Consistent Staffing for Long-Term Care Through On-Call Pools

Transcription

1 Consistent Staffing for Long-Term Care Throgh On-Call Pools Athors names blinded for peer review Nrsing homes managers are increasingly striving to ensre consistency of care, defined as minimizing the nmber of niqe nrse aides who care for a resident dring at least one shift over the corse of one month. Unfortnately, managers often strggle to provide consistent care, primarily de to last-minte nrse aide absences and choosing to staff these absences with aides from an external rental agency. We are the first to stdy the se of an on-call pool aides on staff who may be called in to work in the event of absences as an operational strategy to improve consistency. We provide strctral reslts for the relationship between the nmber of aides in the on-call pool, staffing cost, and inconsistency level. We also show that a restricted on-call pool in which each slot in the pool is restricted to nrse aides from different nits otperforms an open, or nrestricted, on-call pool. We frther demonstrate that converting fll-time positions to part-time positions can improve consistency of care if the part-time aides on-call pool participation rate is sfficiently high. Nmerical reslts for a typical facility demonstrate that an on-call pool can a redce the staffing costs de to absences by 24% while also slightly redcing the inconsistency level, or b significantly redce the inconsistency level withot increasing costs. Key words : health care; absenteeism; nrsing homes; stochastic models; Schr-convexity

2 1. Introdction Recent stdies aimed at improving care in nrsing homes have identified consistent assignment having the same nrse aides care for the same residents over time as an important pillar of effective person-centered care paradigms. National organizations and individal thoght leaders have advocated consistent assignment as a means to elevate both the care that residents receive and the job satisfaction of nrse aides. Nrse aides, who often have the title of Certified Nrsing Assistant CNA, provide the majority of care hors in nrsing homes throgh tasks sch as feeding, bathing, and dressing. The prported benefits of consistent assignment are many. For example, residents do not have to re-explain preferences to new caregivers and also become more comfortable when receiving care. Also, nrse aides tend to develop more meaningfl relationships with residents and can more qickly detect and respond to resident health isses. Some states have even sed consistent assignment goals as part of pay-for-performance programs Roberts et al The Advancing Excellence in America s Nrsing Homes Campaign AE, www. nhqalitycampaign.org, a coalition that incldes 62% of nrsing homes in the United States, promotes consistent assignment as one of for organizational goals. The AE Campaign defines consistency as the nmber of niqe caregivers assigned to each resident in one month, and encorages a goal of 12 or fewer. It even provides a free spreadsheet tool for nrsing homes to calclate and track consistency. The Massachsetts Office of Medicaid provided a $2.8 million prize pool for nrsing homes that sed the AE Campaign s consistency tracking tool and achieved the AE Campaign s stated consistency goal Harris Yet, despite the prominence of consistent assignment as a qality goal for nrsing homes, little attention has been given to the nderlying operational challenges that prsit of this goal entails, specifically how staffing policies may have to be dramatically altered to reach the patient-centered goal of consistent assignment. Even the AE Campaign s own Create Improvement gidelines primarily focs on improving staff commnication, and do not address staffing policies sch as on-call pools. In fact, based on a srvey by Roberts et al. 2015, we are the first to connect patient-centered consistency metrics sch as that of the AE Campaign to tactical managerial staffing decisions. Practitioners literatre does address consistent assignment of corse, bt the most common view, for example in Castle 2013, tends to focs on the staff-centered metric 1

3 2 articlated by Qality Partners of Rhode Island 2007: They measre consistency nder the assmption that each fll-time aide has an assigned nit of residents. The consistency score is then calclated as the percentage of shifts schedled to be worked by fll-time aides in their assigned nit, and the target score is sally set at 80% or 85%. Unfortnately, in or conversations with managers of nrsing homes that had a nearly 100% consistency score sing the staff-centered metric, we still fond significant concern and frstration abot consistency of care. By comparing planned schedles to realized schedles for each shift over several months, we tied this discrepancy to absenteeism i.e., aides calling off a short time before the start of a shift, which occrs with approximately a 5% probability for each aide for each shift Castle Nrse aide call-offs led to redced patient-centered consistency, that was reflected in a steady stream of complaints from residents families: Up to forty different aides cold care for a resident over all shifts in a month, even thogh the planned schedle cold have a perfect staff-centered consistency measre. To try to reconcile this conflict we stdy on-call pools in which fll-time and parttime nrse aides who are on a nrsing home s staff volnteer to be available to work if a schedled nrse aide is absent as an operational strategy. On-call pools may also be referred to as float pools, bt we se the terminology on-call pool as float pool sometimes refers to nrses or nrse aides who are schedled to work a shift in a nit which is only assigned immediately before the start of a shift. We provide strctral reslts for the relationship between the on-call pool size, cost, and consistency metrics in the presence of absenteeism. We also illstrate how the se of part-time aides can lower inconsistency in a system with random participation by aides in on-call pools. Compared to a staffing policy common in practice, we show that tilization of an on-call pool can simltaneosly increase consistency and redce staffing costs. Or findings complement L and L 2016 who empirically find that laws limiting overtime by staff in nrsing homes harm qality partially de to an increased reliance on rental agency staff. Specifically, we make the following contribtions: 1. Model and Framework: To the best of or knowledge, we are the first to consider a metric based on the niqe nmber of servers who provide service over some time horizon, an important metric for long-term care facilities. To do so, or model incldes a strctre with nrsing home residents organized into nits with assigned staff, random absences

