member i is

senior 119877119894= 0 (junior) 1 (senior)

119882

119878

119894 The preference weight of staff member i for work

shift119882

119867

119894 The preference weight of staff member i for day-

off119862 The base of preference weight (C = 2 in this paper)

119875

119878

119894119895 Preference satisfaction of staff member i in

working shift j119875

119867

119894119896 Preference satisfaction of staffmember i in taking

day k

119871

119878

119894119895 Preference rank of staffmember i for shift jwithin

the schedule period 119871119878119894119895isin 119872

119871

119867

119894119896 Preference rank of staff member i for taking day

k off within the schedule period 119871119867119894119896isin 119873

119879

119878

119894119898 In the recent P periods the number of times in

which staff member i has been assigned to the shift ofpreference rankm 0 le 119879119878

119894119898le 119875

119879

119867


which staff member i has been assigned to the day-offof preference rank n 0 le 119879119867

119894119899le (119875 sdot 120573)

120579

119878

119898 Preference score for being assigned to the shift of

preference rankm 120579119878119898=1

= 1 lt 120579

119878

119898=2= 2 lt 120579

119878

119898=3= 3

120579

119867

119899 Preference score for being assigned to the day-off

of preference rank n 120579119867119899=1= 1 lt 120579

119867

119899=2= 3

119863

119895119896 Manpower demand in shift j on day k

120591

min119895119896

Lower bound of the required number of seniorstaff members in shift j on day k119873

max119895119896

The maximum number of staff membersallowed to have day-off in shift j on day k119904

1015840

119894119895 Whether staff member i worked shift j in the

previous schedule periods 1199041015840119894119895isin 0 (no) 1 (yes)


ℎ

1015840

119894119895119896 Whether staff member i had day-off in shift j on

day k in the previous schedule periods ℎ1015840119894119895119896

isin 0 (no)1 (yes)

Decision Variable

119904

119894119895 Whether staff member i is scheduled for shift j

119904

119894119895isin 0 (off shift) 1 (on shift)

ℎ

119894119895119896 Whether staff member i is scheduled for day-off

in shift j on day k ℎ119894119895119896

isin 0 (off shift) 1 (on shift)

332 Mathematical Model The complete mathematicalmodel is constructed as follows

Objective

Max119866 =119868

sum

119894

119869

sum

119895

(119875

119878

119894119895sdot 119904

119894119895) + (

119870

sum

119896

(119875

119867

119894119896sdot ℎ

119894119895119896)) (1)

where

119875

119878

119894119895=

120572119882

119878

119894 if 119871119878

119894119895= 1

119882

119878

119894 if 119871119878

119894119895= 2

0 if 119871119878119894119895= 3

(2)

119875

119867

119894119896=

120572119882

119867

119894 if 119871119867

119894119896= 1

0 if 119871119867119894119896= 3

(3)

119882

119878

119894= 119862

sum119872

119898=1(119879119878

119894119898sdot120579119878

119898) (4)

119882

119867

119894= 119862

sum119873

119899=1((119879119867

119894119899120573)sdot120579119867

119899)

(5)

Constraints

sum

119895isin119869

119904

119894119895= 1 forall119894 isin 119868 (6)

sum

119894isin119868

119904

119894119895= 119863

119895119896 forall119895 isin 119869 119896 isin 119870 (7)

sum

119894isin119868

(119877

119894sdot (119904

119894119895minus ℎ

119894119895119896)) ge 120591

min119895119896 forall119895 isin 119869 119896 isin 119870 (8)

sum

119896isin119870

ℎ

119894119895119896= 119904

119894119895sdot 120573 forall119894 isin 119868 119895 isin 119869 (9)

119873

max119895119896

minus

119868

sum

119894

ℎ

119894119895119896ge 0 forall119895 isin 119869 119896 isin 119870 (10)

119869

sum

119895

(

119896=7

sum

119896=1

ℎ

119894119895119896sdot

119896=14

sum

119896=8

ℎ

119894119895119896) gt 0 forall119894 isin 119868 (11)

119904

1015840

1198943sdot 119904

119894119895le ℎ

1015840

119894314sdot ℎ

119894119895119896 forall119894 isin 119868 (12)

119904

119894119895isin 0 1 forall119894 isin 119868 119895 isin 119869 (13)

ℎ

119894119895119896isin 0 1 forall119894 isin 119868 119895 isin 119869 119896 isin 119870 (14)

Table 1 Shift preference weights of the nursing staff

Staff The assigned shift number in the past Preference weightGood Normal Bad

1 2 0 0 42 2 0 0 43 2 0 0 44 2 0 0 45 1 0 1 166 1 0 1 167 2 0 0 48 2 0 0 49 2 0 0 410 2 0 0 411 2 0 0 412 1 1 0 813 2 0 0 414 2 0 0 415 2 0 0 416 1 1 0 817 1 0 1 1618 1 0 1 1619 1 0 1 1620 2 0 0 4

First the objective function of the model is explained asfollows Equation (1) maximizes the overall preference satis-faction of the nursing staff towards work shifts (sum119868

119894sum

119869

119895(119875

119878

119894119895sdot

119904

119894119895)) and days-off (sum119868

119894sum

119869

119895sum

119870

119896(119875

119867

119894119896sdot ℎ

119894119895119896)) As for the overall

preference satisfaction towards shifts (resp days-off) thepreference satisfaction of each shift preference rank of eachnursing staff is calculated by (2) (resp (3)) in which thepreference weight used in the above preference satisfactioncalculation for shift (resp day-off) is obtained from (4)(resp (5))

We continue to look at the constraints of the model Nineconstraints (from Constraint (6) to (14)) are identified basedon the shift scheduling constraints in which Constraints (6)to (8) are shift constraints Constraints (9) to (12) are day-offconstraints and Constraints (13) to (14) are binary variableconstraints Constraint (6) enforces each staff member tobe assigned to at most one work shift within each scheduleperiod According to Constraints (7) and (8) respectivelymanpower demand and number of senior staff members ofeach shift each day should bemet Constraint (9) enforces thefact that the total number of days-off assigned to each staffmember in a work shift should be the same within a certainschedule period Constraint (10) enforces the maximum totalnumber of nursing staff members allowed to have day-offon each shift each day As the number of patients usuallyfluctuates the schedule planner may adjust the total numberof nursing staff members allowed to be on day-off accordingto the actual number of patients on that particular dayConstraint (11) enforces the fact that each staff member has


