- Email: [email protected]
Cyclical scheduling and allocation of nursing staff
So&-Econ.
Plan.
Sri.
Vol. 7,
pp. 471-487
CYCLICAL
(1973).Pergamon Press.Printedin GreatBritain
SCHEDULING AND ALLOCATION NURSING STAFF
CHRISTOPH MAIER-ROTHE*
OF
and HARRY B. WOLFE
Arthur D. Little, Inc., Acorn Park, Cambridge, Massachusetts
02140
(Received 27 July 1972)
Nursing salaries make up the largest single element in hospital costs. Thus, the effective allocation of available nursing staff to patient care requirements is an important tool in controlling the cost of health care. This paper describes an allocation procedure consisting of two parts : 1. 2.
A cyclical scheduling procedure designed to meet average staffing requirements in a manner consistent with hospital personnel policies and employees’ preferences. A procedure for adjustment of staffing twice daily in accordance with an index of the latest patient care requirements.
The procedure is now in use in a major hospital where it has demonstrably staff allocations.
improved nursing
INTRODUCTION
about onein the hospital budget, typically representing third of the total, is the salaries of the nursing staff. If we are to make any inroads on the current escalation of hospital costs, we should begin with the most effective and efficient possible utilization of the nursing staff. We can identify three stages in determining the utilization of nursing staff: THE LARGEST single cost factor
1.
2.
3.
Forecasting and planning. By this we mean the forecasting of overall nursing staff requirements over an extended period (a year or more), taking into account such considerations as seasonality of patients, staff vacation requirements, and new patient care programs. Scheduliplg. This refers to the preparation of staff schedules, shift-by-shift, taking into account staffing requirements and individual constraints. These schedules are typically developed for periods two to eight weeks ahead. Allocution. This is the process of allocating and reallocating nursing staff on a day-today and shift-to-shift basis, making minor adjustments to the schedule to take into account short-term fluctuations in patient requirements and staff deviations from the schedule.
This paper is concerned with the latter two elements; it describes a formal system for the scheduling and allocation of nursing staff. As indicated in the references [l-13], a number of systems have been developed to facilitate the scheduling and allocation processes; many of these are computer-based. Very few * Currently located at Arthur D. Little, Wiesbaden, Germany. 471
472
CHRISTOPHMAIER-ROTHE and HARRY B. WOLFE
of these systems, however, have won acceptance and been successfully implemented, and the vast majority of hospitals still schedule and allocate nurses on an informal basis. The rejection of more systematic methods appears to reflect in part a set of beliefs that nursing, as a profession, is not amenable to formal scheduling and allocation procedures, that too many variables must be taken into account, that most nurses are non-quantitatively oriented and would therefore resist any system involving computers, and possibly in part the failure of previous efforts to take into account some variables that were important to nurses. In this paper we describe the development, application, and successful implementation of a scheduling and allocation system in the medical department of a large teaching hospital. In addition to the technical aspects of the system, we discuss the variables that were included in its development and how these variables were identified, the steps involved in implementation, and the measured benefits. The system as currently used is entirely manual and operates very satisfactorily in this mode; it could be computerized should this be desirable. THE
SCHEDULING
AND
ALLOCATION
PROBLEM
The basic problem of scheduling is to provide patient care every day around the clock using nurses who generally work five days a week, one shift per day, and prefer to have weekends off. Scheduling is usually done by nursing supervisors for the units of floors for which they are responsible. They estimate patient care requirements and allocate the available nursing staff to the days of the week so that these requirements are approximately satisfied and hospital personnel regulations observed. They try to schedule the nursing staff so that each nurse gets her share of weekends off and none of the nurses is rotated to evenings or night shifts for an unduly long time, and they also try to accommodate individual nurses’ requests for specific days off. A schedule is prepared every other week, specifying work days and days off for each number of the nursing staff over the ensuing two weeks. Preparation of the schedule is a time-consuming task for the nursing supervisor. In addition, most hospitals seem to feel some dissatisfaction with the results: 1.
2.
3. 4. 5.
Coverage tends to fluctuate widely, particularly in the case of RN’s; the number of RN’s assigned to a given unit on a given shift may vary from day to day by a factor of two or more. In their position of total authority over schedules, supervising nurses are apt to be suspected of favoritism, particularly when they try-as is often necessary and desirable -to take into account the desires and constraints of individual nurses. The attempt to resolve scheduling and allocation problems can result in excessive use of overtime. Nurses often feel that there is excessive rotation. Schedules frequently have to be changed on short notice because of changes in patient care requirements, illness of a nurse, etc.
Most of these problems result from the complexity of the scheduling decision and the difficulty of taking all relevant factors into account. We have spent considerable time discussing the scheduling problem with nurses and hospital administrators. From these discussions, the following have emerged as the most important factors in the decision. 1.
The different levels of nursing staff-registered nurses, licensed practical nurses, and nurses’ aides-have different capabilities; LPN’s and aides are legally allowed to per-
Cyclical Scheduling and Allocation of Nursing Staff
2.
3. 4. 5. 6.
7.
form only certain functions. Even within these categories, individuals can be classified by degree of experience and by the amount of responsibility they are able to assume. Nursing coverage must, of course, be provided 24 hours a day, 7 days a week. Nursing requirements are typically lower during the evening and night shifts than during the day shift. Saturday and Sunday requirements tend to be 20-30 per cent lower than weekday requirements, depending on the medical service, due to a lower patient census, fewer new physicians’ orders, or both. Vacations and time off for holidays must be staggered to ensure continued patient coverage and equitable treatment for nurses. Weekend days off are highly prized by the nursing staff, preferably both days in a row. Next in preference is two or more days off in a row in the middle of the week. Long stretches of consecutive working days (usually defined as more than five in a row) are undesirable. Despite a salary differential, the evening and night shifts are more difficult to staff than the day shift. As a result, schedules must provide for “rotation”; daytime staff must work on the other two shifts from time to time. Also, staff on evening and night shifts may sometimes have to work on the day shift in order to attend special programs, training courses, etc. Most nurses prefer to remain on one nursing unit, rather than being “floated” or shifted from one unit to another, partly because of the competence and expertise they are able to develop in a particular unit, and partly because of the camaraderie of continued association with the same group. However, there are some nurses who do not mind, and in fact prefer, floating because of the variety of experience it gives them.
The objectives of a scheduling days off to individual members 1. 2. 3. 4.
473
and allocation procedure are to assign working of the nursing staff so that:
days and
Adequate patient care is assured while overstaffing is avoided. A desirable distribution of days off is achieved. Individual members of the nursing staff are treated fairly. Individuals know well in advance what their schedules are.
In the project described here, different approaches were adopted to the two parts of the problem: the medium-term scheduling task and the short-term allocation task. For scheduling we used a cyclical approach, with a schedule that covers a designated number of weeks (cycle length) and then repeats itself. The cyclical schedule assigns the required registered nurses, licensed nurses, and total staff to each unit in a manner consistent with average patient care requirements, hospital personnel policies, and the nursing staff’s preferences for the distribution of days off. The short-term ,allocation process, for adjusting the cyclical schedule in response to changing patient care requirements and available staff, is based on a system of classifying patients by severity of illness to determine nursing requirements in each unit.
THE PRINCIPLE
OF CYCLICAL
SCHEDULING
Table 1 shows a simple cyclical schedule. There are five nurses and the cycle length of the schedule is five weeks. Days off are denoted by D. The first line indicates working days and days off for the first nurse, the second line for the second nurse and so forth. The first nurse has the first Sunday off, then a Thursday, then she has Tuesday and Wednesday off,
474
CHRJSTOPH MAIER-ROTHHE and HARRY B. WOLFE
Cyclical Scheduling and Allocation of Nursing Staff
415
and so on until the five weeks are up, at which point the entire pattern of working days and days off begins again. The overall pattern of working days and days off is exactly the same for all of the nurses; the only difference is that it begins in a different week for each nurse. Thus, the schedule followed by the first nurse in the first week is followed by the second nurse the following week, the third the week after, and so on. The weeks start with Sunday and end with Saturday. Each nurse works five days a week, works at most four days in a row, and has two out of five weekends off. Four of the five nurses are always present except on Tuesdays and on weekends, when only three are present. In a cyclical schedule, the total number of people present on any given day of the week is the same for all weeks when all cycles of the schedule are filled, that is, when the number of people on the schedule is equal to the length of the cyclical pattern in weeks. This can be shown quite easily as follows: Let a pattern of working days and days off be described by a sequence of variables yi where 1 ” = I0
ifdayiisoff otherwise
i=l,2,...7n
(1)
and II is the number of weeks covered by the pattern. Suppose now that a cyclical schedule is constructed for N individuals such that individualj starts the given pattern rj (1 < P, < n) weeks after day 1. For instance, if r, is equal to 2, individual 3 starts the pattern on day 15. The total number of individuals absent on day i is then simply ..
Di =
c
Yi-7ri
(2)
Yi*
(3)
j=l
where r1 is equal to zero and for any integer k Yi+7kn
The requirement that the total number the same for all weeks means that
=
of individuals
absent
on a given day of a week be
Di+7k = Di
(4)
for any integers i and k. The pattern given by expression (1) is defined such that it cannot be broken down into two or more patterns that repeat themselves. Now, if we assume for simplicity that rj+l = rj+l then requirement
(5)
(4) becomes N c j=l
N Yi+7(k+l-_j)
By leaving out the terms common
=
c j=l
Yi-7(j-1).
(6)
to both sides, this can be reduced to k-
c
I
N-l Yi-7j+7k
=o
=
c
Yi-7j.
