Automating the self-scheduling process of nurses in Swedish healthcare: a pilot study

Abstract

Hospital wards need to be staffed by nurses round the clock, resulting in irregular working hours for many nurses. Over the years, the nurses’ influence on the scheduling has been increased in order to improve their working conditions. In Sweden it is common to apply a kind of self-scheduling where each nurse individually proposes a schedule, and then the final schedule is determined through informal negotiations between the nurses. This kind of self-scheduling is very time-consuming and does often lead to conflicts. We present a pilot study which aims at determining if it is possible to create an optimisation tool that automatically delivers a usable schedule based on the schedules proposed by the nurses. The study is performed at a typical Swedish nursing ward, for which we have developed a mathematical model and delivered schedules. The results of this study are very promising and suggest continued work along these lines.

Appendix

1.1 A.1 The mathematical model

This appendix contains a mathematical description of the model presented in Section 4. First, the parameters, variables, and sets are introduced, and then follows the model, with the constraints categorised as in Section 4.2. In Appendices A.1.3 to A.1.7, the constraints are numbered as refered to in the text, while formulas for calculating parameter values are not numbered.

The most fundamental sets and indices used are:

• The sets of assistant and trained nurses are denoted by I ^a and I ^t, respectively, and I = I ^a ∪ I ^t will denote the set of all nurses.

• The three types of shifts, day, evening, and night, are indexed by j = 1,2,3.

• The days of a scheduling period are indexed by l = k + 7(w − 1), where k = 1,...,7 are the seven weekdays and w = 1,...,sw are the weeks of the scheduling period.

In order to simplify the description in Section 4, we used a basic version of the variables. For instance, x _ijl used there, is in this appendix represented by both X _ijl and AP _ijl . Furthermore, the variables $Y^a_{jl}$ and $Y^t_{jl}$ in the appendix are in the text only represented by y _jl , not specifying the kind of substitute nurse, and the same simplification is made for Q ^a and Q ^t, called Q in the text.

1.1.1 A.1.1 Sets and parameters

Notation:

Description

$b^{rec}_{kw}$ :

Staffing demand at the clinic on weekday k = 1,...,7 in week w = 1,...,sw

$b^{t}_{jl}$ :

The minimum number of trained nurses required on shift j = 1,2,3 on day l = 1,...,7sw

$b^{tot}_{jl}$ :

The exact number of nurses required on shift j = 1,2,3 on day l = 1,...,7sw

bn _mn :

The nurse i ∈ I the predetermined assignment m ∈ mb considers, n = 1

The week w = 1,..., sw the predetermined assignment m ∈ mb considers, n = 2

The weekday k = 1,...,7 the predetermined assignment m ∈ mb considers, n = 3

The shift j = 1,2 the predetermined assignment m ∈ mb considers, n = 4

The number of hours that the predetermined assignment m ∈ mb considers, n = 5

c _ijl :

Scaled desirability grade for nurse i ∈ I at shift j = 1,2,3 on day l = 1,..., 7sw

c′ _ijl :

Desirability grade for nurse i ∈ I at shift j = 1,2,3 on day l = 1,..., 7sw

en :

The average number of nights that a nurse shall work during the period

et _i :

The average number of hours that nurse i ∈ I shall work per week

ew :

The average number of weekends that a nurse shall work during the period

I :

The set of nurses

I^a, I^t:

The sets of assistant and trained nurses, respectively

lbt ^w :

Lower bound on the number of hours to be worked in 1 week, in per cent

lbt ^tot :

Lower bound on the total number of hours to be worked, in per cent

lsan :

The subset of nurses that worked the last Saturday night in the previous period

lsun :

The subset of nurses that worked the last Sunday night in the previous period

lw :

The set of nurses with NM _i  = 1 in the previous period

mb :

The set of predetermined assignments

me1:

A set of exceptions from the set of predetermined assignments mb

me2:

A set of exceptions from the set of predetermined assignments mb

n _i :

= 1 if nurse i ∈ I can work night shifts, and = 0 otherwise

ns _i :

Night bank status for nurse i ∈ I when the period starts

p :

The maximum score per week

pt _i :

Percentage of full-time for nurse i ∈ I

rn :

The set of nurses that work at the clinic, $rn \subseteq I^t$

sw :

Number of weeks in the scheduling period

t _ijk :

The length of shift j = 1,2,3 on weekday k = 1,...,7 for nurse i ∈ I, in hours

t ^rec :

The length of a shift at the clinic, in hours

ts _i :

Time bank status for nurse i ∈ I when the period starts

ubt ^w :

Upper bound on the number of hours to be worked in 1 week, in per cent

ubt ^tot :

Upper bound on the total number of hours to be worked, in per cent

ws _i :

Weekend bank status for nurse i ∈ I when the period starts

1.1.2 A.1.2 Variables

Three types of variables are used, binary, integer, and continous ones, this is specified by respectively using the marks (B), (I), and (C) in the definition.