Table 2 Shift preference rank and preference satisfaction of the nursing staff

StaffShift rotation

Day shift Evening shift Night shiftPreference rank Preference satisfaction Preference rank Preference satisfaction Preference rank Preference satisfaction

1 1 8 2 4 3 02 2 4 1 8 3 03 1 8 2 4 3 04 1 8 2 4 3 05 1 32 2 16 3 06 1 32 2 16 3 07 2 4 1 8 3 08 2 4 1 8 3 09 1 8 2 4 3 010 1 8 2 4 3 011 1 8 2 4 3 012 1 16 2 8 3 013 1 8 2 4 3 014 1 8 2 4 3 015 1 8 2 4 3 016 1 16 2 8 3 017 2 16 1 32 3 018 1 32 2 16 3 019 1 32 2 16 3 020 2 4 3 0 1 8

to be given at least one day-off each week and the intervalbetween two different shifts must be more than 8 hoursConstraint (12) ensures that each staff member has at least an8-hour rest before continuing on to the next shift Accordingto Constraints (13) and (14) the decision variables of workshifts and days-off should be either zero or one

4 Numeric Analysis

The performance and feasibility of the model are evaluatedon the obstetrics and gynecology department of a hospitalThe parameters of the department are stated as follows Thepaper involves a total of 20 nursing staff members in whichthose from1 to 15 are senior while those from16 to 20 arejunior Each staff member is allowed to have a total of 4 days-off over the overall schedule planning period (14 days) A totalof 8 staff members are required for day shift 7 for eveningshift and 5 for night shift During each shift every day thenumber of staff members allowed to have day-off is 3 personsfor the day shift 3 persons for the evening shift and 2 personsfor the night shift and at least one senior staffmust be on dutyduring each shift The schedules of the past two periods arecollected as historical data in which nursing staff members5 6 17 19 and 20 were on night shifts in the previous week

while staff members 6 7 8 9 14 15 and 17 were given day-offon the last day of the previous week

First the shift preference weights and the preferencesatisfaction of three shift preference ranks of each staffmember are calculated and listed in Tables 1 and 2 Table 1shows the number of times in which each shift preferencerank is satisfied in the previous schedule periods and thepreference weight at the current schedule period In Table 1since staff members 5 6 17 18 and 19 were assigned to ldquobadrdquoshifts in the past two schedule periods their shift preferenceweights are set to be larger in the current schedule planningTomaintain fairness of shift rotation those staffmembers aretherefore given higher chances of being assigned to ldquogoodrdquoshifts in the current schedule

Table 2 shows each nursing staff memberrsquos satisfactionfor shift preference ranks A smaller preference rank numberindicates a more preferred shift that is preference rank 1yields the highest degree of preference satisfaction whilepreference rank 3 yields the lowest degree of preferencesatisfaction In general in solving a mathematical model themodel would attempt to satisfy all the shifts with preferencerank 1 in order to enhance the satisfaction of the nursing stafftoward the shift schedule However in the current scheduleplanning period 15 staff members consider day shifts asldquogoodrdquo 4 staff members consider evening shifts as ldquogoodrdquo


and 1 staff member considers night shifts as ldquogoodrdquo Since thetotal number of the nursing staff members who prefer dayshifts exceeds the total manpower demand of 8 persons forday shift the good shift preference ranks of 7 nursing staffmembers cannot be satisfied

Next each staff memberrsquos day-off preference weights andpreference satisfaction for day-off preference ranks are calcu-lated and listed in Tables 3 and 4 Table 3 shows the numberof times in which each day-off preference rank was satisfiedin the past schedule periods and the current weightedpreference It is found from Table 3 that staff member 18 wasgiven the largest number of ldquobadrdquo days-off within the pasttwo schedule periods and hence herhis day-off preferenceweight is set to be larger than the others in the currentschedule To maintain fairness of the schedule the staffmember is therefore prioritized that is given a higher chanceof being assigned ldquogoodrdquo days-off in the current scheduleTable 4 shows each staff memberrsquos preference satisfaction ofthe ldquogoodrdquo day-off preference rank Since each staff memberis entitled to 4 days-off each has 4 days-off with a ldquogoodrdquo day-off preference rank and the same preference satisfaction level

Lastly the numerical example is solved under IBM ILOGCPLEX Studio 124 software on a PC with an Intel i7-3770CPU 390GHz and 16GB RAM which operates in Windows7 (64x) The optimal solution is obtained very frequentlyThe average computing time of generating a solution is about3755 secondsThe generated schedule is listed in Tables 5 and6 which meet the constraints on numbers of senior nursingstaff members shifts and days-off

Note that since both the parameters 119863119895119896

for manpowerdemand in shift j on day k and 119873max

119895119896for the maximum

number of staff members allowed to have day-off in shift jon day k can be adjusted according to the realistic situationsthe generated schedule can be used in the realistic instanceswith different sizes For the schedule with a longer scheduleperiod (with multiple two-week periods) it can be handledby splitting the whole schedule period to multiple two-weekperiods and then using the generated schedule of each two-week period as the history for the next one

Table 5 shows the resultant schedule for shifts in whichthe assigned shifts for 20 nursing staff members are showneach staff member is associated with two entries of 0 (nonas-signed shifts) and one entry of 1 (the assigned shift) theentries with star marks are the shifts that are not preferredFrom Table 5 each staff member is assigned to a fixed shifttype for all working days in the scheduling period of 14 daysand the manpower requirement of each shift is satisfied

From Table 5 we also observe that the schedule satisfiesthe demand of each work shift in which eight staff membersare scheduled to work on the day shift (staff members 3 4 611 12 15 16 and 18) 7 on the evening shift (staffmembers 1 25 7 8 17 and 19) and 5 on the night shift (staffmembers 9 1013 14 and 20) It is also observed from Table 6 that each staffmember is also given sufficient rest time between two shiftsFor example since staff member 5 was scheduled to worknight shift in the previous schedule period and was given