4 3 among staff, a random staffing reqirement, and rles for assigning on-call pool nrse aides to nits when absences occr. 2. Strctral Reslts: a Acconting for overtime wages, bonses paid for enlisting in the on-call pool, and the cost of hiring nrse aides from a rental agency, we establish the convexity of the staffing cost in the nmber of aides in the on-call pool. We also prove that the optimal pool size is of critical fractile type relating the cmlative distribtion fnction of the nmber of absences to a ratio incorporating the cost of overtime aides, the on-call pool bons amont, and the cost of rental agency aides. b We show that the expected level of inconsistency de to absences is convex and non-increasing in the on-call pool size. In other words, there are decreasing retrns of on-call pool size on the consistency level. c We analyze two types of on-call pools: open, in which any aide may enlist in any on-call pool slot, and restricted, in which the home nits of the nrse aides in the on-call pool mst be represented as evenly as possible. The expected inconsistency level de to absences improves as the on-call pool slots are more evenly distribted over the different nits a property that we prove sing the concept of Schr-convexity. Conseqently, the restricted on-call pool otperforms the open pool in terms of consistency. d We show that it may be possible for a nrsing home to replace one fll-time aide with two part-time aides and improve the overall consistency of care, in particlar if the on-call pool participation rate of the part-time aides is sfficiently high. 3. Managerial Insights from Nmerical Stdy: a For a facility with the national median size of 100 beds, we show that it is possible for the implementation of an on-call pool to redce staffing costs de to absences by 24% while simltaneosly redcing the inconsistency de to absences i.e., the redction in consistency de to absences by 23%. Alternately, a nrsing home may redce the inconsistency level de to absences by 70% withot increasing staffing costs. b Enacting an on-call pool may allow a nrsing home to transfer approximately $60,000 per year in wages in hose moving them from rental staffing agencies to overtime wages and on-call pool bonses for staff at the nrsing home. c For the same on-call pool size, inconsistency de to absences is initially lower for smaller nits than for larger nits i.e., 2 nrse aides per shift compared to 4 bt becomes

5 4 lower for larger nits as the on-call pool size increases. Ths the se of an on-call pool, and its size, can inflence whether larger or small nits are preferred, from a consistency of care perspective. The remainder of the paper is organized as follows. Section 2 reviews relevant literatre. Section 3 introdces the nrsing home indstry motivating or model. We define or model for the operation of the on-call pool and establish its strctral properties in Section 4. Nmerical analysis showing the benefits of an on-call pool for a typical nrsing home follows in Section 5. We conclde with a smmary of findings in Section 6. All proofs appear in the appendix. 2. Related Literatre Respondents to a srvey condcted by Miller et al named workforce isses as the foremost challenge facing long-term care providers; not srprisingly, nrse aide staffing policies have received attention in practitioners literatre. Castle and Fergson 2010 provide an overview of qality measres for nrsing homes, inclding the important fivestar rating system of the Centers for Medicare and Medicaid Services. Castle and Engberg 2008 show that higher staffing levels are weakly associated with better qality of care, bt they also noted the importance of the se of rented staff from staffing agencies, the ratio of registered nrses to other caregivers, and trnover rates. Castle 2009 also demonstrates that the se of agency staff has a significant negative association with qality of care. Weech-Maldonado et al propond the importance of sing fll-time registered nrses instead of part-time registered nrses for qality otcomes. And L and L 2016 find that laws prohibiting mandatory overtime reslt in the increased se of contract nrses and nrse trainees, harming qality of care. Brgio et al compare nrsing homes practicing permanent assignment an alternate term for consistent assignment to those practicing rotating assignment. Measring the percentage of time all residents in the facility were cared for by their most freqently assigned nrse aide as the permanency rate, the mean permanency rate was fond to be only only 50% for the facilities with permanent assignment and 26% for those with rotating assignment. They fond significantly higher ratings of residents personal appearance and hygiene and higher reported job satisfaction among nrse aides for permanent assignment schemes, bt otherwise fond few statistically significant differences

6 5 in qality of care otcomes. Castle 2013 shows that nrsing homes that practice consistent assignment experience lower trnover and absence rates, and Castle and Fergson-Rome 2014 link absenteeism to decreases in varios measres of care qality. Defining consistent assignment as in the widely promoted metric by Qality Partners of Rhode Island 2007, they find that 68% of nrsing homes attempt to se consistent assignment while only 28% of nrsing homes achieve the recommended 85% consistency rate. To the best of or knowledge, we are the first to stdy consistent assignment of health care servers over time in the operations management literatre; we are also the first to stdy the relationship between absenteeism and consistent assignment. Researchers have stdied staffing decisions in hospitals and other service settings. Green et al empirically demonstrate that nrse absences at an rban hospital increase in anticipation of higher workloads; they solve a single shift newsvendor problem to avoid nderstaffing. Wang and Gpta 2014 se Schr-convexity to show that nrsing costs are minimized by assigning nrses to home nits in hospitals in a way that maximizes heterogeneity of absentee rates among nrses within each nit. Using a closed qeeing model, de Véricort and Jennings 2011 demonstrate that staffing levels based on simple nrse-to-patient ratios are insfficient for achieving a desired probability of excessive delay; it is necessary to also accont for the total nmber of patients in the system. Otside of healthcare staffing, Fry et al stdy the staffing decision for firefighters incorporating nplanned and extended absences, as well as calendar constraints related to work rles. Researchers have also stdied the vale of workforce flexibility in healthcare and other settings. Wright and Bretthaer 2010 show the vale of workforce flexibility from the se of rental staffing agencies in a hospital. In the context of call centers, Bhandari et al propose an algorithm to compte the nmber of permanent operators, the nmber of temporary operators, and the threshold nmber of cstomers in the system for when to se the temporary servers. Kesavan et al show that sing part-time and temporary workers in retail stores initially improves financial performance bt eventally harms performance if they comprise too mch of the workforce. As a stochastic model, worker absences bear similarities to random yields in prodction planning. Yano and Lee 1995 present the binomial distribtion as the simplest model of random yields and review related literatre.

