A STOCHASTIC APPROACH TO NURSE STAFFING AND SCHEDULING PROBLEMS

Transcription

1 A STOCHASTIC APPROACH TO NURSE STAFFING AND SCHEDULING PROBLEMS Presented by Sera Kahruman & Elif Ilke Gokce Texas A&M University INEN

2 Outline Problem definition Nurse staffing problem Literature and Formulation Computational Study Conclusion and Future study Nurse scheduling problem Formulation Computational Study Conclusion and Future study

3 Problem Definition Patients in a hospital today are in need of highly skilled nursing care In order for this care to be provided in a timely manner, and meet rigorous quality standards, the right number and type of nursing staff must be available when needed demand is unknown health institutions have to provide service 4 hours a day over seven days a week.

4 Problem Definition Poor staffing levels lead to: Burnout Dissatisfaction Desire to leave the job High cost of staff replacement More patient complications Injuries Low quality of service

5 Nurse Skill Types There are types of nurses:. Registered Nurses (RNs). Licensed vocational or licensed practical nurses (LVNs,LPNs). Nurse aides

6 To solve this problem, we need to consider two levels: Strategic level : determining the long-term regular time nursing levels---staffing Tactical level : Daily work/shift assignments of the nurses

7 Outline Problem definition Nurse staffing problem Formulation Computational Study Conclusion and Future study Nurse scheduling problem Formulation Computational Study Conclusion and Future study

8 Nurse staffing problem Hospital administrators determine the regular-time nurse levels annually So, our model should determine the levels of RNs, LPNs(LVNs) and nurse aides for a year Demand is unknown!!! Use -stage recourse model

9 Literature Tien and Kamiyama (98), present a list of personnel scheduling algorithms, which are not restricted to healthcare. Bradley and Martin (990), distinguish three basic decisions in hospital personnel scheduling: staffing, personnel scheduling and allocation Siferd and Benton (99), present an excellent review of the factors influencing hospital staffing and scheduling in the United States

10 Literature Kao, Queyranne(985) : Budgeting Costs of Nursing In Hospital Presents 8 different models Probabilistic or deterministic, different skill types or aggregate skill, single period or multi-period

11 Formulation Major assumption: There are many medical units within a hospital. We assume that each medical unit decides its own workforce levels and we do not allow inter-unit working schedules. The model is same for all units, except for different demand patterns, and parameters.

12 Formulation First stage parameters: c i = total expenses of the hospital for a nurse of type i for the whole year h e = Effective working hours of a nurse for one year AD i = minimum annual demand for each skill class nursing hours (can also be a bound specified by hospital regulations) r,i =Ratio defining the relationship of skill class and i=, (skill class supervises the other types. So there has to be a constraint to make sure that there are enough supervisors)

13 Formulation First stage decision variables: x i : number of regular time nurses of skill type I where i=,, tx i: number of nursing hours of skill type i and higher level skill types which can be used for the demand of lower skill nursing hours i=,.

14 Formulation Second stage parameters: o i = cost per overtime hour of a nurse of type i a i = cost per hour of agency nursing providing a service of skill type i e i,t = percentage of total working time of a nurse of skill class i available on month t {due to vacations and etc, effective nursing hours can change from month to month} O i,t = ratio of the maximum allowed overtime to the effective regular working hours for a nurse of skill type i, on month t d i,t = demand realization in hours for nursing skill of type i on month t.

15 Formulation Second stage decision variables: y i,t = number of overtime hours that nurses of skill type i work on month t. z i,t = number of agency nursing hours of skill type i used for month t. tr i : number of nursing hours of skill type i and higher level skill types which can be used for the demand of lower skill nursing hours i=,.

16 Formulation min s.t. i= h h h x x e e e c i * x * x * x * x i - tx + tx + tx + E{f(x,w ~ )} r, * x r, * x x, x, x AD - tx 0 0 AD AD 0, integer tx, tx 0

17 Formulation where for a realization of w, f(x,ŵ) is defined as: f(x, w ~ ) = min s.t. e e e, t y, t, t i,t * x * x * x x + i, t ( y i= t=, t + y + y, t, t * O + i, t z all variables 0, t + z + z 0 (oi * yi, t + a i * z i,t )), t, t tr + tr + tr d, t tr d, t d, t t t t =,..., =,..., =,..., i =,, and t =,,...,

18 Computational Study Use L-shaped algorithm where master problem is solved as a MIP. Comparison with expected value solution, scenario based solution and worst case solution Comparison with linear relaxation of master MIP Observe the effect of demand fluctuation

19 Conclusion and Future Study a decision making procedure which considers all medical units at the same time allowing staff movements from one unit to the other one as needed. some parameters in the model are much likely to be probabilistic which will make the model more realistic

20 Outline Problem definition Nurse staffing problem Formulation Computational Study Conclusion and Future study Nurse scheduling problem Formulation Computational Study Conclusion and Future study

21 INTRODUCTION Every hospital needs to repeatedly produce duty rosters for its nursing staff. The scheduling of hospital personnel is particularly challenging because of different staffing needs on different days and shifts. Because of time-consuming manual scheduling NSP has attracted much attention.