(7)
j=N-k
Requirement (7) is true for arbitrary integers i and k only if N is an integer multiple In this case, expression (3) can be used to reduce the condition to an identity.
of II.
476
CHRISTOPH MAIER-ROTHE and HARRYB. WOLFE
In the example of Table 1, we did not distinguish between charge nurses, registered nurses, licensed practical nurses, and aides. Since there must always be a nurse present who can take charge of the unit and since there must always be present a certain number of registered nurses and licensed nurses regardless of the number of aides, the assignment of individual nurses to the cycles of a schedule must be planned with care. STAFFING
REQUIREMENTS
AND
PERSONNEL
POLICY
A schedule must meet staffing requirements and must be consistent with the hospital’s personnel policies. Patient care requirements fluctuate from day to day and even from hour to hour; minimum staffing requirements consist of the minimum number and the mix of nursing staff required to accommodate average patient care requirements. There is no sure way of determining minimum staffing requirements objectively without extensive work sampling and analysis of a great number of different activities. Such an extensive investigation may not be justified solely for the purpose of staff scheduling, in which case one must rely on expert opinion, and on obser\lation. It is not always easy to obtain agreement on minimum staffing requirements. Estimates from nursing supervisors tend to be somewhat biased in favor of more people, while estimates from the hospital administration may be biased in the opposite direction. It is thus advisable to consult both sources. If patient care requirements fluctuate widely from day to day, it may also be worthwhile to analyze actual staff coverage over an extended period of time and to conduct interviews to find out what happened on days with a particularly large or small number of nurses on duty. In the present case we combined the above approaches, working jointly with the Assistant Director of Nursing. The agreed-upon absolute minimum staff per unit for a weekday shift was two registered nurses, two licensed practical nurses, and three aides, in addition to the head nurse. On Sundays, because of the generally reduced level of activity, this could be reduced to one head nurse, one registered nurse, one licensed practical nurse, and two to three aides. The workload appeared to be heaviest immediately before and after the weekend, tapering off somewhat in midweek. The requirements for Saturdays were somewhere between the regular weekday requirements and those for Sundays. Requirements for evening and night shifts, worked out in the same way, were much lower than for any of the day shifts. Hospital personnel policies usually reflect agreements reached between the administration and the nursing staff regarding the distribution of days off for nurses. These agreements specify, among other things, the precise beginning of the work week, the number of days per week that must be given off, the maximum number of working days in a row, how many holidays are to be given off, the duration of vacations, and how many weekends are to be given off. They limit the types of cyclical patterns that are feasible. For instance, if four days off every two weeks are specified instead of two days off each week, the possible range of cyclical patterns is considerably increased. In our case, the work week started with Sunday morning and ended with Saturday night. Each nurse had to have two days off every week and these had to be a weekend at least every third week. Charge nurses and alternate charge nurses were given every other weekend off. After a working weekend, the following Monday was to be given off if possible. The nurses preferred two days off in a row over two single days off; if single days were to be given off, they preferred Fridays and perhaps Thursdays over other week days. Supervisors, registered nurses, and licensed practical nurses were eligible for 20 days of vacation per year. Other employees were eligible for 15 days annual vacation during the
477
Cyclical Scheduling and Allocation of Nursing Staff
first 8 years of service and 20 days thereafter. Each nurse was allowed one holiday per month, and could save up to three holidays to take later. In addition, all employees were entitled to paid sick leave, earned at the rate of one day each month of employment. On the average, each nurse spent 6 days per year in a training program. Thus, a licensed nurse (that is, a registered nurse or a licensed practical nurse) was available for floor duty 211 days per year and an aide was available 216 days per year. Under the circumstances, assuming that holidays, sick leaves, and training are distributed evenly through the year, the total number of licensed nurses required to ensure the agreed-upon minimum coverage of a shift is given by: l/7 X 3651211 2 Ri = 0.247 2 RI i=1 i=l and the total number
of aides by: 117 X 3651216 2 Ri = 0.241 2 Ri i=l i=l
where R, is the required
number TABLE
2.
of licensed
nurses or aides on duty on day i of the week.
STAFFING REQUIREMENTSFOR A DAY SHIFT
Type of nursing staff
Weekdays
Registered nurses Total licensed nurses Total staff including aides
3 4 I
Staff to be present on Saturday Sunday 2 4 7
2 3 6
Minimum total staff requirement 4.68 6.91 11.98
Table 2 shows the results for the day shift. Equivalent figures were constructed for the evening and night shifts. A minimum of twelve people was necessary to ensure full coverage of a day shift. At least five out of the twelve had to be registered nurses and at least seven altogether had to be licensed. The minimum therefore consisted of five registered nurses, two licensed practical nurses, and five aides. SELECTION
OF THE
CYCLICAL
PATTERN
Before a cyclical schedule can actually be assembled, patterns of working days and days off must be found which are consistent with requirements of the kind outlined above. In this case, at least twelve people were required to staff a day shift. If the cycle length were set at 12 weeks, then each member of the staff could be assigned to a separate cycle. Since no shift required more than twelve people, there was no need to consider cycles more than 12 weeks long. On the other hand, since we knew that, on a fair number of days, there would be fewer than twelve people available, in view of the uneven distribution of vacations, illness, training, and holidays throughout the year, we also considered the possibility of a IO-week cycle. We decided to examine scheduling alternatives for three basic alternatives: a IO-week cycle with four weekends off, a 12-week cycle with five weekends off, and a 12-week cycle with four weekends off. Feasible patterns of working days and days off could be found through some algorithmic approach such as integer programming. However, since it is not very difficult to find S.B.P.S. 7/5-F
CHRISTOPH MAIER-ROTHE and HARRY B. WOLFE
478
feasible solutions by trial and error and since there were in this case no clear trade-offs among long weekends, the number of two-day periods off, the number of working days in a row, and so forth, it was decided to use a heuristic approach to generate a number of feasible alternatives. In formulating these alternatives we tried to anticipate preferences of the nursing staff. We then presented the alternatives to the nurses and let them decide which they preferred. The finding of feasible alternatives is facilitated by formalizing some of the constraints. The variables Ui (i = I, 2, . . . , 7) denote the number of times during one cycle that day i of the week is off as a single day, ui (i = 1,2, . . . , 6) denote the number of times during one cycle that days i and (i+l) of the week are off in a pair. The variables y, and y, denote the number of three-day weekends off including a Monday or Friday respectively, and z denotes the number of 4-day weekends off. Since at least every third weekend must be given off, the entire pattern must consist of two and three-week stretches between weekends off. The parameter pz denotes the number per cycle of three-week stretches between two successive weekends off and the parameter p denotes the total number of weekends off during one cycle. One cycle contains n weeks. Table 3 shows the constraints that the variables Ui, vi, yi, and z must satisfy in order to form a feasible pattern. These constraints are explained below. Although they are not sufficient to determine a feasible pattern, they are necessary, and they proved valuable in the systematic exploration of feasible alternatives. A complete cycle of n weeks must contain 2n days off. Of these, 2p are on weekends and therefore 2(n-p) must be on weekdays. Constraint (1) of Table 3 expresses this relation. Constraints (2) through (8) are simply balance equations for the individual weekdays. Constraint (9) is a balance equation for weekends. Since only a three-week stretch (between TABLE 3. CONSTRAINTSET
111111112 2 1 1 3 4 5 6 7 8 9 10 11 -1 12 13 -1 14 -1 -1 15 -1 -1 -1 16 -1 17 -1 -1 -1 18 19 20 21 22 23 24
FOR MAXIMUMOF
2 1 1 1
1 1
1
5 WORKING
2
2
DAYS
2
IN A ROW
2
1 1
1 1
1
1 1
1
1 1
1 1
1
1 1
1
1
1
-1
1
-1 -1
-1 -1 -1
-1 -1 -1
1 1 1 1 1 1
-1 -1
-1 -1 -1 -1
-1 -1 -1 -1 -1
1
-1
-1
-1 -1 -1 -1
1
-1 -1 -1 -1 -1 1
1 1 1 1 1
1 1
1 1 1 1
1 1 1
1 1
1 1 1
1 1 1 1
1 1 1 1 1
1 1 1 1 1 1 1
-1
1
2 = 2(n-p) = n--p-c1 1 = n-c, = n-CJ = n-C4 = n-C5 1 1 = n-C6 = n-p-c, 1 1 =p = Pz l=O 1 l=O = 0 =O =o =o =o -1 --1 =O =o =o =o =o =o -1 =o
479
Cyclical Scheduling and Allocation of Nursing Staff
weekends off) can contain a two-day period off, the total number of two-day periods off is less than or equal to pz ; constraint (10) states this. Constraints (11) through (24) relate specifically to the requirement that not more than five working days in a row are allowed in this case (for the preparation of the feasible alternatives that contain stretches of six working days in a row, a different version of constraints (11) through (24) was used). Constraint (11) states that a four-day weekend and a three-day weekend including Monday must always be followed either by a single Sunday off or by a 2-day period off including Sunday and Monday (it cannot be followed by a weekend off). Constraint (12) expresses a similar requirement for Fridays. If a 2-day period including Sunday and a Monday is off, then if not more than five working days in a row are allowed, the following Sunday must be off. Therefore, constraint (13) states that the total number of 2-day periods off including Sunday and Monday (ur) must be less than the total number of single Sundays (~1~)off. These requirements are repeated for the individual days of the week, constraints (11-18) relating to the subsequent days off and constraints (19-24) relating to preceding days off. Fourteen feasible alternatives are shown in Table 4. The first four describe a IO-week cycle with a total of four weekends off, the next four a 12-week cycle with a total of five TABLE 4. FOURTEEN ALTERNATIVE CYCLICALPAITERNS
Length of cycle (weeks) No. of alternative No. of 4-day weekends off Relative frequency (%) No. of 3-day weekends off Relative frequency (%) Total No. of weekends off Relative frequency (%) No. of 2-day periods off Relative frequency (%) No. of single days off Relative frequency (%) No. of 6-workday stretches Relative frequency (%) No. of S-workday stretches Relative frequency (%)
10
12
1 234 1 000 10 0 0 0 0210 0 20 10 0 4444 40 40 40 40 2222 20 20 20 20 6678 60 60 10 80 2210 20 20 10 0 2210 20 20 10 0
5678 1 000 8 0 0 0 0210 0 17 8 0 5555 42 42 42 42 2222 17 17 17 17 8910 8 67 67 75 83 2210 17 17 8 0 2210 17 17 8 0
12 9 10 1 0 800000 122110 8 17 444444 33 33 3 4 25 33 768798 58 50 420100 33 17 343222 25 33
11 0
12 0
13 0
14 0
17
8
8
0
33 3 25
33 4 33
33 3 25
33 4 33
67
58
75
67
0
8
0
0
25
17
17
17
weekends off, and the last a 1Zweek cycle with four weekends off. For easier comparison of the individual alternatives, the relative frequencies of occurrence of four-day weekends off, three-day weekends off, and so forth are shown. Table 5 shows the pattern of days off for four of the alternatives. The nurses preferred the cycles with more weekends off and therefore rejected the 12week schedule with four weekends off. They were initially quite excited about the possibility of long weekends, a feature offered only rarely under the old scheduling procedures, but their enthusiasm was quickly dampened when they realized, however, that these could usually only be achieved by allowing longer stretches of working days in a row. The nurses were originally shown an earlier version of the figure in which some of the alternatives involved stretches of seven working days in a row. These were rejected immediately, and one group, consisting primarily of somewhat older staff members, considered also unacceptable those alternatives involving six days.