7 6 3. Nrsing Home Indstry Overview The Centers for Medicare & Medicaid Services project nrsing care facilities and contining care retirement commnities to be a $176.1 billion indstry in 2016, with $90.8 billion of this amont by Medicare and Medicaid Keehan et al Ot-of-pocket payments are projected to constitte $52.6 billion of this spending, while the remaining $32.7 billion is divided among private insrance, other government insrance programs, and other thirdparty payers. In its National Stdy of Long-Term Care Providers, the Centers for Disease Control and Prevention s 2013 report lists 15,700 nrsing homes in the United States serving 1,383,700 residents, while 22,200 residential care commnities serve 713,300 residents Harris-Kojetin et al This stdy reports that nrsing homes employ 952,100 flltime eqivalent nrsing workers, of whom 65% are nrse aides. Aides provide the majority of care in nrsing homes; nrse aides perform an average of 2.46 hors of care per resident per day, while registered nrses and licensed practical nrses provide 0.52 and 0.85 hors, respectively. According to the Nrsing Home Compare dataset Centers for Medicare and Medicaid Services 2015, the median facility size is exactly 100 beds, and the national bed occpancy rate is 82%. Of all nrsing homes, 69% have for-profit ownership, 24% are not-for-profit, and 7% are owned by varios government agencies. Approximately 70% of nrsing home residents live in facility with at least 100 beds, and 15% live in a facility with at least 200 beds. Long Term Care Facilities srveys of the Pennsylvania Department of Health 2015 report a total of 701 nrsing homes with 88,063 licensed beds that serve 79,297 residents in Pennsylvania. The median price per day for a private room which incldes nrsing care, meals, and tilities is $301. The median daily Medicare reimbrsement rate across facilities is $446, which may reflect both additional services received by residents and a willingness by Medicare to be a higher payor. The nmbers of fll-time and part-time nrse aides employed by the facilities are 29,029 and 11,342, respectively. Herzenberg 2015 reports the median horly wage of a nrse aide in Pennsylvania as $ We interviewed managers and analyzed nrse aide schedles of two non-profit nrsing homes in sothwestern Pennsylvania, each of which has between 100 and 200 licensed beds and between 50 and 100 nrse aides on staff. Both had previosly adopted a consistent assignment regime in which each nrse aide almost exclsively served one nit of either approximately 15 or 30 residents. Both facilities reported between 2.7 and 3.0 nrse aide

8 7 Table 1 Notation Symbol Description y Staffing level; i.e., the nmber of nrse aides schedled or on-call for one shift Nmber of nits i.e., grops of residents in the nrsing home q Nmber of nrse aides reqired to be schedled per nit per shift exogenosly determined R k Random nmber of nrse aides reqired for nit k g U l Probability mass fnction for R k, l 0, 1, 2,... g T m Probability mass fnction for the total reqirement over all nits, m 0, 1, 2,... y 0 Minimm nmber of aides to schedle for one shift y 0 = q γ Probability that any one schedled nrse aide is absent on any one shift N S Random nmber of schedled nrse aides ot of y 0 who show p to work a shift w c Marginal cost of an on-call aide working one shift compared to a schedled aide w a Marginal cost of a rental agency aide working one shift compared to a schedled aide b Bons paid to on-call aide who is not called in d Cost of cancelling a schedled shift for a nrse aide Cy Expected cost of staffing de to absences for one shift with staffing level y L Oy Expected inconsistency level for one nit for one shift with an open on-call pool and staffing level y L Ry Expected inconsistency level for one nit for one shift with a restricted on-call pool and staffing level y κ Nmber of shifts in span of time over which inconsistency level is measred e.g., one month S k Random variable representing net staffing shortage in nit k on one shift H ρ k y Random variable for aides in on-call pool having y aides and sign-p rle ρ with nit k as home nit σ k Net staffing shortage for nit k η j The home nit of on-call pool aide j x k The nmber of on-call pool aides for some arbitrary nit k on a sample path χ k The home nit corresponding to x k Bk; n, p Binomial probability mass fnction Hk; N, K, n Hypergeometric probability mass fnction fy First forward difference operator; i.e., fy = fy 1 fy staffing hors per patient per day. Bt the two nrsing homes adopted qite different approaches towards sing part-time aides: part-time aides comprised approximately 15% of aides on staff at one nrsing home and 40% at the other. 4. The Operation of On-Call Pools Residents of the nrsing home are groped into symmetric nits, which are also sometimes called commnities, neighborhoods, or hoseholds. For each shift the nmber of aides reqired to be schedled for each nit is q; q is exogenosly determined and insensitive to small changes in the resident censs. Choosing q is typically a longer-term strategic decision related to facility layot; we consider that decision to be otside the scope of the paper. The total nmber of nrse aides reqired to be schedled per shift is then y 0 = q. However, we allow the actal nmber of nrses reqired in each nit for a shift to vary randomly immediately before the start of a shift: Extra aides might be needed for varios reasons, inclding a large nmber of admissions from hospitals, an otbreak of an illness among residents, or an nannonced inspection. In contrast, nrsing home managers and

9 8 consltants have explained to s that the staffing reqirement on a nit wold almost never decrease at the last minte. Nevertheless, to generalize or model and frther its applicability to other settings, we allow the staffing reqirement of each nit to increase or decrease randomly at the start of each shift. as For convenience, we denote the probability mass fnction of the binomial distribtion n! k!n k! Bk; n, γ = γn k 1 γ k for 0 k n, 0 otherwise. and the probability mass fnction of the hypergeometric distribtion as K k N K n k for max0, n K N k minn, K, Hk; N, K, n = N n 0 otherwise. We also denote maxx, 0 by x and max x, 0 by x. The seqence of events for any shift is as follows: 1. The nrsing home enlists a total of y y 0 nrse aides who volnteer to work dring the shift nder consideration; y 0 of the aides are schedled and the remaining y y 0 aides are on-call. 2. Of the y 0 schedled nrse aides, a total of N S aides show p to work. Each aide is absent with probability γ; these absences are independent and identically distribted. 3. A random nmber R k of nrse aides are needed as the actal staffing reqirement in each nit k for a shift. R k follows an independent probability mass fnction g U l, l 0, 1, 2,..., and reflects any last-minte changes to the patient censs or other instittional tasks dring the shift that wold increase or decrease the nmber of staff needed in the nit. The random staffing reqirement over all nits for the shift, R k, follows the probability mass fnction g T m, m 0, 1, 2,..., which is the convoltion of the nit independent probability mass fnctions identical to g U l. 4. We define S k as the random variable representing the net staffing shortage in nit k considering the random staffing reqirement and absences by schedled aides in that nit. More precisely, P r S k = i = q g U q i jbj; q, γ 1