Table 3 Day-off preference weights of the nursing staff

Staff The assigned number in the past Preference weightGood Bad

1 7 1 56569 (asymp57)

2 8 0 4

3 5 3 113137 (asymp11)

4 8 0 4

5 5 3 113137 (asymp11)

6 8 0 4

7 8 0 4

8 8 0 4

9 7 1 56569 (asymp57)

10 8 0 4

11 8 0 4

12 7 1 56569 (asymp57)

13 6 2 8

14 7 1 56569 (asymp57)

15 7 1 56569 (asymp57)

16 7 1 56569 (asymp57)

17 8 0 4

18 3 5 226274 (asymp23)

19 8 0 4

20 8 0 4

1 day-off on the last day heshe is therefore not scheduledfor ldquogoodrdquo (day) shift even though heshe has a large shiftpreference weight In addition as for shift preference 3 staffmembers are scheduled for the ldquonormalrdquo shift (staff members1 5 and 19) 4 for the ldquobadrdquo shifts (staff members 9 10 13and 14) and the rest for the ldquogoodrdquo shift It can be found thatthose scheduled for ldquobadrdquo shifts have a small shift preferenceweight indicating that the schedule is formulated fairly

Table 6 shows the resultant schedule for days-off inwhichthe four days-off of each staff member are marked with 1 theentries with star marks are the days-off that are not preferredFrom Table 6 the day-off constraints of all the nursing staffare satisfied and the total number of nonpreferred days-offis small In addition note that the total number of days-offof each staff member within the schedule period is the sameand each staff member is given at least 1 day-off each weekTherefore the number of nursing staff members allowed totake a day-off and the minimum number of senior nursingstaff members on duty each shift each day are fulfilled Asfor day-off preference only a few staff members are not givenldquogoodrdquo days-off (staff members 4 6 11 and 20) and all thesehave low day-off preference weights


Table 4 Day-off preference and preference satisfaction of the nursing staff

Staff Schedule period1 2 3 4 5 6 7 8 9 10 11 12 13 14

1 57 57 57 572 4 4 4 43 11 11 11 114 4 4 4 45 11 11 11 116 4 4 4 47 4 4 4 48 4 4 4 49 57 57 57 5710 4 4 4 411 4 4 4 412 57 57 57 5713 8 8 8 814 57 57 57 5715 57 57 57 5716 57 57 57 5717 4 4 4 418 23 23 23 2319 4 4 4 420 4 4 4 4

Table 5 The resultant schedule for shifts

Staff Number of shiftsDay shift Evening shift Night shift

1 0 1 (normal) 02 0 1 03 1 0 04 1 0 05 0 1 (normal) 06 1 0 07 0 1 08 0 1 09 0 0 1 (bad)10 0 0 1 (bad)11 1 0 012 1 0 013 0 0 1 (bad)14 0 0 1 (bad)15 1 0 016 1 0 017 0 1 018 1 0 019 0 1 (normal) 020 0 0 1

5 Conclusion

Nurse-scheduling is a difficult and time-consuming task forschedule planners Since there are a lot of hard constraintsimposed by the government and the hospital and multiplepreference ranks of the nursing staff for work shifts and days-off are considered it is rather impossible to obtain a perfectnurse shift schedule manuallyTherefore this paper performsa mathematical programming approach to establish a modelto effectively make a nurse shift schedule The results of theschedule show that all hard constraints are filled completelyand the preferred shift and days-off of the nursing staff areassigned fairly

In the future we intend to extend our work along thefollowing two lines First for the objective function of themathematical model proposed in this paper the preferencesatisfactions of the nursing staff for work shifts and days-off of the schedule could be integrated Since the numberof days-off of the schedule is greater than the number ofshifts the proposed model is scheduled in two stages inthis paper (first days-off and then work shifts) Therefore itwould be of interest to integrate them to be solved in a singlestage to enlarge possible solutions to increase the preferencesatisfaction of the nursing staff


Table 6 The resultant schedule for days-off


1 1 1 1 12 1 1 1 13 1 1 1 14 1 1 1lowast 15 1 1 1 16 1lowast 1 1 17 1 1 18 1 1 1 19 1 1 1 110 1 1 1 111 1lowast 1 1 112 1 1 1 113 1 1 1 114 1 1 1 115 1 1 1 116 1 1 1 117 1 1 1 118 1 1 1 119 1 1 1 120 1 1lowast 1 1lowastEach entry with a star mark in this schedule indicates the day-off that is not preferred by the nursing staff member

Second various work shift patterns for each nursing staffmember could be taken into account In spite ofmost nursingstaff members like a fixed shift type for all working days ofthe scheduling period some staff members might want moreflexible work patterns that is various work shift patterns indifferent working days of the scheduling period Therefore itwould be of interest to deal with the flexible work patterns

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank the anonymous referees for the commentsthat improved the content as well as the presentation of thispaperThiswork has been supported in part byNSC 101-2628-E-009-025-MY3 Taiwan

References

[1] J J Bartholdi III ldquoA guaranteed-accuracy round-off algorithmfor cyclic scheduling and set coveringrdquoOperations Research vol29 no 3 pp 501ndash510 1981

[2] M Hadwan M Ayob N R Sabar and R Qu ldquoA harmonysearch algorithm for nurse rostering problemsrdquo InformationSciences vol 233 no 6 pp 126ndash140 2013

[3] G Felici and C Gentile ldquoA polyhedral approach for the staffrostering problemrdquoManagement Science vol 50 no 3 pp 381ndash393 2004

[4] A T Ernst H Jiang M Krishnamoorthy and D Sier ldquoStaffscheduling and rostering a review of applications methods andmodelsrdquo European Journal of Operational Research vol 153 no1 pp 3ndash27 2004