CHRISTOPH MAIER-ROTHE and HARRY B. WOLFE
480
TABLE 5. DAYS OFF IN PATTERNS 1,4,
1
SIlrKi2V
-------, ~-~
4
/ (,1;
1
4
(,
.
1
11 I
Monday
1___
II Wednesday!
I
’
II
-6 ~-
I
4
~_. ._ _.~__~
6
A
i-
4 II
Tuesday
1
A
-11
6 AND 1 I
6
-1
--
6-,I4
II
6
6
II _~___
1 4
4
II
6 II
II Friday ___ Saturday__
1 611
4
Thursday _ ___~__
( 1
1
6
II
4
4 II,
h ii
I 4 6
14, 6
I
II
11
6
6
4
6 II
6 II
The advantage of presenting to the nurses alternative cyclical patterns such as those in Table 4 is that the trade-offs between desirable distributions of days off on the one hand and undesirable stretches of working days on the other are made quite explicit, and the choice is made by the nurses rather than imposed upon them. After several discussions with the nursing staff, it was agreed that alternative (4) was to be used for some floors and alternative (8) for others. At the same time, cyclical patterns for the evening and night shifts were introduced. In the latter two cases, the selection process was considerably simpler because of the comparatively small number of feasible alternatives. Once each cyclical pattern was selected, the individual nurses of a floor had to be assigned to the different cycles of the selected pattern. This process is described in the following section for alternative (4), a IO-week schedule with four weekends off. ASSEMBLY
OF
A SCHEDULE
The first step in assembling the schedule was to assign individual nurses to the cycles of alternative (4) in a manner that would provide the mix of staff specified in Table 2. A careless assignment would probably not have provided a satisfactory distribution of registered or licensed practical nurses. In cycle scheduling, the best way to avoid bad assignments is to begin by finding cycles that have as few common days off as possible during the full 10 (in this case) weeks. If four registered nurses are available in addition to a head nurse and an alternative charge nurse, then these four should be assigned to four cycles for which there is a minimum of duplication in days off and, if possible, no day off common to all four. Similar optimum combinations of two and three cycles must be found for staff categories in which only two or three people are available. Only after the cycle combinations have been established can individuals be assigned to any of the cycles. If, as in this case, weekends off are more frequent than one in three, and, at the same time, staffing requirements on weekends are not substantially lower than during weekdays, the
481
Cyclical Scheduling and Allocation of Nursing Staff
days are likely to be Saturday and Sunday. Cycle combinations that do not provide adequate coverage on weekends need not be considered even if weekday coverage happens to be very good. No easy analytical method was found to determine the best and other good combinations of two, three, and four cycles. Therefore, the desirable cycle combinations were determined by enumeration. This is not as cumbersome as it sounds because there are fewer distinct cycle combinations that one might intuitively think. All permutations of a given cycle combination are, of course, equivalent. If the ten-week pattern of alternative (4), as shown in Table 5, is designated cycle 1 and the same pattern starting one week later is designated cycle 2, and so forth until cycle 10, then we can describe any combination of m cycles by an m-tuple (i1, iz, . . . , i,,,). Thus the 3-tuple (1, 5, 7) describes the combination of the three cycles 1, 5 and 7. It is clear that, if we add the same number to each of the elements, we obtain an equivalent combination of cycles. When we obtain a number greater than ten, we recall that cycles i and (i+lO) are exactly the same. Therefore, the cycle combinations (3, 7, 9), (1, 5, 9), and (3, 5, 9) are all equivalent to (1, 5, 7). They are simply different arrangements of the same combination. For a lo-week cycle, there are exactly five distinct two-cycle combinations, twelve distinct three-cycle combinations and twenty-two distinct four-cycle combinations. For an n-week cyclical pattern there are always at least (n-l) !/(m-n)! m ! m-cycle combinations and at most n of these are equivalent. critical
TABLE 6. DETERMINATION OF THE BESTTHREE-CYCLE COMBINATION
Distinct 3-cycle combination 1 1 1 1
1 1 1 1
1 1 1 1
2 2 2 2 2 2 2 3 3 3 3 4
3 4 5 6 7 8 9 5 6 7 8 7
Weekends with 0 1 2 3 (people) 0 0 0 0 0 0 0 0 2 0 2 0
2 4 2 4 4 2 4 4 2
Weekdays with 0 1 2 3 (people)
8
0
0
1
8
0
0
6
8
0
0
6
8
0
0
2
34
15
30
18
4
Table 6 lists all distinct three-cycle combinations, showing for each combination how often all three people are present, only two people, only one person, and no one at all, for weekdays and weekends over one entire IO-week cycle. These evaluations were made with the help of transparent overlays, which later also facilitated the actual scheduling process. Coverage posed more of a problem on weekends than on weekdays, so combinations with poor weekend coverage were not further examined. As the table shows, the two combinations (1, 2, 3) and (1, 3, 7) provide almost equal coverage. The coverage on weekends is the same, and since the coverage on weekdays is slightly better for the former, this was selected as the best three-cycle combination. Table 7 shows the coverage it provides for
CHRISTOPHMAIER-ROTHE and HARRY B. WOLFE
482
TABLE 7. DAILY COVERAGEPROVIDEDBY THE BESTTHREE-CYCLECOMBINATION u,2,
3)
Week Day Sunday Monday Tuesday Wednesday Thursday Friday Saturday
1
2
3
4
5
6
7
8
9
10
2 3 3 1 2 3 2
2 2 2 2 3 3 I
1 2 3 2 3 2 2
2 2 2 2 3 2 2
2 2 2 2 3 2 2
2 2 2 2 2 3 2
2 2 2 3 2 3 1
1 3 2 3 2 2 2
2 3 2 2 2 2 2
2 3 2 2 2 2 2
each day of the 10 weeks. The number in each field corresponds to the number of people present during that particular day of the cycle if one person is assigned to each cycle. The best two-cycle and four-cycle combinations were similarly calculated. The fourcycle combination was particularly important since four registered nurses and four aides were available on the floor in question; Table 8 shows the coverage provided by the best combination.
TABLE 8. DAILY COVERAGEPROVIDEDBY THE BESTFOUR-CYCLECOMBINATION (1, 3, 5,7)
Week Day Sunday Monday Tuesday Wednesday Thursday Friday Saturday
1
2
3
4
5
6
7
8
9
10
3 3 3 2 3 4 2
2 3 3 4 3 3 2
2 4 3 2 3 3 3
3 3 2 3 4 3 2
2 3 4 2 3 3 3
3 3 2 3 3 4 2
2 3 3 3 4 3 2
2 3 3 3 3 3 3
3 3 3 3 3 3 2
2 4 2 3 3 3 3
After the necessary combinations were identified, the schedule was easy to assemble. Exactly twelve people were available for the day shift. Since the head nurse and the alternative charge nurse were allowed every other weekend off, they were not included in the cyclical schedule. Four registered nurses were assigned to the day shift in addition to the head nurse and alternative charge nurse. Two practical nurses and four nursing aides completed the staff for this floor. By comparing these numbers with the requirements of Table 2, we can see that the staffing requirements were met. The resulting schedule is shown in Table 9. The four registered nurses were assigned to cycle combination (1, 3, 5, 7), the best four-cycle combination. The two practical nurses were given cycles 9 and 10, and combination (2, 4, 6, S), which is equivalent to the combination used for registered nurses, remained for the nursing aides. In this particular case, we did not need the best three-cycle combination but for other floors we did. The only step that remained to make this schedule operational was to determine which individual would be assigned to which cycle within each of the three combinations. We
Cyclical Scheduling and Allocation of Nursing Staff
LL
c 3
483
CHRISTOPHMAIER-ROTHEand HARRY B. WOLFE
484
avoided starting anyone at a point that would resuh in either two successive weekends off or two successive working weekends. Thus, no one felt unfairly treated.