10 9 and y 0 P r S k = i = g T y 0 i jbj; y 0, γ. 2 A vale of S k < 0 represents a staffing srpls in nit k. When the staffing reqirement is deterministically q aides, S det k Binomialq, γ. The total nmber of positions that need to be filled across all nits is then S k ; these are replaced as follows from highest to lowest priority: Reallocated schedled aides: min S k, S k schedled aides are not needed in the nit for which they were schedled and are reallocated to other nits. If R k < N S, then the shifts for N S R k nrse aides are cancelled at a cost of d per shift. On-call aides: min y y 0, S k on-call aides are called in to work at an additional cost of w c per aide per shift beyond what a schedled aide wold have cost. The remaining [y R k y 0 N S ] aides who are not called in receive a bons b, b 0, per aide per shift for being on call. Rental agency aides: S k y y 0 rental agency aides are employed for the shift at an additional cost of w a per aide per shift beyond what a schedled aide wold have cost. All notation is presented in Table 1. Or analysis separates the consistency of care metric for the day, evening, and night shifts, and or model can be applied to any one of these shifts. We assme that w c w a ; i.e., it is less expensive to se an on-call aide than an agency aide. Otherwise, the optimal soltion wold be to have no on-call pool. We also assme that the nmber of aides willing to participate in on-call pools is relatively large a realistic assmption if aides are allowed to sign p for on-call pool slots of other shifts and any aide participating in the pool is eqally likely to have any nit as his or her home nit. Frthermore, or modeling emphasis on marginal cost allows s to ignore paid sick leave for a call off as a constant vale by which the costs of every strategy shifts. We now mst consider the system dynamics within each of the symmetric nits to express the consistency of care metric. Each nit has a set of nrse aides who may be flltime or part-time employees assigned to it; this set of nrse aides has sfficient capacity to staff the nit. Reflecting what we have observed in practice, each aide is assigned to only one nit. These aides provide a baseline level of inconsistency; this is the minimm

11 10 inconsistency level for the nit and best consistency score that residents can experience for the specified shift or shifts. For example, a nit which reqires q = 4 nrse aides for each day shift might have seven fll-time nrse aides to cover all day shifts in a month, which corresponds to a baseline inconsistency level of seven. Treating the baseline inconsistency level as fixed, we focs on the expected nmber of additional aides that work in the nit given a total staffing level y, defining this as the inconsistency de to absenteeism, or inconsistency level. If the nit experiences six absences in a month and relies on rental agency aides as sbstittes i.e., y = y 0, the inconsistency level de to absenteeism is six. From or observations of nrsing home schedles it is nlikely that the same rental agency aide will sbstitte for the same nit on mltiple occasions in one month. The total inconsistency level, which corresponds to the AE Campaign s metric for consistency of care, is then 13 aides. We let H ρ k y 0, 1,..., y y 0 represent the random nmber of aides in the on-call pool ot of the y y 0 aides in the on-call pool that have nit k as their home nit given an on-call pool with y aides and some enlistment rle ρ. Aides from the on-call pool that serve in their native nit do not increase the inconsistency level. Bt each reallocated schedled aide and each rental agency aide increases the inconsistency level by one; we assme that the probability that these aides wold serve in the same nit otside their home nit mltiple times within the interval over which consistency of care is measred is sfficiently low. Likewise, all aides from the on-call pool that serve in nits different from their home nit each increase the inconsistency level by one. Becase a nrsing home prefers to have an aide work a schedled shift rather than dismiss the schedled aide and bring in an on-call aide to work at an overtime rate, the reallocated schedled aides have the highest priority for filling shortages. Contingent pon this, on-call aides with a shortage in their native nits are assigned to their native nits as long as all srpls schedled aides can be reallocated to satisfy other shortages. Ths, min min H ρ k y, S k, S k on-call aides serve in their native nit. The expected inconsistency level L ρ y experienced by each of the symmetric nits is the expected difference between the total nmber of shortages and the nmber of on-call aides serving in their native nit; i.e., [ L ρ y = E S k min min H ρ k y, ] S k, S k /. 3

12 The expected inconsistency de to absences over κ shifts can then be compted as κl ρ y; e.g., κ = 30 for the day, evening, or night shift over a 30-day month. We characterize two types of on-call pools based on the rles governing which aides may join on-call pools: open ρ = O and restricted ρ = R. In the open on-call pool, any aide may fill any of the y y 0 positions, which means that each nit is eqally likely to be the home nit of the aide filling any position and P r H O k y = i = Bi; y y 0, 1/ for any k 1, 2,...,. When the staffing reqirement is deterministic, we can write the inconsistency level from the perspective of any one nit acconting for the possible vales of S k and Hk O y as 0 L det O y = Bi; y y 0, 1/ 11 q Bj; q, γj i, 4 where the index i refers to the nmber of on-call aides in the pool from the nit and the index j refers to the nmber of absences within the nit. With a restricted on-call pool, aides are eqally likely to come from any nit sbject to the restriction that no nit may have more than one more aide in the on-call pool than any other nit; each nit has either y y 0 / or y y 0 / 1 aides in the on-call pool. For the restricted on-call pool, y y 0 / y y 0 / if i = y y 0 / 1, P r Hk R y = i = 1 y y 0 / y y 0 / if i = y y 0 /, 0 otherwise. The expected inconsistency level with a deterministic staffing reqirement is q y L det y0 y y0 y y0 R y = Bi; q, γ i i= 0 /1 q = Bi; q, γ i y y 0, 5 i= 0 /1 which we can alternatively write as q L det R y = Bi; q, γ i y y 0. 6 j=i