[5] J V Bergh J Belien P D Bruecker E Demeulemeester and LD Boeck ldquoPersonnel scheduling a literature reviewrdquo EuropeanJournal of Operational Research vol 226 no 3 pp 367ndash3852013

[6] C Valouxis and E Housos ldquoHybrid optimization techniquesfor the workshift and rest assignment of nursing personnelrdquoArtificial Intelligence in Medicine vol 20 no 2 pp 155ndash1752000

[7] HW Purnomo and J F Bard ldquoCyclic preference scheduling fornurses using branch and pricerdquoNaval Research Logistics vol 54no 2 pp 200ndash220 2007

[8] P A Clark K Leddy M Drain and D Kaldenberg ldquoStatenursing shortages and patient satisfaction more RNmdashbetterpatient experiencesrdquo Journal of Nursing Care Quality vol 22no 2 pp 119ndash127 2007

[9] R MrsquoHallah and A Alkhabbaz ldquoScheduling of nurses a casestudy of a Kuwaiti health care unitrdquo Operations Research ForHealth Care vol 2 no 1-2 pp 1ndash19 2013

[10] J F Bard andHW Purnomo ldquoPreference scheduling for nursesusing column generationrdquo European Journal of OperationalResearch vol 164 no 2 pp 510ndash534 2005

[11] U Aickelin and K A Dowsland ldquoAn indirect genetic algorithmfor a nurse-scheduling problemrdquo Computers and OperationsResearch vol 31 no 5 pp 761ndash778 2004


[12] B Maenhout and M Vanhoucke ldquoAn integrated nurse staffingand scheduling analysis for loner-term nursing staff allocationproblemsrdquo Omega vol 41 no 2 pp 485ndash499 2013

[13] S Topaloglu and H Selim ldquoNurse scheduling using fuzzymodeling approachrdquo Fuzzy Sets and Systems vol 161 no 11 pp1543ndash1563 2010

[14] MNAzaiez and S S Al Sharif ldquoA 0-1 goal programmingmodelfor nurse schedulingrdquo Computers and Operations Research vol32 no 3 pp 491ndash507 2005

[15] S Topaloglu ldquoA multi-objective programming model forscheduling emergency medicine residentsrdquo Computers andIndustrial Engineering vol 51 no 3 pp 375ndash388 2006

[16] S Topaloglu ldquoA shift scheduling model for employees withdifferent seniority levels and an application in healthcarerdquoEuropean Journal of Operational Research vol 198 no 3 pp943ndash957 2009

[17] J Belien andEDemeulemeester ldquoA branch-and-price approachfor integrating nurse and surgery schedulingrdquo European Journalof Operational Research vol 189 no 3 pp 652ndash668 2008

[18] C A Glass and R A Knight ldquoThe nurse rostering problem acritical appraisal of the problem structurerdquo European Journal ofOperational Research vol 202 no 2 pp 379ndash389 2010

[19] C Valouxis C Gogos G Goulas P Alefragis and EHousos ldquoAsystematic two phase approach for the nurse rostering problemrdquoEuropean Journal of Operational Research vol 219 no 2 pp425ndash433 2012

[20] C C Tsai and S H A Li ldquoA two-stage modeling with geneticalgorithms for the nurse scheduling problemrdquo Expert Systemswith Applications vol 36 no 5 pp 9506ndash9512 2009

[21] S J Sadjadi R SoltaniM Izadkhah F Saberian andMDarayildquoA new nonlinear stochastic staff scheduling modelrdquo ScientiaIranica vol 18 no 3 pp 699ndash710 2011

[22] W J Gutjahr and M S Rauner ldquoAn ACO algorithm fora dynamic regional nurse-scheduling problem in AustriardquoComputers and Operations Research vol 34 no 3 pp 642ndash6662007

[23] Z Lu and J K Hao ldquoAdaptive neighborhood search for nurserosteringrdquo European Journal of Operational Research vol 218no 3 pp 865ndash876 2012

[24] M V Chiaramonte and L M Chiaramonte ldquoAn agent-basednurse rostering system under minimal staffing conditionsrdquoInternational Journal of Production Economics vol 114 no 2 pp697ndash713 2008

[25] M Vanhoucke and B Maenhout ldquoOn the characterization andgeneration of nurse scheduling problem instancesrdquo EuropeanJournal of Operational Research vol 196 no 2 pp 457ndash4672009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


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Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


with the objective of minimizing the penalty cost for notfulfilling the preferences of the nursing staff Maenhout andVanhoucke [12] investigated the penalty costs with multipleconstraints (including the nursing staff rsquos preferences andsome specific combinations of work shifts and days-off)Topaloglu and Selim [13] considered a variety of uncertainfactors in nurse shift scheduling to propose a fuzzy mul-tiobjective integer programming model which takes intoconsideration the fuzziness of the objective and the nursingstaff rsquos preferences

From the literature we discover that in the penalty cost(or satisfaction) due to the nursing staff rsquos preferences in theobjective function all the previous works only focused onthe most preferred work shift and the number of satisfactorydays-off of the shift schedule at the current schedule periodand further investigated neither different preference ranks(such as three ranks good normal or bad) of the nursingstaff toward different work shifts or planned days-off nor thenumber of times in which their preferences are satisfied in theprevious shift schedules and days-off Since the constraintsof the schedule lead to the fact that the most preferred workshifts and days-off of each staff member cannot be fulfilledcompletely we believe that if the penalty cost (or satisfactionlevel) due to preferences of the nursing staff does not considerdifferent preference ranks and the historical data of previousschedules the long-term fairness and justice of the nurseshift schedules is affected For example if each nursing staffmember has three different preference ranks (good normalor bad) towards a 3-shift rotation those scheduled to a badwork shift would have a larger preference penalty cost (orlower preference satisfaction level) as compared with thosescheduled to work a normal shift and should be assigneda higher priority to their higher preference rank in theirschedule at the next scheduling period