ALLOCATION
PROCEDURE
Patient care requirements change from day to day and even from one hour to the next. The cyclical schedule guarantees a certain minimum coverage but does not take into account ~uctuations in patient care requirements. It is evident that the reiative ~u~tuations on one floor are larger than those of an entire division consisting of several floors. Thus, these fluctuations can be accommodated by short-term assignment of nurses from a “float pool” to those floors that need them most. This is the only way to provide extra coverage without permanent overstaffing. In special situations when peaks of patient care requirements on one or severaI floors cannot be accommodated from the float pool, individuals may be temporarily shifted from floors with lighter requirements to those with heavy requirements. However, this should be considered only as a last resort since it reduces coverage below desirable levels on the floors from which the extra staff are obtained; also, nurses generally dislike being “pulled”. Since these adjustments in the schedule are based on patient care requirements, we needed a means of assessing these requirements. The approach had to be simple enough so that requirements could be established at least once a day without substantial effort. In cooperation with the nursing staff, we developed a simple classification scheme for patients. According to this scheme, patients are classified into four categories: acute care, complete care, partial care, and ambulatory care. Table 10 defines these categories. The differences TABLE 10. PATIENTCLA~IFI~ATION
Acute care:
Needs constant observation; frequent vital signs, constant monitoring, and/or suctioning; may be on dialysis; may need respirator; warrants close medical supervision and continuous nursing care.
Complete care:
Needs assistance with entire care (may require special treatment, dressing, vital signs, monitor); may or may not be able to feed self (some may be incontinent).
Partial care:
Some assistance required with physical care; may need special treatments procedures done by nursing staff.
Ambulatory
care:
or other
Physically able to care for his personaf needs.
in patient care requirements among categories are quantified by assigning relative values: 4, 3, 2 and 1 for acute, compIete, partial, and ambulatory care, respectively. If a private duty nurse is assigned to a patient, it is assumed that only haif the normally required care is provided by the regular staff. In addition, discharges are given a relative value of 0.5. The total relative value per floor is then calculated by multiplying the number of patients in each category by the relative value of the category and adding the resulting products. The system operates as follows. Patients are classified twice a day, at 6 a.m. and 2 p.m., to provide guidance for the day and evening staffing respectively. The patient classification forms {Table II> are filled out by the unit floor supervisor (it typically takes only a few
485
Cyclical Scheduling and Allocation of Nursing Staff TABLE 11. PATIENT CLASSIFICATIONFORM
No.of patients
Classification
Specials*
Rel. value
Total value
4 3 2 1 0.5
Acute Complete Partial Ambulatory Expected discharges
* For patients with specials subtract half the relative value from the total.
minutes to fill them out) and forwarded to the nursing supervisor for all medical units. The nursing supervisor or her secretary or assistant copies the relative patient values for each unit onto a form (Table 12), notes the number of available staff (number of assigned staff TABLE 12. STAFF ALLOCATION PROCEDURE
Floor
Relative patient values
Available staff
Calculated staff
Assigned staff
V= Float Total staff
R=s&fi= Values/staff
I. 2. 3. 4. 5. 6.
Copy the total relative patient values from the individual floor sheets to column 1 of the staff allocation form. Total these to obtain V,the total patient values in the house. Write down the available assigned staff on each floor in column 2. Add the available float staff to obtain total available staff. Calculate R = total relative values available staff = average value per staff. Calculate the required staff per floor by dividing relative patient value for the floor by R. Round off the nearest integer. The total of the calculated staff should equal total available staff. There may be a discrepancy of 1 due to the rounding off. If so, add or subtract 1 from any appropriate floor. In the final column write down the staff actually assigned to each floor.
per unit minus no-shows), totals these, and adds the sum to the float staff available to get total available staff. She then divides ‘total relative values by total available staff to find the workload index: the number of relative value points per staff member. By then dividing the total relative value for each floor by the workload index, she determines how many nurses should be assigned to each floor to equalize the index. The staffing supervisors then assign the float staff to the individual floors, taking into account their knowledge of the
486
CHRISTOPH MAIER-ROTHE and HARRY B. WOLFE
type and quality of assigned staff on each floor and of the float pool. If further adjustment is necessary, nursing staff may be “pulled” from one floor and assigned to another. In order to provide the desirable flexibility while minimizing the amount of “pulling” from one floor to another, it is desirable to build up a large float pool. Part-time staff can be automatically assigned to the float pool, nurses who enjoy floating should be encouraged to remain in the pool, and new nursing staff should whenever possible be hired for the pool rather than for specific floor assignments. RESULTS
In the hospital where we worked, most nurses, especially the supervisors who did the nurse scheduling, recognized the need for improved scheduling. Most also knew something about cyclical scheduling before we started our assignement. With few exceptions, they initially had a fairly strong resistance to cyclical scheduling, primarily because they associated the idea of a repetitive pattern with extreme inflexibility. The joint sessions during which we constructed schedules for alternatives (4) and (8) convinced them of the contrary. Once they learned to take advantage of the self-balancing effect of the different cycles of the selected pattern, they found not only that the scheduling process was much simpler than before, but also that it provided for more even coverage and a more just distribution of days off. Since the cyclical schedule has a certain built-in safety margin above the minimum staffing requirements, small deviations from the schedule can usually be accommodated. When this is not the case, a simple switching of cycles between two individuals of the same category will provide the desired changes. The net effect of cyclical scheduling is probably to increase flexibility rather than to decrease it. To assess the effectiveness of the cyclical scheduling procedures, cyclical schedules were developed retrospectively for all three shifts of two floors for a period of eight weeks. These were then compared with the actual plans as originally prepared by the nursing supervisors. An effort was made to incorporate into the cyclical schedule all changes in the staffing of a floor that were reported on the scheduling sheets. Days spent in orientation, for instance, were counted as regular nurse-days under both scheduling schemes. The eight-week period covered by the comparison contained 168 shifts (days, evenings, and nights) for each of the two floors. According to the scheduling sheets and the minimum staffing requirements, forty-four shifts of one floor and seventy-four shifts of the other floor could be considered understaffed. The cyclical schedules reduced these cases of understaffing from forty-four to two on the first floor and from seventy-four to forty-nine on the second. They also, of course, reduced the number of shifts that could be considered overstaffed in view of the available nursing staff. In addition, the average number of weekends off offered to the total staff was slightly larger under the cyclical schedules than under the plan indicated by the scheduling sheets. To determine the impact of the new short-term allocation procedure on the workload index, we compared the actual workload index on a few days without any daily allocation, chosen at random, with the workload index that would have resulted had the daily allocation as described above been in effect. The range of the workload index on a particular day, that is, the difference between the highest and lowest workload index of all floors, is a suitable measure of the degree to which the workload is unbalanced on that day. A perfect balance (range of zero) is not possible even with the new procedure, since we assumed that an individual’s assignment cannot be split among floors, that is, an assignment is valid for the entire shift.
Cyclical Scheduling and Allocation of Nursing Staff
487
During the test period the workload index varied by about 50 per cent from floor to floor on any given day. The allocation system reduced this variation to 15 per cent. The day-to-day variations in index for a given floor were reduced by about 60 per cent. These reductions assume that individual staff members can be rather freely and temporarily reassigned to another floor. In practice, the supervisors hesitated to “pull” and tended to accept a somewhat unbalanced workload. The results of the comparison motivated a deliberate effort to increase the pool of floating nurses to gain additional flexibility. REFERENCES 1. V. M. DOMINICK,An automated solution to the problem of nursing staff allocation, Presented at the Institute on Computer Scheduling of Hospital Functions sponsored by the American Hospital Association, 18 February (1970). 2. DAVID H. HARRIS, Nursing-staffing requirements, Hospitals 44, 64-70, April 16 (1970). 3. JOHN P. HOWELL, Cyclical scheduling of nursing personnel, Hospitals 40, 77-85, 16 January (1966). 4. RICHARDM. JACKSONand CAROLYNKORTGE,Automated proficiency reports, Hospitals 45, 7678, 1 January (1971). 5. RICHARDA. MCCARTNEY,BARBARAMCKEE and LEE D. CAUY, Nurse staffing systems, Hospitals 44, 102-105, 16 November (1970). 6. ARTHURR. MORRISHand ANNA R. O’COPI’NOR,Cyclic scheduling, Hospitals 44, 66-71, 16 February (1970). 7. GERALDINEPARDEE,Classifying patients to predict staff requirements, Am. J. Nursing 68 (3), 517-520 (1968). 8. ELMINAM. PRICE, Techniques to improve staffing, Anl. /. Nursing 70 (IO), 2112-2115 (1970). 9. MARYANN FRANCIS, Implementing a program of cyclical scheduling of nursing personnel, Hospirals 40, 108, 16 July (1966). 10. JUNE B. SOMERS, A computerized nursing care system, Hospitals 45, 93 16 April (1971). 11. RICHARD E. WENDELLand GORDON P. WRIGHT, A simulation-optimization model of nursing care in intensive care units, Design and Development Center, The Technological Institute, Northwestern University, Evanston, III. DDC-68-11-3 (1968). 12. HARVEYWOLFEand JOHN P. YOUNG, Staffing the nursing unit. Part I. Controlled variable staffing, Nursing Rex 14 (3), 236-243 (1965). 13. HARVEYWOLFEand JOHN P. YOUNG, Staffing the nursing unit. Part II. The multiple assignment technique, Nursing Res. 14 (4), 299-303 (1965).