13 12 One detail from practice different from or model assmptions can occr if the same rental agency aide covers for an aide who calls off for a mltiple-day absence. Exclding the absences following the initial call-off when estimating γ can help ensre that L O y and L R y accrately represent the consistency metric. Another potentially relevant practical detail that we ignore is that any bons pay b wold be considered nondiscretionary, and ths may affect each employee s reglar rate of pay, which is sed to determine the overtime pay rate. For more details, see Rotman Cost of Absenteeism We define the cost of absenteeism as the difference between the staffing cost with absenteeism and the cost withot absenteeism. Ths, minimizing the cost of absenteeism also minimizes the total staffing cost. The expected facility-wide cost of absenteeism Cy for a single shift is [ ] Cy =de S k be y y 0 S k [ ] [ ] w c E min y y 0, S k w a E S k y y 0, which sing 2 we can alternatively express as 1 y0 Cy = g T y 0 i jbj; y 0, γ by y 0 di i= y 0 0 y0 i= 0 1 g T y 0 i jbj; y 0, γ w c i by y 0 i y0 g T y 0 i jbj; y 0, γ w a i y y 0 w c y y 0. 7 The following reslt demonstrates the optimal staffing policy for or model, which has the strctre of a newsvendor model with a piecewise linear cost fnction. For analytical ease, we define the first forward difference fnction for any fnction f as fy = fy 1 fy. Proposition 1. The minimizer of 7 is given by 0 y 0 y = min y N y y 0, g T y 0 i jbj; y 0, γ w a w c. 8 b w a w c

14 When the staffing reqirement is not sbject to random variation, we can state the cost-minimizing staffing level y det fnction for a binomial random variable. by more sccinctly sing only the cmlative distribtion Corollary 1. When R k = q for every nit k, k = 1,...,, the minimizer of 7 is given y det = min y N y y 0, 4.2. Inconsistency De to Absenteeism 0 Bj; y 0, γ w a w c b w a w c The complexity of the system s operational dynamics particlarly the reallocation of schedled aides leads s to se a sample path approach as described by Lindvall 2002 to demonstrate properties of the inconsistency fnction. We define a sample path sing two pieces of information: 1. The net staffing shortages σ = σ 1,..., σ with σ k being the shortage in nit k or srpls of σ k aides for σk < 0 which is mapped to one of the nits pon realization of the sample path. This allows s to take the expectation which we denote with the sbscript σ of the! combinations of assignments of net shortage qantities σ k,..., σ k to the nits. 2. The home nits of on-call pool aides η = η 1,..., η 0 with η i representing the home nit sing the same frame of reference i.e., indexing for nits as in σ of each aide i = 1,..., y y 0 in the on-call pool. We can rewrite 3 on a sample path as L ρ y; η, σ = σk min min 0 which can be alternatively written as σk L ρ y; η, σ = 0 i=1 1 η i = k 1 σ k > i 1 j=1 1 η j = k i=1 1η i = k, σ k, σ k, 10 1 σ k > i 1 j=1 l=1 1 η j = l 11

15 14 Ths, the vale of a marginal aide in the on-call pool is L ρ y; η, σ = L ρ y 1; η, η 0 1, σ L ρ y; η, σ 1 η 0 1 = k 1 σ k > 0 j=1 1 η j = k 1 σ k > 0 j=1 l=1 1 η j = l = 12 We can then characterize the expected change in the inconsistency level de to one additional aide in the on-call pool with expectation taken over the assignment of nits from η and the home nit η 0 1 of the marginal aide in the on-call pool as E η,η0 1 [L ρ y; η, σ] = [ E 1 η 0 1 = k, σ k > 0 j=1 1 η j = k, 0 σ k > j=1 ] 1 η j = l /. Becase the home nit of any on-call aide i.e., η i depends on the on-call pool enlistment rle, we present separate reslts for the open and restricted on-call pools. We then show that the restricted on-call pool otperforms the open on-call pool in terms of the inconsistency. Proposition 2. For a system with an open on-call pool, the inconsistency level L O y is convex and non-increasing in the staffing level y. Corollary 2. When the staffing reqirement is deterministic, enlisting one additional aide in an open on-call pool redces the inconsistency level by q j=1 Bj; q, γ j 1 L det O y = l=1 13 Bi; y y 0, 1/. 14 We analyze the restricted on-call pool in a similar fashion, and take advantage of the properties of the restricted on-call pool that allow s to write the vale of a marginal aide in the on-call pool more explicitly in or proofs. Proposition 3. For a system with a restricted on-call pool, the inconsistency level L R y is convex and non-increasing in the staffing level y. If the staffing reqirement is not sbject to randomness, we can more precisely characterize the relationship between the inconsistency and the staffing level.