In light of the above this paper proposes a binary integerlinear programming model for the nurse shift schedule withdifferent levels of satisfaction preference and a differentpriority ordering of the nursing staff for planning their shiftschedule The objective of this model is to maximize theoverall satisfaction level of the nursing staff towards theirwork shifts and days-off schedule In addition some commonnurse-scheduling constraints are considered including thelimited number of nursing staff members work shift lim-itation and day-off limitation Our mathematical model iscapable of effectively helping schedule planners to design anurse shift schedule that attempts to within all schedulingconstraints satisfy the shift and day-off preferences of mostof the nursing staff members in a fair manner and to achievethe highest overall satisfaction level

Themain contributions of this paper are stated as followsIn the proposed mathematical model for nurse schedule thework shift and day-off preferences of the nursing staff arecategorized into different levels and are then integrated to besolved In addition in the case where the preferred number ofshifts or days-off exceeds the actual shifts or days-off available(due to schedule constraints) a priority ordering mechanismof the nursing staff when planning their shift schedule isapplied to solve the contradictory situations among theirpreferences

The rest of this paper is organized as follows Section 2gives the literature review of our work Section 3 firstdescribes our concerned problem and then constructsits mathematical programming model Section 4 gives anumeric example to analyze the performance of the shiftschedule constructed by our mathematical model Finally aconclusion is made in Section 5

2 Literature Review

Thenurse-scheduling problems have been solved by a varietyof methods which are mainly introduced by mathematicalprogramming heuristics and others in this section

First the mathematical programming approaches for thenurse-scheduling problem are introduced Maenhout andVanhoucke [12] proposed an integrated analysis method tosolve human resource planning and shift scheduling problemof nurses on the long run Azaiez and Al Sharif [14] solvedthe nurse-scheduling problemwith a 0-1 linear programmingmodel which takes into consideration the ratio of nursesworking night shifts or having days-off on weekends andtries to avoid unnecessary overtime so that hospital costscan be reduced Topaloglu [15] proposed a multiobjectiveprogrammingmodel to tackle the nurse-scheduling problemsof house physicians in the emergency room Based on theAHP method soft constraints in the model are weighted inaccordance with their relative importance and this becomesthe basis for weighting objective functions Topaloglu [16]proposed a multiobjective scheduling model for planningthe shifts of resident physicians in which seniority of aresident physician is used for the weight setting Empiricalanalysis showed that the model is far superior to the manualscheduling method in terms of efficiency and time-saving

Belien and Demeulemeester [17] proposed an integratedscheduling method for nurses and surgeons They appliedthe branch-and-price method to perform integration whichis one of the most common methods to generate explicitsolutions among the column generation techniques GlassandKnight [18] identified four categories of nurse-schedulingproblems and solved them with mixed-integer linear pro-gramming Results showed that the optimal solutions wereproduced in all benchmarking examples within 30 minutesThe characteristic of this model is that it reduces the collec-tion space based on the structure of the problem so that theproblem-solving efficiency can be enhanced Valouxis et al[19] proposed a 2-stage solution to solve the nurse-schedulingproblem where the workload and days-off of each nurse arefirst determined before the shifts are planned

Second the heuristics for nurse-scheduling problems areintroduced Hadwan et al [2] proposed a harmony searchalgorithm for the nurse-scheduling problem which wastested in hospitals in Malaysia and was shown to be betterthan the genetic algorithm approach as well as most heuristicapproaches Aickelin andDowsland [11] proposed an indirectgenetic algorithm for the nurse-scheduling problem inwhicha chromosome encoding is performed recombination isconducted with a heuristic approach evolution is conductedvia mixed-crossover to locate better solutions The method






3 Methodology














331 Symbols


Parameters




119882

119878


shift119882

119867



119875

119878



119867


day k

119871

119878



119871

119867



119879

119878



119894119898le 119875

119879

119867



119894119899le (119875 sdot 120573)

120579

119878



= 1 lt 120579

119878

119898=2= 2 lt 120579

119878

119898=3= 3

120579

119867



119867

119899=2= 3

119863


120591

min119895119896


max119895119896


1015840




ℎ

1015840



isin 0 (no)1 (yes)

Decision Variable

119904


119904


ℎ





Objective

Max119866 =119868

sum

119894

119869

sum

119895

(119875

119878

119894119895sdot 119904

119894119895) + (

119870

sum

119896

(119875

119867

119894119896sdot ℎ

119894119895119896)) (1)

where

119875

119878

119894119895=

120572119882

119878

119894 if 119871119878

119894119895= 1

119882

119878

119894 if 119871119878

119894119895= 2

0 if 119871119878119894119895= 3

(2)

119875

119867

119894119896=

120572119882

119867

119894 if 119871119867

119894119896= 1

0 if 119871119867119894119896= 3

(3)

119882

119878

119894= 119862

sum119872

119898=1(119879119878

119894119898sdot120579119878

119898) (4)

119882

119867

119894= 119862

sum119873

119899=1((119879119867

119894119899120573)sdot120579119867

119899)

(5)

Constraints

sum

119895isin119869

119904

119894119895= 1 forall119894 isin 119868 (6)

sum

119894isin119868

119904

119894119895= 119863

119895119896 forall119895 isin 119869 119896 isin 119870 (7)

sum

119894isin119868

(119877

119894sdot (119904

119894119895minus ℎ

119894119895119896)) ge 120591


sum

119896isin119870

ℎ

119894119895119896= 119904


119873

max119895119896

minus

119868

sum

119894

ℎ


119869

sum

119895

(

119896=7

sum

119896=1

ℎ

119894119895119896sdot

119896=14

sum

119896=8

ℎ

119894119895119896) gt 0 forall119894 isin 119868 (11)

119904

1015840

1198943sdot 119904

119894119895le ℎ

1015840

119894314sdot ℎ

119894119895119896 forall119894 isin 119868 (12)

119904


ℎ




1 2 0 0 42 2 0 0 43 2 0 0 44 2 0 0 45 1 0 1 166 1 0 1 167 2 0 0 48 2 0 0 49 2 0 0 410 2 0 0 411 2 0 0 412 1 1 0 813 2 0 0 414 2 0 0 415 2 0 0 416 1 1 0 817 1 0 1 1618 1 0 1 1619 1 0 1 1620 2 0 0 4