Plan.
Sri.
Vol. 7,
pp. 471-487
CYCLICAL
(1973).Pergamon Press.Printedin GreatBritain
SCHEDULING AND ALLOCATION NURSING STAFF
CHRISTOPH MAIER-ROTHE*
OF
and HARRY B. WOLFE
Arthur D. Little, Inc., Acorn Park, Cambridge, Massachusetts
02140
(Received 27 July 1972)
Nursing salaries make up the largest single element in hospital costs. Thus, the effective allocation of available nursing staff to patient care requirements is an important tool in controlling the cost of health care. This paper describes an allocation procedure consisting of two parts : 1. 2.
A cyclical scheduling procedure designed to meet average staffing requirements in a manner consistent with hospital personnel policies and employees’ preferences. A procedure for adjustment of staffing twice daily in accordance with an index of the latest patient care requirements.
The procedure is now in use in a major hospital where it has demonstrably staff allocations.
improved nursing
INTRODUCTION
about onein the hospital budget, typically representing third of the total, is the salaries of the nursing staff. If we are to make any inroads on the current escalation of hospital costs, we should begin with the most effective and efficient possible utilization of the nursing staff. We can identify three stages in determining the utilization of nursing staff: THE LARGEST single cost factor
1.
2.
3.
Forecasting and planning. By this we mean the forecasting of overall nursing staff requirements over an extended period (a year or more), taking into account such considerations as seasonality of patients, staff vacation requirements, and new patient care programs. Scheduliplg. This refers to the preparation of staff schedules, shift-by-shift, taking into account staffing requirements and individual constraints. These schedules are typically developed for periods two to eight weeks ahead. Allocution. This is the process of allocating and reallocating nursing staff on a day-today and shift-to-shift basis, making minor adjustments to the schedule to take into account short-term fluctuations in patient requirements and staff deviations from the schedule.
This paper is concerned with the latter two elements; it describes a formal system for the scheduling and allocation of nursing staff. As indicated in the references [l-13], a number of systems have been developed to facilitate the scheduling and allocation processes; many of these are computer-based. Very few * Currently located at Arthur D. Little, Wiesbaden, Germany. 471
472
CHRISTOPHMAIER-ROTHE and HARRY B. WOLFE
of these systems, however, have won acceptance and been successfully implemented, and the vast majority of hospitals still schedule and allocate nurses on an informal basis. The rejection of more systematic methods appears to reflect in part a set of beliefs that nursing, as a profession, is not amenable to formal scheduling and allocation procedures, that too many variables must be taken into account, that most nurses are non-quantitatively oriented and would therefore resist any system involving computers, and possibly in part the failure of previous efforts to take into account some variables that were important to nurses. In this paper we describe the development, application, and successful implementation of a scheduling and allocation system in the medical department of a large teaching hospital. In addition to the technical aspects of the system, we discuss the variables that were included in its development and how these variables were identified, the steps involved in implementation, and the measured benefits. The system as currently used is entirely manual and operates very satisfactorily in this mode; it could be computerized should this be desirable. THE
SCHEDULING
AND
ALLOCATION
PROBLEM
The basic problem of scheduling is to provide patient care every day around the clock using nurses who generally work five days a week, one shift per day, and prefer to have weekends off. Scheduling is usually done by nursing supervisors for the units of floors for which they are responsible. They estimate patient care requirements and allocate the available nursing staff to the days of the week so that these requirements are approximately satisfied and hospital personnel regulations observed. They try to schedule the nursing staff so that each nurse gets her share of weekends off and none of the nurses is rotated to evenings or night shifts for an unduly long time, and they also try to accommodate individual nurses’ requests for specific days off. A schedule is prepared every other week, specifying work days and days off for each number of the nursing staff over the ensuing two weeks. Preparation of the schedule is a time-consuming task for the nursing supervisor. In addition, most hospitals seem to feel some dissatisfaction with the results: 1.
2.
3. 4. 5.
Coverage tends to fluctuate widely, particularly in the case of RN’s; the number of RN’s assigned to a given unit on a given shift may vary from day to day by a factor of two or more. In their position of total authority over schedules, supervising nurses are apt to be suspected of favoritism, particularly when they try-as is often necessary and desirable -to take into account the desires and constraints of individual nurses. The attempt to resolve scheduling and allocation problems can result in excessive use of overtime. Nurses often feel that there is excessive rotation. Schedules frequently have to be changed on short notice because of changes in patient care requirements, illness of a nurse, etc.
Most of these problems result from the complexity of the scheduling decision and the difficulty of taking all relevant factors into account. We have spent considerable time discussing the scheduling problem with nurses and hospital administrators. From these discussions, the following have emerged as the most important factors in the decision. 1.
The different levels of nursing staff-registered nurses, licensed practical nurses, and nurses’ aides-have different capabilities; LPN’s and aides are legally allowed to per-
Cyclical Scheduling and Allocation of Nursing Staff
2.
3. 4. 5. 6.
7.
form only certain functions. Even within these categories, individuals can be classified by degree of experience and by the amount of responsibility they are able to assume. Nursing coverage must, of course, be provided 24 hours a day, 7 days a week. Nursing requirements are typically lower during the evening and night shifts than during the day shift. Saturday and Sunday requirements tend to be 20-30 per cent lower than weekday requirements, depending on the medical service, due to a lower patient census, fewer new physicians’ orders, or both. Vacations and time off for holidays must be staggered to ensure continued patient coverage and equitable treatment for nurses. Weekend days off are highly prized by the nursing staff, preferably both days in a row. Next in preference is two or more days off in a row in the middle of the week. Long stretches of consecutive working days (usually defined as more than five in a row) are undesirable. Despite a salary differential, the evening and night shifts are more difficult to staff than the day shift. As a result, schedules must provide for “rotation”; daytime staff must work on the other two shifts from time to time. Also, staff on evening and night shifts may sometimes have to work on the day shift in order to attend special programs, training courses, etc. Most nurses prefer to remain on one nursing unit, rather than being “floated” or shifted from one unit to another, partly because of the competence and expertise they are able to develop in a particular unit, and partly because of the camaraderie of continued association with the same group. However, there are some nurses who do not mind, and in fact prefer, floating because of the variety of experience it gives them.
The objectives of a scheduling days off to individual members 1. 2. 3. 4.
473
and allocation procedure are to assign working of the nursing staff so that:
days and
Adequate patient care is assured while overstaffing is avoided. A desirable distribution of days off is achieved. Individual members of the nursing staff are treated fairly. Individuals know well in advance what their schedules are.
In the project described here, different approaches were adopted to the two parts of the problem: the medium-term scheduling task and the short-term allocation task. For scheduling we used a cyclical approach, with a schedule that covers a designated number of weeks (cycle length) and then repeats itself. The cyclical schedule assigns the required registered nurses, licensed nurses, and total staff to each unit in a manner consistent with average patient care requirements, hospital personnel policies, and the nursing staff’s preferences for the distribution of days off. The short-term ,allocation process, for adjusting the cyclical schedule in response to changing patient care requirements and available staff, is based on a system of classifying patients by severity of illness to determine nursing requirements in each unit.
THE PRINCIPLE
OF CYCLICAL
SCHEDULING
Table 1 shows a simple cyclical schedule. There are five nurses and the cycle length of the schedule is five weeks. Days off are denoted by D. The first line indicates working days and days off for the first nurse, the second line for the second nurse and so forth. The first nurse has the first Sunday off, then a Thursday, then she has Tuesday and Wednesday off,
474
CHRJSTOPH MAIER-ROTHHE and HARRY B. WOLFE
Cyclical Scheduling and Allocation of Nursing Staff
415
and so on until the five weeks are up, at which point the entire pattern of working days and days off begins again. The overall pattern of working days and days off is exactly the same for all of the nurses; the only difference is that it begins in a different week for each nurse. Thus, the schedule followed by the first nurse in the first week is followed by the second nurse the following week, the third the week after, and so on. The weeks start with Sunday and end with Saturday. Each nurse works five days a week, works at most four days in a row, and has two out of five weekends off. Four of the five nurses are always present except on Tuesdays and on weekends, when only three are present. In a cyclical schedule, the total number of people present on any given day of the week is the same for all weeks when all cycles of the schedule are filled, that is, when the number of people on the schedule is equal to the length of the cyclical pattern in weeks. This can be shown quite easily as follows: Let a pattern of working days and days off be described by a sequence of variables yi where 1 ” = I0
ifdayiisoff otherwise
i=l,2,...7n
(1)
and II is the number of weeks covered by the pattern. Suppose now that a cyclical schedule is constructed for N individuals such that individualj starts the given pattern rj (1 < P, < n) weeks after day 1. For instance, if r, is equal to 2, individual 3 starts the pattern on day 15. The total number of individuals absent on day i is then simply ..
Di =
c
Yi-7ri
(2)
Yi*
(3)
j=l
where r1 is equal to zero and for any integer k Yi+7kn
The requirement that the total number the same for all weeks means that
=
of individuals
absent
on a given day of a week be
Di+7k = Di
(4)
for any integers i and k. The pattern given by expression (1) is defined such that it cannot be broken down into two or more patterns that repeat themselves. Now, if we assume for simplicity that rj+l = rj+l then requirement
(5)
(4) becomes N c j=l
N Yi+7(k+l-_j)
By leaving out the terms common
=
c j=l
Yi-7(j-1).
(6)
to both sides, this can be reduced to k-
c
I
N-l Yi-7j+7k
=o
=
c
Yi-7j.