16 15 Corollary 3. When the staffing reqirement is deterministic for a system with a restricted on-call pool and a staffing level y, enlisting one additional on-call aide redces the inconsistency level by L det R y = q i= 0 1 Bi; q, γ/. 15 Taken together, Propositions 1, 2, and 3 cold inform a manager s choice of the on-call pool size considering both cost and consistency. Becase inconsistency is non-increasing in y, a manager seeking to minimize costs sbject to a constraint on the inconsistency level cold find the vale of y that minimizes costs and compare this with the lowest vale of y for which the inconsistency is below the maximm allowable level. This latter vale cold be fond by bisection search over y sing 3 for inconsistency. Alternatively, if a manager cold assign a monetary non-negative penalty vale to the inconsistency level, the reslting cost fnction i.e., the sm of the staffing cost and the inconsistency penalty wold be convex; the sm of a convex fnction and a non-negative vale mltiplied by a convex fnction. This cold then be optimized sing standard techniqes. We next slightly modify 3 to stdy how the allocation of on-call nrse aides across the varios nits affects the expected inconsistency de to absences. Instead of a sample path defined by the home nit η j of each aide in the on-call pool, we se x = x 1,..., x where x k is the nmber of on-call pool aides for some arbitrary nit k that is mapped to a specific nit on a sample path with the variable χ k 1,...,. Specifically, we now have L ρ x; σ = E χ [ with σ k min l=1 min ] σ k, x l 1 χl = k, σ k /, 16 L ρ x = E σ [L ρ x; σ]. 17 We se the concepts of majorization and Schr-convexity to assess the performance in terms of the expected inconsistency level de to absences of different on-call pool enlistment rles. We se the notation x [1] to represent the largest element in x, x [2] to represent the second-largest element, and so on, and se the notation x x to say that x majorizes x.

17 16 Table 2 Additional Notation for Section 4.3 Symbol Description Nmber of fll-time aides reqired for one nit across κ shifts if only fll-time aides may be sed δ Nmber of part-time aides employed in the nit τ Ratio of schedled hors per week for a part-time aide to fll-time aide α Probability that any one nschedled fll-time aide enlists in on-call pool for one shift β Probability that any one nschedled part-time aide enlists in on-call pool for one shift Cδ Expected vale of consistency metric with δ part-time aides φ Probability that native aide wold get on-call slot if native aide is available Proposition 4. The expected inconsistency level de to absences L ρ x is Schr-convex in x; i.e., L ρ x L ρ x if x x. Becase the restricted on-call pool prodces an allocation that cannot be majorized i.e., x [1] x [] 1 for the restricted on-call pool, we have the following reslt: Corollary 4. The restricted on-call pool minimizes the expected inconsistency de to absences L ρ x Workforce Mix and Consistency of Care We now consider a variation on or model that acconts for random participation in the on-call pool by nrse aides and a decision to choose the mix of part-time and fll-time workers. Employing a higher percentage of fll-time aides means that fewer niqe aides are needed to cover all shifts in one month. However, part-time aides typically are eager to participate in an on-call pool, ths having additional part-time aides in a nit may increase the probability that at least one native aide enlists in the on-call pool for a shift. We focs or analysis on the restricted on-call pool model de to the improved consistency it offers compared to the open model. New notation introdced in this section is listed in Table 2. We assme that all shifts over the corse of κ schedling periods can be served by fll-time nrse aides. In other words, withot absenteeism the patients wold experience niqe fll-time aides over the corse of κ shifts. Each fll-time aide may be replaced, however, by part-time aides who work a fraction τ, 0 < τ < 1, of the hors of a fll-time aide. The decision variable in this model is then the nmber of part-time aides to employ in one nit, which we denote by δ, 0 δ /τ. For analytical tractability, we assme that the fraction of the q schedled aides on each shift who are part-time aides is constant and eqivalent to the proportion of total schedled shifts assigned to part-time aides. That is, τδ/ of all shifts are worked by fll-time aides, and the remaining τδ/ shifts are worked by part-time aides. Ths, the nmber of

18 17 fll-time aides not working on a given shift and ths eligible to volnteer for the on-call pool is τδ τδq/, and the nmber of part-time aides eligible is δ τδq/. For example, if = 6 fll-time aides are reqired to cover all shifts and if part-time aides are schedled for τ = 0.5 as many hors as fll-time aides, choosing δ = 2 corresponds to a system with two part-time aides and τδ = 5 fll-time aides. To simplify or analysis, we only consider a system with y y 0 ; there may only be p to one aide on-call per nit. For the cases that we consider, y y 0 incldes all vales for the on-call pool size that redce inconsistency while not increasing staffing costs compared to a system withot an on-call pool. Given this constraint, it becomes necessary to define rles for choosing among nrse aides in the event that mltiple aides from different nits wish to enlist in the on-call pool on a shift. In sch a case, specifically when y y 0 <, we assme that the nit nder consideration has the opportnity to provide a nrse aide to the on-call pool on a fraction φ of all shifts. On the remaining fraction 1 φ of shifts, enogh aides from otside the nit have volnteered for the on-call pool slot and have priority over any native aide. In practice, a system in which on-call pool enlistment priority rotates among different nits corresponds to this model. For example, sppose one on-call pool position is open to aides in two different nits, Unit A and Unit B. A nrse aide from Unit A may occpy the on-call pool slot if Unit A has priority over Unit B on that shift or if Unit B has priority bt no aide from Unit B enlists. If the probability that no aide from a specific nit enlists in the on-call pool is 50%, then φ = 75%. We evalate the expected nmber of niqe nrse aides who provide care in a nit over κ schedled shifts when the nit has δ part-time aides on staff: Cδ = τδ δ κe [ S H ], which acconts for the τδ fll-time aides, δ part-time aides, and effect of absences over κ shifts. As before, we assme that non-native aides who sbstitte for absent aides are different each time that a sbstittion is reqired. With S H, we only need to consider cases when S H recall H 1: q Cδ = τδ δ κ ip r S = i P r H = i=1