119894sum

119869

119895(119875

119878

119894119895sdot

119904

119894119895)) and days-off (sum119868

119894sum

119869

119895sum

119870

119896(119875

119867

119894119896sdot ℎ






StaffShift rotation


1 1 8 2 4 3 02 2 4 1 8 3 03 1 8 2 4 3 04 1 8 2 4 3 05 1 32 2 16 3 06 1 32 2 16 3 07 2 4 1 8 3 08 2 4 1 8 3 09 1 8 2 4 3 010 1 8 2 4 3 011 1 8 2 4 3 012 1 16 2 8 3 013 1 8 2 4 3 014 1 8 2 4 3 015 1 8 2 4 3 016 1 16 2 8 3 017 2 16 1 32 3 018 1 32 2 16 3 019 1 32 2 16 3 020 2 4 3 0 1 8


4 Numeric Analysis

















1 7 1 56569 (asymp57)

2 8 0 4

3 5 3 113137 (asymp11)

4 8 0 4

5 5 3 113137 (asymp11)

6 8 0 4

7 8 0 4

8 8 0 4

9 7 1 56569 (asymp57)

10 8 0 4

11 8 0 4

12 7 1 56569 (asymp57)

13 6 2 8

14 7 1 56569 (asymp57)

15 7 1 56569 (asymp57)

16 7 1 56569 (asymp57)

17 8 0 4

18 3 5 226274 (asymp23)

19 8 0 4

20 8 0 4






1 57 57 57 572 4 4 4 43 11 11 11 114 4 4 4 45 11 11 11 116 4 4 4 47 4 4 4 48 4 4 4 49 57 57 57 5710 4 4 4 411 4 4 4 412 57 57 57 5713 8 8 8 814 57 57 57 5715 57 57 57 5716 57 57 57 5717 4 4 4 418 23 23 23 2319 4 4 4 420 4 4 4 4




5 Conclusion










Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














3 Methodology














331 Symbols


Parameters




119882

119878


shift119882

119867



119875

119878



119867


day k

119871

119878



119871

119867



119879

119878



119894119898le 119875

119879

119867



119894119899le (119875 sdot 120573)

120579

119878



= 1 lt 120579

119878

119898=2= 2 lt 120579

119878

119898=3= 3

120579

119867



119867

119899=2= 3

119863


120591

min119895119896


max119895119896


1015840




ℎ

1015840



isin 0 (no)1 (yes)

Decision Variable

119904


119904


ℎ





Objective

Max119866 =119868

sum

119894

119869

sum

119895

(119875

119878

119894119895sdot 119904

119894119895) + (

119870

sum

119896

(119875

119867

119894119896sdot ℎ

119894119895119896)) (1)

where

119875

119878

119894119895=

120572119882

119878

119894 if 119871119878

119894119895= 1

119882

119878

119894 if 119871119878

119894119895= 2

0 if 119871119878119894119895= 3

(2)

119875

119867

119894119896=

120572119882

119867

119894 if 119871119867

119894119896= 1

0 if 119871119867119894119896= 3

(3)

119882

119878

119894= 119862

sum119872

119898=1(119879119878

119894119898sdot120579119878

119898) (4)

119882

119867

119894= 119862

sum119873

119899=1((119879119867

119894119899120573)sdot120579119867

119899)

(5)

Constraints

sum

119895isin119869

119904

119894119895= 1 forall119894 isin 119868 (6)

sum

119894isin119868

119904

119894119895= 119863

119895119896 forall119895 isin 119869 119896 isin 119870 (7)

sum

119894isin119868

(119877

119894sdot (119904

119894119895minus ℎ

119894119895119896)) ge 120591


sum

119896isin119870

ℎ

119894119895119896= 119904


119873

max119895119896

minus

119868

sum

119894

ℎ


119869

sum

119895

(

119896=7

sum

119896=1

ℎ

119894119895119896sdot

119896=14

sum

119896=8

ℎ

119894119895119896) gt 0 forall119894 isin 119868 (11)

119904

1015840

1198943sdot 119904

119894119895le ℎ

1015840

119894314sdot ℎ

119894119895119896 forall119894 isin 119868 (12)

119904


ℎ




1 2 0 0 42 2 0 0 43 2 0 0 44 2 0 0 45 1 0 1 166 1 0 1 167 2 0 0 48 2 0 0 49 2 0 0 410 2 0 0 411 2 0 0 412 1 1 0 813 2 0 0 414 2 0 0 415 2 0 0 416 1 1 0 817 1 0 1 1618 1 0 1 1619 1 0 1 1620 2 0 0 4


119894sum

119869

119895(119875

119878

119894119895sdot

119904

119894119895)) and days-off (sum119868

119894sum

119869

119895sum

119870

119896(119875

119867

119894119896sdot ℎ






StaffShift rotation


1 1 8 2 4 3 02 2 4 1 8 3 03 1 8 2 4 3 04 1 8 2 4 3 05 1 32 2 16 3 06 1 32 2 16 3 07 2 4 1 8 3 08 2 4 1 8 3 09 1 8 2 4 3 010 1 8 2 4 3 011 1 8 2 4 3 012 1 16 2 8 3 013 1 8 2 4 3 014 1 8 2 4 3 015 1 8 2 4 3 016 1 16 2 8 3 017 2 16 1 32 3 018 1 32 2 16 3 019 1 32 2 16 3 020 2 4 3 0 1 8


4 Numeric Analysis

















1 7 1 56569 (asymp57)

2 8 0 4

3 5 3 113137 (asymp11)

4 8 0 4

5 5 3 113137 (asymp11)

6 8 0 4

7 8 0 4

8 8 0 4

9 7 1 56569 (asymp57)

10 8 0 4

11 8 0 4

12 7 1 56569 (asymp57)

13 6 2 8

14 7 1 56569 (asymp57)

15 7 1 56569 (asymp57)

16 7 1 56569 (asymp57)

17 8 0 4

18 3 5 226274 (asymp23)