(7)
j=N-k
Requirement (7) is true for arbitrary integers i and k only if N is an integer multiple In this case, expression (3) can be used to reduce the condition to an identity.
of II.
476
CHRISTOPH MAIER-ROTHE and HARRYB. WOLFE
In the example of Table 1, we did not distinguish between charge nurses, registered nurses, licensed practical nurses, and aides. Since there must always be a nurse present who can take charge of the unit and since there must always be present a certain number of registered nurses and licensed nurses regardless of the number of aides, the assignment of individual nurses to the cycles of a schedule must be planned with care. STAFFING
REQUIREMENTS
AND
PERSONNEL
POLICY
A schedule must meet staffing requirements and must be consistent with the hospital’s personnel policies. Patient care requirements fluctuate from day to day and even from hour to hour; minimum staffing requirements consist of the minimum number and the mix of nursing staff required to accommodate average patient care requirements. There is no sure way of determining minimum staffing requirements objectively without extensive work sampling and analysis of a great number of different activities. Such an extensive investigation may not be justified solely for the purpose of staff scheduling, in which case one must rely on expert opinion, and on obser\lation. It is not always easy to obtain agreement on minimum staffing requirements. Estimates from nursing supervisors tend to be somewhat biased in favor of more people, while estimates from the hospital administration may be biased in the opposite direction. It is thus advisable to consult both sources. If patient care requirements fluctuate widely from day to day, it may also be worthwhile to analyze actual staff coverage over an extended period of time and to conduct interviews to find out what happened on days with a particularly large or small number of nurses on duty. In the present case we combined the above approaches, working jointly with the Assistant Director of Nursing. The agreed-upon absolute minimum staff per unit for a weekday shift was two registered nurses, two licensed practical nurses, and three aides, in addition to the head nurse. On Sundays, because of the generally reduced level of activity, this could be reduced to one head nurse, one registered nurse, one licensed practical nurse, and two to three aides. The workload appeared to be heaviest immediately before and after the weekend, tapering off somewhat in midweek. The requirements for Saturdays were somewhere between the regular weekday requirements and those for Sundays. Requirements for evening and night shifts, worked out in the same way, were much lower than for any of the day shifts. Hospital personnel policies usually reflect agreements reached between the administration and the nursing staff regarding the distribution of days off for nurses. These agreements specify, among other things, the precise beginning of the work week, the number of days per week that must be given off, the maximum number of working days in a row, how many holidays are to be given off, the duration of vacations, and how many weekends are to be given off. They limit the types of cyclical patterns that are feasible. For instance, if four days off every two weeks are specified instead of two days off each week, the possible range of cyclical patterns is considerably increased. In our case, the work week started with Sunday morning and ended with Saturday night. Each nurse had to have two days off every week and these had to be a weekend at least every third week. Charge nurses and alternate charge nurses were given every other weekend off. After a working weekend, the following Monday was to be given off if possible. The nurses preferred two days off in a row over two single days off; if single days were to be given off, they preferred Fridays and perhaps Thursdays over other week days. Supervisors, registered nurses, and licensed practical nurses were eligible for 20 days of vacation per year. Other employees were eligible for 15 days annual vacation during the
477
Cyclical Scheduling and Allocation of Nursing Staff
first 8 years of service and 20 days thereafter. Each nurse was allowed one holiday per month, and could save up to three holidays to take later. In addition, all employees were entitled to paid sick leave, earned at the rate of one day each month of employment. On the average, each nurse spent 6 days per year in a training program. Thus, a licensed nurse (that is, a registered nurse or a licensed practical nurse) was available for floor duty 211 days per year and an aide was available 216 days per year. Under the circumstances, assuming that holidays, sick leaves, and training are distributed evenly through the year, the total number of licensed nurses required to ensure the agreed-upon minimum coverage of a shift is given by: l/7 X 3651211 2 Ri = 0.247 2 RI i=1 i=l and the total number
of aides by: 117 X 3651216 2 Ri = 0.241 2 Ri i=l i=l
where R, is the required
number TABLE
2.
of licensed
nurses or aides on duty on day i of the week.
STAFFING REQUIREMENTSFOR A DAY SHIFT
Type of nursing staff
Weekdays
Registered nurses Total licensed nurses Total staff including aides
3 4 I
Staff to be present on Saturday Sunday 2 4 7
2 3 6
Minimum total staff requirement 4.68 6.91 11.98
Table 2 shows the results for the day shift. Equivalent figures were constructed for the evening and night shifts. A minimum of twelve people was necessary to ensure full coverage of a day shift. At least five out of the twelve had to be registered nurses and at least seven altogether had to be licensed. The minimum therefore consisted of five registered nurses, two licensed practical nurses, and five aides. SELECTION
OF THE
CYCLICAL
PATTERN
Before a cyclical schedule can actually be assembled, patterns of working days and days off must be found which are consistent with requirements of the kind outlined above. In this case, at least twelve people were required to staff a day shift. If the cycle length were set at 12 weeks, then each member of the staff could be assigned to a separate cycle. Since no shift required more than twelve people, there was no need to consider cycles more than 12 weeks long. On the other hand, since we knew that, on a fair number of days, there would be fewer than twelve people available, in view of the uneven distribution of vacations, illness, training, and holidays throughout the year, we also considered the possibility of a IO-week cycle. We decided to examine scheduling alternatives for three basic alternatives: a IO-week cycle with four weekends off, a 12-week cycle with five weekends off, and a 12-week cycle with four weekends off. Feasible patterns of working days and days off could be found through some algorithmic approach such as integer programming. However, since it is not very difficult to find S.B.P.S. 7/5-F
CHRISTOPH MAIER-ROTHE and HARRY B. WOLFE
478
feasible solutions by trial and error and since there were in this case no clear trade-offs among long weekends, the number of two-day periods off, the number of working days in a row, and so forth, it was decided to use a heuristic approach to generate a number of feasible alternatives. In formulating these alternatives we tried to anticipate preferences of the nursing staff. We then presented the alternatives to the nurses and let them decide which they preferred. The finding of feasible alternatives is facilitated by formalizing some of the constraints. The variables Ui (i = I, 2, . . . , 7) denote the number of times during one cycle that day i of the week is off as a single day, ui (i = 1,2, . . . , 6) denote the number of times during one cycle that days i and (i+l) of the week are off in a pair. The variables y, and y, denote the number of three-day weekends off including a Monday or Friday respectively, and z denotes the number of 4-day weekends off. Since at least every third weekend must be given off, the entire pattern must consist of two and three-week stretches between weekends off. The parameter pz denotes the number per cycle of three-week stretches between two successive weekends off and the parameter p denotes the total number of weekends off during one cycle. One cycle contains n weeks. Table 3 shows the constraints that the variables Ui, vi, yi, and z must satisfy in order to form a feasible pattern. These constraints are explained below. Although they are not sufficient to determine a feasible pattern, they are necessary, and they proved valuable in the systematic exploration of feasible alternatives. A complete cycle of n weeks must contain 2n days off. Of these, 2p are on weekends and therefore 2(n-p) must be on weekdays. Constraint (1) of Table 3 expresses this relation. Constraints (2) through (8) are simply balance equations for the individual weekdays. Constraint (9) is a balance equation for weekends. Since only a three-week stretch (between TABLE 3. CONSTRAINTSET
111111112 2 1 1 3 4 5 6 7 8 9 10 11 -1 12 13 -1 14 -1 -1 15 -1 -1 -1 16 -1 17 -1 -1 -1 18 19 20 21 22 23 24
FOR MAXIMUMOF
2 1 1 1
1 1
1
5 WORKING
2
2
DAYS
2
IN A ROW
2
1 1
1 1
1
1 1
1
1 1
1 1
1
1 1
1
1
1
-1
1
-1 -1
-1 -1 -1
-1 -1 -1
1 1 1 1 1 1
-1 -1
-1 -1 -1 -1
-1 -1 -1 -1 -1
1
-1
-1
-1 -1 -1 -1
1
-1 -1 -1 -1 -1 1
1 1 1 1 1
1 1
1 1 1 1
1 1 1
1 1
1 1 1
1 1 1 1
1 1 1 1 1
1 1 1 1 1 1 1
-1
1
2 = 2(n-p) = n--p-c1 1 = n-c, = n-CJ = n-C4 = n-C5 1 1 = n-C6 = n-p-c, 1 1 =p = Pz l=O 1 l=O = 0 =O =o =o =o -1 --1 =O =o =o =o =o =o -1 =o
479
Cyclical Scheduling and Allocation of Nursing Staff
weekends off) can contain a two-day period off, the total number of two-day periods off is less than or equal to pz ; constraint (10) states this. Constraints (11) through (24) relate specifically to the requirement that not more than five working days in a row are allowed in this case (for the preparation of the feasible alternatives that contain stretches of six working days in a row, a different version of constraints (11) through (24) was used). Constraint (11) states that a four-day weekend and a three-day weekend including Monday must always be followed either by a single Sunday off or by a 2-day period off including Sunday and Monday (it cannot be followed by a weekend off). Constraint (12) expresses a similar requirement for Fridays. If a 2-day period including Sunday and a Monday is off, then if not more than five working days in a row are allowed, the following Sunday must be off. Therefore, constraint (13) states that the total number of 2-day periods off including Sunday and Monday (ur) must be less than the total number of single Sundays (~1~)off. These requirements are repeated for the individual days of the week, constraints (11-18) relating to the subsequent days off and constraints (19-24) relating to preceding days off. Fourteen feasible alternatives are shown in Table 4. The first four describe a IO-week cycle with a total of four weekends off, the next four a 12-week cycle with a total of five TABLE 4. FOURTEEN ALTERNATIVE CYCLICALPAITERNS
Length of cycle (weeks) No. of alternative No. of 4-day weekends off Relative frequency (%) No. of 3-day weekends off Relative frequency (%) Total No. of weekends off Relative frequency (%) No. of 2-day periods off Relative frequency (%) No. of single days off Relative frequency (%) No. of 6-workday stretches Relative frequency (%) No. of S-workday stretches Relative frequency (%)
10
12
1 234 1 000 10 0 0 0 0210 0 20 10 0 4444 40 40 40 40 2222 20 20 20 20 6678 60 60 10 80 2210 20 20 10 0 2210 20 20 10 0
5678 1 000 8 0 0 0 0210 0 17 8 0 5555 42 42 42 42 2222 17 17 17 17 8910 8 67 67 75 83 2210 17 17 8 0 2210 17 17 8 0
12 9 10 1 0 800000 122110 8 17 444444 33 33 3 4 25 33 768798 58 50 420100 33 17 343222 25 33
11 0
12 0
13 0
14 0
17
8
8
0
33 3 25
33 4 33
33 3 25
33 4 33
67
58
75
67
0
8
0
0
25
17
17
17
weekends off, and the last a 1Zweek cycle with four weekends off. For easier comparison of the individual alternatives, the relative frequencies of occurrence of four-day weekends off, three-day weekends off, and so forth are shown. Table 5 shows the pattern of days off for four of the alternatives. The nurses preferred the cycles with more weekends off and therefore rejected the 12week schedule with four weekends off. They were initially quite excited about the possibility of long weekends, a feature offered only rarely under the old scheduling procedures, but their enthusiasm was quickly dampened when they realized, however, that these could usually only be achieved by allowing longer stretches of working days in a row. The nurses were originally shown an earlier version of the figure in which some of the alternatives involved stretches of seven working days in a row. These were rejected immediately, and one group, consisting primarily of somewhat older staff members, considered also unacceptable those alternatives involving six days.