22 NURSE SCHEDULING PROBLEM (NSP) The NSP involves producing a periodic duty roster for nursing staff, subject to a variety of constraints. A key feature of real NSP is that the planned nurse schedule usually has to be changed to deal with unforeseen circumstances such as staff sickness and emergencies.

23 NURSE SCHEDULING PROBLEM (NSP) Solution approaches that have been proposed to solve NSP is classified in three main categories: Optimization-mathematical programming single-objective (Warner (976), Rosenbloom, E.S. andgoertzen, (987)) multi-objective (Arthur and Ravidran,98) Heuristics (Cheang et al., 00). Artificial intelligence

24 NSP FORMULATION Indices: g =,, (nurse grade index) k =, N (nurse index for grade nurse) l =, N (nurse index for grade nurse) m =, N (nurse index for grade nurse) j =, (shift index) j =, 4, 7, 0,, 6, 9 corresponds to 7am-pm shifts j =, 5, 8,, 4, 7, 0 corresponds to pm-pm shifts j =, 6, 9,, 5, 8, corresponds to pm-7am shifts

25 NSP FORMULATION First Stage Parameters: N: number of first grade nurses N: number of second grade nurses N: number of third grade nurses Pkj: preference cost of first grade nurse k working at shift j Plj: preference cost of second grade nurse l working at shift j Pmj: preference cost of third grade nurse m working at shift j ADjg: average demand for g th grade nurses at shift j

26 NSP FORMULATION First Stage Decision Variable: x kjg = if the first grade nurse k is assigned to shift j as a grade g 0 o/w nurse y ljg = if thesecondgradenursel is assignedtoshift j as a gradeg 0 o/w nurse z mj = if the third grade nurse i is assigned to shift j 0 o/w

27 D ~ jg NSP FORMULATION Second Stage Parameters: ocg: over time cost for nurse grade type g ucg: unsatisfied demand cost for nurse grade type g Djg: demand realization for nurse grade type g

28 NSP FORMULATION Second Stage Decision Variable: ox kjg if the first grade nurse i is assigned to shift j as a grade g = 0 o/w nurse oy ljg = if the second grade nursei is assigned to shift j as a grade g 0 o/w nurse oz mj = if the third grade nurse i is assigned to shift j 0 o/w ujg: unsatisfied demand for nurse grade type g at shift j

29 NSP FORMULATION Min N N P kj x kjg + P lj y ljg + k = j= g = l= j= g = m= j= N P mj z mj + E ( f ( x, y, ~ z, D )) Subject to: N k = x kj AD j j =,... N k= x kj N + l= y lj AD j j =,... N k = x kj N N + ylj + l = m= z mj AD j j =,... j= g= x = 6 k,... N kjg =

30 NSP FORMULATION j= g = y ljg = = 6 l,... N z mj = m = j = kj + xk ( j+ ) + xk( j+ ) + xkj + xk ( j+ ) + xk( j+ ) + xkj + xk ( j+ ) + xk( j+ ) x 6,... N k =,... N j =,4,7,0,,6,9 ylj + yl ( j+ ) + yl ( j+ ) + ylj + yl ( j+ ) + yl ( j+ ) l =,... N j =,4,7,0,,6,9 z mj + z m + ( j+ ) + z m( j ) m =,... N j =, 4, 7,0,,6,9 x kjg { 0,} k =,... N j =,,... g =,, { 0,} m =,... N j =,,... g =, z mj { 0,} m =,... N j =,,... y mjg

31 NSP FORMULATION E( f ~ ( x, D) = Min N N ocoxkjg + ocoyljg + ocoz mj + k= j= g = l= j= g = m= j= j= g= N uc g u jg Subject to: N k = x kj N + k= ox kj + u j D j j =,... N k = x kj N N N ylj oxkj l= k = l= oy lj + u j D j j =,... N k= x kj N N N N N ylj zmj oxkj oylj l= m= k = l= m= oz mj + u j D j j =,...

32 NSP FORMULATION ox kj + oxkj + oxkj xk ( j ) xk ( j ) xk ( j ) 0 k =,... N j =,... oy lj + oylj yl( j ) + yl ( j ) 0 l =,... N j =,... oz mj z m ( j ) 0 m =,... N j =,... j= g= ox k,... N kjg = j= g= j= oz oy l,... N ljg = m,... N mj = oy mjg { 0,} m =,... N j =,,... g =, ox kjg { 0,} k =,... N j =,,... g =,, { 0,} m =,... N j =,,... oz mj 0,int j =,... g =,, u jg

33 COMPUTATIONAL STUDY L Algorithm Data Set Number of Nurses (0-5-0) Number of Scenarios Number of Grade one Nurses Computation Time Solve the problem using CPLEX

34 CONCLUSION and FUTURE STUDY Improve cut in L Algorithm Change demand type Add new constraints

35 THANKS.