19 8 0 4

20 8 0 4






1 57 57 57 572 4 4 4 43 11 11 11 114 4 4 4 45 11 11 11 116 4 4 4 47 4 4 4 48 4 4 4 49 57 57 57 5710 4 4 4 411 4 4 4 412 57 57 57 5713 8 8 8 814 57 57 57 5715 57 57 57 5716 57 57 57 5717 4 4 4 418 23 23 23 2319 4 4 4 420 4 4 4 4




5 Conclusion










Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















331 Symbols


Parameters




119882

119878


shift119882

119867



119875

119878



119867


day k

119871

119878



119871

119867



119879

119878



119894119898le 119875

119879

119867



119894119899le (119875 sdot 120573)

120579

119878



= 1 lt 120579

119878

119898=2= 2 lt 120579

119878

119898=3= 3

120579

119867



119867

119899=2= 3

119863


120591

min119895119896


max119895119896


1015840




ℎ

1015840



isin 0 (no)1 (yes)

Decision Variable

119904


119904


ℎ





Objective

Max119866 =119868

sum

119894

119869

sum

119895

(119875

119878

119894119895sdot 119904

119894119895) + (

119870

sum

119896

(119875

119867

119894119896sdot ℎ

119894119895119896)) (1)

where

119875

119878

119894119895=

120572119882

119878

119894 if 119871119878

119894119895= 1

119882

119878

119894 if 119871119878

119894119895= 2

0 if 119871119878119894119895= 3

(2)

119875

119867

119894119896=

120572119882

119867

119894 if 119871119867

119894119896= 1

0 if 119871119867119894119896= 3

(3)

119882

119878

119894= 119862

sum119872

119898=1(119879119878

119894119898sdot120579119878

119898) (4)

119882

119867

119894= 119862

sum119873

119899=1((119879119867

119894119899120573)sdot120579119867

119899)

(5)

Constraints

sum

119895isin119869

119904

119894119895= 1 forall119894 isin 119868 (6)

sum

119894isin119868

119904

119894119895= 119863

119895119896 forall119895 isin 119869 119896 isin 119870 (7)

sum

119894isin119868

(119877

119894sdot (119904

119894119895minus ℎ

119894119895119896)) ge 120591


sum

119896isin119870

ℎ

119894119895119896= 119904


119873

max119895119896

minus

119868

sum

119894

ℎ


119869

sum

119895

(

119896=7

sum

119896=1

ℎ

119894119895119896sdot

119896=14

sum

119896=8

ℎ

119894119895119896) gt 0 forall119894 isin 119868 (11)

119904

1015840

1198943sdot 119904

119894119895le ℎ

1015840

119894314sdot ℎ

119894119895119896 forall119894 isin 119868 (12)

119904


ℎ




1 2 0 0 42 2 0 0 43 2 0 0 44 2 0 0 45 1 0 1 166 1 0 1 167 2 0 0 48 2 0 0 49 2 0 0 410 2 0 0 411 2 0 0 412 1 1 0 813 2 0 0 414 2 0 0 415 2 0 0 416 1 1 0 817 1 0 1 1618 1 0 1 1619 1 0 1 1620 2 0 0 4


119894sum

119869

119895(119875

119878

119894119895sdot

119904

119894119895)) and days-off (sum119868

119894sum

119869

119895sum

119870

119896(119875

119867

119894119896sdot ℎ






StaffShift rotation


1 1 8 2 4 3 02 2 4 1 8 3 03 1 8 2 4 3 04 1 8 2 4 3 05 1 32 2 16 3 06 1 32 2 16 3 07 2 4 1 8 3 08 2 4 1 8 3 09 1 8 2 4 3 010 1 8 2 4 3 011 1 8 2 4 3 012 1 16 2 8 3 013 1 8 2 4 3 014 1 8 2 4 3 015 1 8 2 4 3 016 1 16 2 8 3 017 2 16 1 32 3 018 1 32 2 16 3 019 1 32 2 16 3 020 2 4 3 0 1 8


4 Numeric Analysis

















1 7 1 56569 (asymp57)

2 8 0 4

3 5 3 113137 (asymp11)

4 8 0 4

5 5 3 113137 (asymp11)

6 8 0 4

7 8 0 4

8 8 0 4

9 7 1 56569 (asymp57)

10 8 0 4

11 8 0 4

12 7 1 56569 (asymp57)

13 6 2 8

14 7 1 56569 (asymp57)

15 7 1 56569 (asymp57)

16 7 1 56569 (asymp57)

17 8 0 4

18 3 5 226274 (asymp23)

19 8 0 4

20 8 0 4






1 57 57 57 572 4 4 4 43 11 11 11 114 4 4 4 45 11 11 11 116 4 4 4 47 4 4 4 48 4 4 4 49 57 57 57 5710 4 4 4 411 4 4 4 412 57 57 57 5713 8 8 8 814 57 57 57 5715 57 57 57 5716 57 57 57 5717 4 4 4 418 23 23 23 2319 4 4 4 420 4 4 4 4




5 Conclusion










Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










ℎ

1015840



isin 0 (no)1 (yes)

Decision Variable

119904


119904


ℎ





Objective

Max119866 =119868

sum

119894

119869

sum

119895

(119875

119878

119894119895sdot 119904

119894119895) + (

119870

sum

119896

(119875

119867

119894119896sdot ℎ

119894119895119896)) (1)

where

119875

119878

119894119895=

120572119882

119878

119894 if 119871119878

119894119895= 1

119882

119878

119894 if 119871119878

119894119895= 2

0 if 119871119878119894119895= 3

(2)

119875

119867

119894119896=

120572119882

119867

119894 if 119871119867

119894119896= 1

0 if 119871119867119894119896= 3

(3)

119882

119878

119894= 119862

sum119872

119898=1(119879119878

119894119898sdot120579119878

119898) (4)

119882

119867

119894= 119862

sum119873

119899=1((119879119867

119894119899120573)sdot120579119867

119899)

(5)