CHRISTOPH MAIER-ROTHE and HARRY B. WOLFE
480
TABLE 5. DAYS OFF IN PATTERNS 1,4,
1
SIlrKi2V
-------, ~-~
4
/ (,1;
1
4
(,
.
1
11 I
Monday
1___
II Wednesday!
I
’
II
-6 ~-
I
4
~_. ._ _.~__~
6
A
i-
4 II
Tuesday
1
A
-11
6 AND 1 I
6
-1
--
6-,I4
II
6
6
II _~___
1 4
4
II
6 II
II Friday ___ Saturday__
1 611
4
Thursday _ ___~__
( 1
1
6
II
4
4 II,
h ii
I 4 6
14, 6
I
II
11
6
6
4
6 II
6 II
The advantage of presenting to the nurses alternative cyclical patterns such as those in Table 4 is that the trade-offs between desirable distributions of days off on the one hand and undesirable stretches of working days on the other are made quite explicit, and the choice is made by the nurses rather than imposed upon them. After several discussions with the nursing staff, it was agreed that alternative (4) was to be used for some floors and alternative (8) for others. At the same time, cyclical patterns for the evening and night shifts were introduced. In the latter two cases, the selection process was considerably simpler because of the comparatively small number of feasible alternatives. Once each cyclical pattern was selected, the individual nurses of a floor had to be assigned to the different cycles of the selected pattern. This process is described in the following section for alternative (4), a IO-week schedule with four weekends off. ASSEMBLY
OF
A SCHEDULE
The first step in assembling the schedule was to assign individual nurses to the cycles of alternative (4) in a manner that would provide the mix of staff specified in Table 2. A careless assignment would probably not have provided a satisfactory distribution of registered or licensed practical nurses. In cycle scheduling, the best way to avoid bad assignments is to begin by finding cycles that have as few common days off as possible during the full 10 (in this case) weeks. If four registered nurses are available in addition to a head nurse and an alternative charge nurse, then these four should be assigned to four cycles for which there is a minimum of duplication in days off and, if possible, no day off common to all four. Similar optimum combinations of two and three cycles must be found for staff categories in which only two or three people are available. Only after the cycle combinations have been established can individuals be assigned to any of the cycles. If, as in this case, weekends off are more frequent than one in three, and, at the same time, staffing requirements on weekends are not substantially lower than during weekdays, the
481
Cyclical Scheduling and Allocation of Nursing Staff
days are likely to be Saturday and Sunday. Cycle combinations that do not provide adequate coverage on weekends need not be considered even if weekday coverage happens to be very good. No easy analytical method was found to determine the best and other good combinations of two, three, and four cycles. Therefore, the desirable cycle combinations were determined by enumeration. This is not as cumbersome as it sounds because there are fewer distinct cycle combinations that one might intuitively think. All permutations of a given cycle combination are, of course, equivalent. If the ten-week pattern of alternative (4), as shown in Table 5, is designated cycle 1 and the same pattern starting one week later is designated cycle 2, and so forth until cycle 10, then we can describe any combination of m cycles by an m-tuple (i1, iz, . . . , i,,,). Thus the 3-tuple (1, 5, 7) describes the combination of the three cycles 1, 5 and 7. It is clear that, if we add the same number to each of the elements, we obtain an equivalent combination of cycles. When we obtain a number greater than ten, we recall that cycles i and (i+lO) are exactly the same. Therefore, the cycle combinations (3, 7, 9), (1, 5, 9), and (3, 5, 9) are all equivalent to (1, 5, 7). They are simply different arrangements of the same combination. For a lo-week cycle, there are exactly five distinct two-cycle combinations, twelve distinct three-cycle combinations and twenty-two distinct four-cycle combinations. For an n-week cyclical pattern there are always at least (n-l) !/(m-n)! m ! m-cycle combinations and at most n of these are equivalent. critical
TABLE 6. DETERMINATION OF THE BESTTHREE-CYCLE COMBINATION
Distinct 3-cycle combination 1 1 1 1
1 1 1 1
1 1 1 1
2 2 2 2 2 2 2 3 3 3 3 4
3 4 5 6 7 8 9 5 6 7 8 7
Weekends with 0 1 2 3 (people) 0 0 0 0 0 0 0 0 2 0 2 0
2 4 2 4 4 2 4 4 2
Weekdays with 0 1 2 3 (people)
8
0
0
1
8
0
0
6
8
0
0
6
8
0
0
2
34
15
30
18
4
Table 6 lists all distinct three-cycle combinations, showing for each combination how often all three people are present, only two people, only one person, and no one at all, for weekdays and weekends over one entire IO-week cycle. These evaluations were made with the help of transparent overlays, which later also facilitated the actual scheduling process. Coverage posed more of a problem on weekends than on weekdays, so combinations with poor weekend coverage were not further examined. As the table shows, the two combinations (1, 2, 3) and (1, 3, 7) provide almost equal coverage. The coverage on weekends is the same, and since the coverage on weekdays is slightly better for the former, this was selected as the best three-cycle combination. Table 7 shows the coverage it provides for
CHRISTOPHMAIER-ROTHE and HARRY B. WOLFE
482
TABLE 7. DAILY COVERAGEPROVIDEDBY THE BESTTHREE-CYCLECOMBINATION u,2,
3)
Week Day Sunday Monday Tuesday Wednesday Thursday Friday Saturday
1
2
3
4
5
6
7
8
9
10
2 3 3 1 2 3 2
2 2 2 2 3 3 I
1 2 3 2 3 2 2
2 2 2 2 3 2 2
2 2 2 2 3 2 2
2 2 2 2 2 3 2
2 2 2 3 2 3 1
1 3 2 3 2 2 2
2 3 2 2 2 2 2
2 3 2 2 2 2 2
each day of the 10 weeks. The number in each field corresponds to the number of people present during that particular day of the cycle if one person is assigned to each cycle. The best two-cycle and four-cycle combinations were similarly calculated. The fourcycle combination was particularly important since four registered nurses and four aides were available on the floor in question; Table 8 shows the coverage provided by the best combination.
TABLE 8. DAILY COVERAGEPROVIDEDBY THE BESTFOUR-CYCLECOMBINATION (1, 3, 5,7)
Week Day Sunday Monday Tuesday Wednesday Thursday Friday Saturday
1
2
3
4
5
6
7
8
9
10
3 3 3 2 3 4 2
2 3 3 4 3 3 2
2 4 3 2 3 3 3
3 3 2 3 4 3 2
2 3 4 2 3 3 3
3 3 2 3 3 4 2
2 3 3 3 4 3 2
2 3 3 3 3 3 3
3 3 3 3 3 3 2
2 4 2 3 3 3 3
After the necessary combinations were identified, the schedule was easy to assemble. Exactly twelve people were available for the day shift. Since the head nurse and the alternative charge nurse were allowed every other weekend off, they were not included in the cyclical schedule. Four registered nurses were assigned to the day shift in addition to the head nurse and alternative charge nurse. Two practical nurses and four nursing aides completed the staff for this floor. By comparing these numbers with the requirements of Table 2, we can see that the staffing requirements were met. The resulting schedule is shown in Table 9. The four registered nurses were assigned to cycle combination (1, 3, 5, 7), the best four-cycle combination. The two practical nurses were given cycles 9 and 10, and combination (2, 4, 6, S), which is equivalent to the combination used for registered nurses, remained for the nursing aides. In this particular case, we did not need the best three-cycle combination but for other floors we did. The only step that remained to make this schedule operational was to determine which individual would be assigned to which cycle within each of the three combinations. We
Cyclical Scheduling and Allocation of Nursing Staff
LL
c 3
483
CHRISTOPHMAIER-ROTHEand HARRY B. WOLFE
484
avoided starting anyone at a point that would resuh in either two successive weekends off or two successive working weekends. Thus, no one felt unfairly treated.