19 18 Each fll-time aide who is not schedled volnteers for the on-call pool with independent and identical probability α, and each part-time aide who is not schedled volnteers with independent and identical probability β. Ths, the probability that the on-call pool incldes a native aide is τδq δ P r H = 1 = φ 1 1 β 1 α τδ τδq, which we sbstitte into 18 to get the overall inconsistency level: q τδq δ Cδ = τδ δ κ Bi; q, γ i φ 1 1 β 1 α τδ τδq. i=1 Becase q i=1 ibi; q, γ = q ibi; q, γ = qγ is the expected vale of a binomial random variable and q i=1 Bi; q, γ = 1 B0; q, γ, we rewrite this expression as Cδ = τδ δ κqγ κ 1 B0; q, γ φ = τδ δ κqγ κφ 1 B0; q, γ τδq δ 1 1 β 1 α τδ τδq τδq δ κφ 1 B0; q, γ 1 β 1 α τδ τδq. 19 Using 19 we can evalate the effect of the nmber of part-time workers on consistency. Proposition 5. The overall inconsistency level may decrease in the nmber of parttime workers δ; specifically, δ close to zero and a sfficiently high part-time worker on-call pool participation rate β ensre Cδ decreases in δ. Proposition 5 provides the managerial insight that converting a fll-time position into mltiple part-time positions can actally decrease the expected nmber of niqe nrse aides caring for residents. Ths, sbject to constraints on the nmber of hors worked that correspond to part-time stats, managers shold hire part-time aides who are willing to participate in the on-call pool at a relatively high rate. The increasing se of part-time aides in on-call pools has another significant benefit; part-time aides in the on-call pool typically wold not be working enogh hors to earn overtime pay. 5. Nmerical Reslts Motivated by or collaboration with a nrsing home to implement an on-call pool, we introdce model parameters representing typical nrsing homes in Section 5.1. In Section 5.2 we stdy the impact of the size of the on-call pool on both the cost of absenteeism and inconsistency level for facilities with different nmbers of beds and with different nmbers of residents per nit. We analyze the workforce mix decision in Section 5.3.

20

21 20 $2,496 less paid to workers who are absent. As shown in Figre 1, an on-call pool of one nrse aide minimizes the expected monthly cost, and prodces a savings of $609, or 24% of costs de to absences. Using an on-call pool of three aides for the staff of 16 aides provides a soltion that has approximately the same cost as the system withot an on-call pool, bt keeps most of the payments in-hose to aides on staff in the forms of on-call wages and bons pay. Figre 1 also shows that the monthly inconsistency level de to absences is convex and non-increasing in the on-call pool size. Each of the first aides in the on-call pool redces the inconsistency level for the day or evening shift by a constant amont for restricted pools: 1.39 fewer niqe aides each month per on-call aide for a large-nit facility and 0.37 per aide for a small-nit facility. As indicated by Proposition 4, the restricted on-call pool otperforms the open on-call pool. The maximm difference between restricted and open systems occrs when y = y 0. This difference in performance shows that nrsing homes shold manage on-call pool enlistment so that participation on any shift is spread as evenly as possible across the nits. The on-call pool sizes in Figre 1 that are identified as cost-netral i.e., approximately the same as when y = y 0 can significantly redce the expected monthly inconsistency level. A restricted on-call pool with three aides redces the inconsistency de to absences by 70% for a facility with large nits and 37% for a facility with small nits. The inconsistency level de to absences is initially higher for large nit facilities compared to small nit facilities, bt eventally becomes lower for larger pool sizes. For example, the inconsistency level is lower for large nits than small nits when a restricted on-call pool is operated with at least three aides. The additional staff needed for a large nit compared to the small nit means that the overall consistency score i.e., the nmber of niqe aides per resident per month will initially be lower for the small nit. However, the vale of having small nits diminishes when absences are considered, becase it is easier to replace an absent aide with a native aide from the on-call pool when the nit size is larger and ths there are fewer nits. For a system withot an on-call pool, it is possible to decrease both the cost and inconsistency by adding one. That the dominated strategy of relying solely on rental agency aides is commonly sed in practice reflects the challenge that many managers have making decisions given the stochastic natre of absences and overtime wages combined with the

22

23 22 per month that is lower than if there were δ = 0 part-time aides on staff. In other words, replacing one fll-time aide with two part-time aide both increases the nmber of niqe caregivers on staff by 1.0 and lowers the inconsistency level de to absences by at least 1.0. As the fll-time aide participation rate increases, the minimm part-time aide participation rate also mst increase becase part-time aides mst be available for more shifts to get the same redction in the inconsistency level. The reqired vale of β is 50-65% higher for the small nit facility than the large nit facility, which reflects the lower probability of an on-call nrse to be native with the same on-call pool size. 6. Conclding Remarks Nrse aides play a critical role in the delivery of care to nrsing home residents. Nrsing home managers make concerted efforts to avoid having a high nmber of niqe aides caring for each resident; i.e., to attain high consistency of care. This paper analyzes both staffing cost and care consistency when nrse aides are sbject to random absences on each shift. One key featre of or model is the explicit acconting for the expected nmber of niqe nrse aides who care for a resident in a month a metric that is increasingly promoted by advocates of patient-centered care in nrsing homes. We provide strctral reslts for decisions related to the on-call pool s size and organization, and show nmerically that an on-call pool can redce both the staffing cost de to absences and improve consistency of care. We also show that sing part-time aides in conjnction with on-call pools can frther improve consistency of care if the part-time aides on-call pool participation rate is sfficiently high. We recognize that or assmption of a constant fraction of part-time and fll-time aides working on each shift and eligible for the on-call pool may not be strictly implementable. However, we believe that this model still provides important insights on how adding part-time nrse aides to a nit can actally decrease the total nmber of niqe nrse aides with whom residents in that nit interact in a month. Or work points to an opportnity for resorce-constrained nrsing homes to improve nrse aide job satisfaction, nrse aide pay, the personal comfort of nrsing home residents, the medical care that they receive, and nrse aide staffing costs. Ftre work can examine nrse aide schedling policies in more detail to gide decisions abot pay differentials for weekend and night shifts and promises made to employees to satisfy labor rles or provide schedle predictability. For example, some nrsing homes