Constraints

sum

119895isin119869

119904

119894119895= 1 forall119894 isin 119868 (6)

sum

119894isin119868

119904

119894119895= 119863

119895119896 forall119895 isin 119869 119896 isin 119870 (7)

sum

119894isin119868

(119877

119894sdot (119904

119894119895minus ℎ

119894119895119896)) ge 120591


sum

119896isin119870

ℎ

119894119895119896= 119904


119873

max119895119896

minus

119868

sum

119894

ℎ


119869

sum

119895

(

119896=7

sum

119896=1

ℎ

119894119895119896sdot

119896=14

sum

119896=8

ℎ

119894119895119896) gt 0 forall119894 isin 119868 (11)

119904

1015840

1198943sdot 119904

119894119895le ℎ

1015840

119894314sdot ℎ

119894119895119896 forall119894 isin 119868 (12)

119904


ℎ




1 2 0 0 42 2 0 0 43 2 0 0 44 2 0 0 45 1 0 1 166 1 0 1 167 2 0 0 48 2 0 0 49 2 0 0 410 2 0 0 411 2 0 0 412 1 1 0 813 2 0 0 414 2 0 0 415 2 0 0 416 1 1 0 817 1 0 1 1618 1 0 1 1619 1 0 1 1620 2 0 0 4


119894sum

119869

119895(119875

119878

119894119895sdot

119904

119894119895)) and days-off (sum119868

119894sum

119869

119895sum

119870

119896(119875

119867

119894119896sdot ℎ






StaffShift rotation


1 1 8 2 4 3 02 2 4 1 8 3 03 1 8 2 4 3 04 1 8 2 4 3 05 1 32 2 16 3 06 1 32 2 16 3 07 2 4 1 8 3 08 2 4 1 8 3 09 1 8 2 4 3 010 1 8 2 4 3 011 1 8 2 4 3 012 1 16 2 8 3 013 1 8 2 4 3 014 1 8 2 4 3 015 1 8 2 4 3 016 1 16 2 8 3 017 2 16 1 32 3 018 1 32 2 16 3 019 1 32 2 16 3 020 2 4 3 0 1 8


4 Numeric Analysis

















1 7 1 56569 (asymp57)

2 8 0 4

3 5 3 113137 (asymp11)

4 8 0 4

5 5 3 113137 (asymp11)

6 8 0 4

7 8 0 4

8 8 0 4

9 7 1 56569 (asymp57)

10 8 0 4

11 8 0 4

12 7 1 56569 (asymp57)

13 6 2 8

14 7 1 56569 (asymp57)

15 7 1 56569 (asymp57)

16 7 1 56569 (asymp57)

17 8 0 4

18 3 5 226274 (asymp23)

19 8 0 4

20 8 0 4






1 57 57 57 572 4 4 4 43 11 11 11 114 4 4 4 45 11 11 11 116 4 4 4 47 4 4 4 48 4 4 4 49 57 57 57 5710 4 4 4 411 4 4 4 412 57 57 57 5713 8 8 8 814 57 57 57 5715 57 57 57 5716 57 57 57 5717 4 4 4 418 23 23 23 2319 4 4 4 420 4 4 4 4




5 Conclusion










Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











StaffShift rotation


1 1 8 2 4 3 02 2 4 1 8 3 03 1 8 2 4 3 04 1 8 2 4 3 05 1 32 2 16 3 06 1 32 2 16 3 07 2 4 1 8 3 08 2 4 1 8 3 09 1 8 2 4 3 010 1 8 2 4 3 011 1 8 2 4 3 012 1 16 2 8 3 013 1 8 2 4 3 014 1 8 2 4 3 015 1 8 2 4 3 016 1 16 2 8 3 017 2 16 1 32 3 018 1 32 2 16 3 019 1 32 2 16 3 020 2 4 3 0 1 8


4 Numeric Analysis

















1 7 1 56569 (asymp57)

2 8 0 4

3 5 3 113137 (asymp11)

4 8 0 4

5 5 3 113137 (asymp11)

6 8 0 4

7 8 0 4

8 8 0 4

9 7 1 56569 (asymp57)

10 8 0 4

11 8 0 4

12 7 1 56569 (asymp57)

13 6 2 8

14 7 1 56569 (asymp57)

15 7 1 56569 (asymp57)

16 7 1 56569 (asymp57)

17 8 0 4

18 3 5 226274 (asymp23)

19 8 0 4

20 8 0 4






1 57 57 57 572 4 4 4 43 11 11 11 114 4 4 4 45 11 11 11 116 4 4 4 47 4 4 4 48 4 4 4 49 57 57 57 5710 4 4 4 411 4 4 4 412 57 57 57 5713 8 8 8 814 57 57 57 5715 57 57 57 5716 57 57 57 5717 4 4 4 418 23 23 23 2319 4 4 4 420 4 4 4 4




5 Conclusion










Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















1 7 1 56569 (asymp57)

2 8 0 4

3 5 3 113137 (asymp11)

4 8 0 4

5 5 3 113137 (asymp11)

6 8 0 4

7 8 0 4

8 8 0 4

9 7 1 56569 (asymp57)

10 8 0 4

11 8 0 4

12 7 1 56569 (asymp57)

13 6 2 8

14 7 1 56569 (asymp57)

15 7 1 56569 (asymp57)

16 7 1 56569 (asymp57)

17 8 0 4

18 3 5 226274 (asymp23)

19 8 0 4

20 8 0 4






1 57 57 57 572 4 4 4 43 11 11 11 114 4 4 4 45 11 11 11 116 4 4 4 47 4 4 4 48 4 4 4 49 57 57 57 5710 4 4 4 411 4 4 4 412 57 57 57 5713 8 8 8 814 57 57 57 5715 57 57 57 5716 57 57 57 5717 4 4 4 418 23 23 23 2319 4 4 4 420 4 4 4 4




5 Conclusion










Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1 57 57 57 572 4 4 4 43 11 11 11 114 4 4 4 45 11 11 11 116 4 4 4 47 4 4 4 48 4 4 4 49 57 57 57 5710 4 4 4 411 4 4 4 412 57 57 57 5713 8 8 8 814 57 57 57 5715 57 57 57 5716 57 57 57 5717 4 4 4 418 23 23 23 2319 4 4 4 420 4 4 4 4




5 Conclusion










Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article modelling a nurse shift schedule with...

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