ALLOCATION
PROCEDURE
Patient care requirements change from day to day and even from one hour to the next. The cyclical schedule guarantees a certain minimum coverage but does not take into account ~uctuations in patient care requirements. It is evident that the reiative ~u~tuations on one floor are larger than those of an entire division consisting of several floors. Thus, these fluctuations can be accommodated by short-term assignment of nurses from a “float pool” to those floors that need them most. This is the only way to provide extra coverage without permanent overstaffing. In special situations when peaks of patient care requirements on one or severaI floors cannot be accommodated from the float pool, individuals may be temporarily shifted from floors with lighter requirements to those with heavy requirements. However, this should be considered only as a last resort since it reduces coverage below desirable levels on the floors from which the extra staff are obtained; also, nurses generally dislike being “pulled”. Since these adjustments in the schedule are based on patient care requirements, we needed a means of assessing these requirements. The approach had to be simple enough so that requirements could be established at least once a day without substantial effort. In cooperation with the nursing staff, we developed a simple classification scheme for patients. According to this scheme, patients are classified into four categories: acute care, complete care, partial care, and ambulatory care. Table 10 defines these categories. The differences TABLE 10. PATIENTCLA~IFI~ATION
Acute care:
Needs constant observation; frequent vital signs, constant monitoring, and/or suctioning; may be on dialysis; may need respirator; warrants close medical supervision and continuous nursing care.
Complete care:
Needs assistance with entire care (may require special treatment, dressing, vital signs, monitor); may or may not be able to feed self (some may be incontinent).
Partial care:
Some assistance required with physical care; may need special treatments procedures done by nursing staff.
Ambulatory
care:
or other
Physically able to care for his personaf needs.
in patient care requirements among categories are quantified by assigning relative values: 4, 3, 2 and 1 for acute, compIete, partial, and ambulatory care, respectively. If a private duty nurse is assigned to a patient, it is assumed that only haif the normally required care is provided by the regular staff. In addition, discharges are given a relative value of 0.5. The total relative value per floor is then calculated by multiplying the number of patients in each category by the relative value of the category and adding the resulting products. The system operates as follows. Patients are classified twice a day, at 6 a.m. and 2 p.m., to provide guidance for the day and evening staffing respectively. The patient classification forms {Table II> are filled out by the unit floor supervisor (it typically takes only a few
485
Cyclical Scheduling and Allocation of Nursing Staff TABLE 11. PATIENT CLASSIFICATIONFORM
No.of patients
Classification
Specials*
Rel. value
Total value
4 3 2 1 0.5
Acute Complete Partial Ambulatory Expected discharges
* For patients with specials subtract half the relative value from the total.
minutes to fill them out) and forwarded to the nursing supervisor for all medical units. The nursing supervisor or her secretary or assistant copies the relative patient values for each unit onto a form (Table 12), notes the number of available staff (number of assigned staff TABLE 12. STAFF ALLOCATION PROCEDURE
Floor
Relative patient values
Available staff
Calculated staff
Assigned staff
V= Float Total staff
R=s&fi= Values/staff
I. 2. 3. 4. 5. 6.
Copy the total relative patient values from the individual floor sheets to column 1 of the staff allocation form. Total these to obtain V,the total patient values in the house. Write down the available assigned staff on each floor in column 2. Add the available float staff to obtain total available staff. Calculate R = total relative values available staff = average value per staff. Calculate the required staff per floor by dividing relative patient value for the floor by R. Round off the nearest integer. The total of the calculated staff should equal total available staff. There may be a discrepancy of 1 due to the rounding off. If so, add or subtract 1 from any appropriate floor. In the final column write down the staff actually assigned to each floor.
per unit minus no-shows), totals these, and adds the sum to the float staff available to get total available staff. She then divides ‘total relative values by total available staff to find the workload index: the number of relative value points per staff member. By then dividing the total relative value for each floor by the workload index, she determines how many nurses should be assigned to each floor to equalize the index. The staffing supervisors then assign the float staff to the individual floors, taking into account their knowledge of the
486
CHRISTOPH MAIER-ROTHE and HARRY B. WOLFE
type and quality of assigned staff on each floor and of the float pool. If further adjustment is necessary, nursing staff may be “pulled” from one floor and assigned to another. In order to provide the desirable flexibility while minimizing the amount of “pulling” from one floor to another, it is desirable to build up a large float pool. Part-time staff can be automatically assigned to the float pool, nurses who enjoy floating should be encouraged to remain in the pool, and new nursing staff should whenever possible be hired for the pool rather than for specific floor assignments. RESULTS
In the hospital where we worked, most nurses, especially the supervisors who did the nurse scheduling, recognized the need for improved scheduling. Most also knew something about cyclical scheduling before we started our assignement. With few exceptions, they initially had a fairly strong resistance to cyclical scheduling, primarily because they associated the idea of a repetitive pattern with extreme inflexibility. The joint sessions during which we constructed schedules for alternatives (4) and (8) convinced them of the contrary. Once they learned to take advantage of the self-balancing effect of the different cycles of the selected pattern, they found not only that the scheduling process was much simpler than before, but also that it provided for more even coverage and a more just distribution of days off. Since the cyclical schedule has a certain built-in safety margin above the minimum staffing requirements, small deviations from the schedule can usually be accommodated. When this is not the case, a simple switching of cycles between two individuals of the same category will provide the desired changes. The net effect of cyclical scheduling is probably to increase flexibility rather than to decrease it. To assess the effectiveness of the cyclical scheduling procedures, cyclical schedules were developed retrospectively for all three shifts of two floors for a period of eight weeks. These were then compared with the actual plans as originally prepared by the nursing supervisors. An effort was made to incorporate into the cyclical schedule all changes in the staffing of a floor that were reported on the scheduling sheets. Days spent in orientation, for instance, were counted as regular nurse-days under both scheduling schemes. The eight-week period covered by the comparison contained 168 shifts (days, evenings, and nights) for each of the two floors. According to the scheduling sheets and the minimum staffing requirements, forty-four shifts of one floor and seventy-four shifts of the other floor could be considered understaffed. The cyclical schedules reduced these cases of understaffing from forty-four to two on the first floor and from seventy-four to forty-nine on the second. They also, of course, reduced the number of shifts that could be considered overstaffed in view of the available nursing staff. In addition, the average number of weekends off offered to the total staff was slightly larger under the cyclical schedules than under the plan indicated by the scheduling sheets. To determine the impact of the new short-term allocation procedure on the workload index, we compared the actual workload index on a few days without any daily allocation, chosen at random, with the workload index that would have resulted had the daily allocation as described above been in effect. The range of the workload index on a particular day, that is, the difference between the highest and lowest workload index of all floors, is a suitable measure of the degree to which the workload is unbalanced on that day. A perfect balance (range of zero) is not possible even with the new procedure, since we assumed that an individual’s assignment cannot be split among floors, that is, an assignment is valid for the entire shift.
Cyclical Scheduling and Allocation of Nursing Staff
487
During the test period the workload index varied by about 50 per cent from floor to floor on any given day. The allocation system reduced this variation to 15 per cent. The day-to-day variations in index for a given floor were reduced by about 60 per cent. These reductions assume that individual staff members can be rather freely and temporarily reassigned to another floor. In practice, the supervisors hesitated to “pull” and tended to accept a somewhat unbalanced workload. The results of the comparison motivated a deliberate effort to increase the pool of floating nurses to gain additional flexibility. REFERENCES 1. V. M. DOMINICK,An automated solution to the problem of nursing staff allocation, Presented at the Institute on Computer Scheduling of Hospital Functions sponsored by the American Hospital Association, 18 February (1970). 2. DAVID H. HARRIS, Nursing-staffing requirements, Hospitals 44, 64-70, April 16 (1970). 3. JOHN P. HOWELL, Cyclical scheduling of nursing personnel, Hospitals 40, 77-85, 16 January (1966). 4. RICHARDM. JACKSONand CAROLYNKORTGE,Automated proficiency reports, Hospitals 45, 7678, 1 January (1971). 5. RICHARDA. MCCARTNEY,BARBARAMCKEE and LEE D. CAUY, Nurse staffing systems, Hospitals 44, 102-105, 16 November (1970). 6. ARTHURR. MORRISHand ANNA R. O’COPI’NOR,Cyclic scheduling, Hospitals 44, 66-71, 16 February (1970). 7. GERALDINEPARDEE,Classifying patients to predict staff requirements, Am. J. Nursing 68 (3), 517-520 (1968). 8. ELMINAM. PRICE, Techniques to improve staffing, Anl. /. Nursing 70 (IO), 2112-2115 (1970). 9. MARYANN FRANCIS, Implementing a program of cyclical scheduling of nursing personnel, Hospirals 40, 108, 16 July (1966). 10. JUNE B. SOMERS, A computerized nursing care system, Hospitals 45, 93 16 April (1971). 11. RICHARD E. WENDELLand GORDON P. WRIGHT, A simulation-optimization model of nursing care in intensive care units, Design and Development Center, The Technological Institute, Northwestern University, Evanston, III. DDC-68-11-3 (1968). 12. HARVEYWOLFEand JOHN P. YOUNG, Staffing the nursing unit. Part I. Controlled variable staffing, Nursing Rex 14 (3), 236-243 (1965). 13. HARVEYWOLFEand JOHN P. YOUNG, Staffing the nursing unit. Part II. The multiple assignment technique, Nursing Res. 14 (4), 299-303 (1965).