24 23 reqire nrse aides to work every other weekend. Other nrsing homes have nrse aides who have fll-time stats bt only work extended shifts on weekends. These qestions for ftre research are especially important when connected to the overall staffing decision. Appendix Proof of Proposition 1. The first forward difference of the cost of absences Cy = Cy 1 Cy can be written sing 7 and cancelling identical terms: 0 1 y0 Cy = g T y 0 i jbj; y 0, γ w c i by y 0 1 i Collecting terms, Cy =b =b 0 i= y0 g T y 0 i jbj; y 0, γ w a i y 1 y 0 w c y y 0 1 y0 g T y 0 i jbj; y 0, γ w c i by y 0 i i= 0 1 y0 g T y 0 i jbj; y 0, γ w a i y y 0 w c y y 0. y0 y0 g T y 0 i jbj; y 0, γ g T y 1 jbj; y 0, γ w c y y 0 1 y0 g T y 1 jbj; y 0, γ w a w c y y 0 i= y0 y0 g T y 0 i jbj; y 0, γ w a w c g T y 0 i jbj; y 0, γ w a w c y 0 i= 0 1 y0 g T y 0 i jbj; y 0, γ. Next, we demonstrate that Cy is convex in y by showing that Cy Cy 1 for any y y 0, which is eqivalent to 0 b g T y 0 i jbj; y 0, γ w a w c b 0 1 y0 g T y 0 i jbj; y 0, γ w a w c i= 0 1 i= 0 2 y0 g T y 0 i jbj; y 0, γ y0 g T y 0 i jbj; y 0, γ. Rearranging terms, we have y0 y0 b g T y 1 jbj; y 0, γ w a w c g T y 1 jbj; y 0, γ, which redces to the relationship, w a w c b, that we have assmed regarding the costs. De to the convexity of Cy in y, we know that y = min y N y y 0, Cy 0

25 24 minimizes 7. We rewrite Cy 0 as 0 y0 b g T y 0 i jbj; y 0, γ w a w c b 0 y0 g T y 0 i jbj; y 0, γ w a w c 0 y 0 Ths the minimizer of 7 is given by y = min Proof of Proposition 2. y N y y 0, g T y 0 i jbj; y 0, γ 0 y 0 0 y 0 i= y0 g T y 0 i jbj; y 0, γ y0 g T y 0 i jbj; y 0, γ b w a w c w a w c g T y 0 i jbj; y 0, γ w a w c b w a w c. g T y 0 i jbj; y 0, γ w a w c b w a w c We compare sample paths to show that. E η,ω [L O y; η, σ] E η,ψ,ω [L O y; η ψ, σ], 20 where ω is the vale being copled representing the home nit for the marginal aide in the on-call pool i.e., η 0 1 = ω on the left-hand side of 20 and η 0 2 = ω on the right-hand side. The two sample paths being compared wold also share η, and η 0 1 on the right-hand side of 20 cold take any vale ψ 1,...,. To illstrate with an example for y y 0 = 3, the sample path cold have η = 6, 2, 1 and both η 0 1 on the left-hand side and η 0 2 on the right-hand side eqal to 4. We show that 20 holds becase it holds for the sample paths being paired, which by sbstittion sing 13 and becase η 0 1 on the left-hand side of 20 is eqivalent to η 0 2 on the right-hand side is eqivalent to σ k > 1 η j = k, σ k > 1 η j = l 1 σ k > j=1 0 1 j=1 1 η j = k, j=1 l=1 0 1 σ k > j=1 1 η j = l, 0 1 which holds becase j=1 1 η j = k 0 j=1 1 η 0 1 j = k and j=1 l=1 1 η j = l l=1 1 η j = l on any sample path. Becase this comparison holds on all pairings for the home nits 0 j=1 η 0 1 and η 0 2 of the marginal aides, it holds in expectation. Proof of Corollary 2. Taking the first forward difference of L det O y sing 4, we have L det O y =L det y 1 L O y O 0 1 = Bi; y y 0 1, 1/ Bi; y y 0, 1/ 0 l=1 q Bj; q, γj i j=i q Bj; q, γj i. 21 j=i

26 25 By the definition of the binomial distribtion, we note that Bi; y y 0 1, 1/ = 1 Bi; y y 0, 1/ 1 Bi 1; y y 0, 1/, which we sbstitte into 21 to get L det O y = Bi; y y 0, 1/ 1 q Bi 1; y y 0, 1/ Bj; q, γj i Bi; y y 0, 1/ 0 q Bj; q, γj i. Noting that By y 0 1; y y 0, 1/ = 0 and sbtracting terms, 0 L det O y = j=i 1 Bi; y y 0, 1/ q Bj; q, γj i j=i Bi 1; y y 0, 1/ j=i q Bj; q, γj i. Becase Bi; y y 0, 1/ = 0 for i < 0, we can change the smmation index to get 0 L det O y = 1 Bi; y y 0, 1/ 0 1 Bi; y y 0, 1/ j=i q Bj; q, γj i j=i q j=i1 We combine terms, noting that Bj; q, γj i = 0 for j = i, to get 0 L det O y = Changing the order of smmation, we have Proof of Proposition as where 1 Bi; y y 0, 1/ q j=i1 Bj; q, γj i 1. Bj; q, γj i j i 1 = 0 Bi; y y 0, 1/ q j=i1 Bj; q, γ. L det O y = q j=1 Bj; q, γ j 1 Bi; y y 0, 1/. De to the operational characteristics of the restricted on-call pool, we can write E [L η,η0 1 Ry; η, σ] = 1 σ k > 0 H i; 1, 1 σ k > 1σ k > 0 y y0 σ k 1 min 1, y y 0 σ k, 0 y y0 /, 22 represents the probability that the home nit of the marginal aide in the on-call pool has a shortage when there are y y 0 1 aides in the on-call pool. The remainder of the expression represents the probability that the marginal aide is not blocked by reallocated schedled aides for the case that the marginal aide s home nit does have a shortage. The marginal aide is blocked if the nmber of