Notation:

Description

AP _ijl :

= 1 if nurse i ∈ I is working shift j = 1,2,3 on day l = 1,...,7sw without contributing to the staffing demand $b^{tot}_{jl}$, and = 0 otherwise, (B)

AT _iw :

The number of hours nurse i ∈ I works without contributing to the staffing demand $b^{tot}_{jl}$ in week w = 1,...,sw, (C)

MP _ijl :

= 1 if nurse i ∈ I is on holiday or performs an individual task on shift j = 1,2,3 on day l = 1,...,7sw, and = 0 otherwise, (B)

MT _iw :

The number of hours that nurse i ∈ I spends on holiday or individual tasks in week w = 1,...,sw , (C)

NM _i :

= 1 if nurse i ∈ I shall work the first Monday the next period, and = 0 otherwise, (B)

NNS _i :

Night bank status for nurse i ∈ I, (C)

P _i :

Score for nurse i ∈ I, (C)

Q^a, Q^t:

Lowest score for the nurses in I ^a and I ^t, respectively, (C)

R _ikw :

= 1 if nurse $i \in rn \subseteq I^t$ works at the clinic on day k = 1,...,7 in week w = 1,...,sw, and = 0 otherwise, (B)

RP _ijl :

= 1 if nurse i ∈ I works at the clinic on shift j = 1,2,3 on day l = 1,...,7sw, and = 0 otherwise, (B)

RT _iw :

The number of hours nurse i ∈ I works at the clinic in week w = 1,...,sw, (C)

TNS _iw :

Time bank status for nurse i ∈ I and week w = 1,...,sw, (C)

W _iw :

= 1 if nurse i ∈ I works the weekend in week w = 1,...,sw, and = 0 otherwise, (B)

WNS _i :

Weekend bank status for nurse i ∈ I, (C)

X _ijl :

= 1 if nurse i ∈ I contributes to the staffing demand $b^{tot}_{jl}$ on shift j = 1,2,3 on day l = 1,...,7sw, and = 0 otherwise, (B)

$Y^{a}_{jl}$ :

Number of assistant substitute nurses on shift j = 1,2,3, on day l = 1,...,7sw, (I)

$Y^{t}_{jl}$ :

Number of trained substitute nurses on shift j = 1,2,3, on day l = 1,...,7sw, (I)

1.1.3 A.1.3 The objective function

$\displaystyle\max \quad z = \sum\limits_{i \in I}P_i + \alpha \big ( Q^{a} + Q^{t} \big ) - \beta \sum\limits_{l=1}^{7sw}\sum\limits_{j=1}^{3} \big (Y_{jl}^a+Y_{jl}^t \big )$

1.1.4 A.1.4 Staffing demand

$$\sum\limits_{i \in I^t}X_{ijl}+Y^t_{jl} \geq b^t_{jl}, \quad j=1,2,3, ~ l=1,\ldots,7sw$$

(1)

$$\sum\limits_{i \in I}X_{ijl} + Y^t_{jl}+ Y^a_{jl} = b^{tot}_{jl} \quad j=1,2,3, ~ l=1,\ldots,7sw $$

(2)

$$X_{i3l} = 0, \quad i \in I: n_i = 0, ~ l=1,\ldots,7sw$$

(3)

$$\begin{array}{rll} MT_{iw}&=& \sum\limits_{m \in m'} bn_{m5}, \kern6pt m'=\{m:bn_{m1}=i, ~ bn_{m2}=w \},\nonumber\\ i &\in& I, ~ w=1,\ldots,sw \end{array}$$

(4)

$$\begin{array}{rll} me1 & := & \{m:bn_{m2}\!=\!1 \textrm{ and } bn_{m3}\!=\! 1,2\} \cup \{m:bn_{m4}\!=\!2 \} \\ me2 & := & \{m:bn_{m2}\!=\!1 \textrm{ and } bn_{m3}\!=\! 1\} \end{array}$$

$$X_{bn_{m1},3,bn_{m3}-2+7(bn_{m2}-1)} = 0, \quad m \in mb \setminus me1 $$

(5)

$$X_{bn_{m1},3,bn_{m3}-1+7(bn_{m2}-1)} = 0, \quad m \in mb \setminus me2 $$

(6)

$$X_{bn_{m1},1,bn_{m3}+7(bn_{m2}-1)} = 0, \quad m \in mb $$

(7)

$$X_{bn_{m1},2,bn_{m3}+7(bn_{m2}-1)} = 0, \quad m \in mb $$

(8)

$$X_{bn_{m1},3,bn_{m3}+7(bn_{m2}-1)} = 0, \quad m \in mb $$

(9)

$$\sum_{i \in rn} R_{ikw} = b^{rec}_{kw}, \quad k=1,\ldots,5, ~ w=1,...,sw $$

(10)

$$\begin{array}{lll}\frac{1}{|rn|} \sum\limits_{w=1}^{sw} \sum\limits_{k=1}^{5} b^{rec}_{kw} - 3 &\leq& \sum\limits_{w=1}^{sw} \sum\limits_{k=1}^{5} R_{ikw} \\&\leq&\frac{1}{|rn|} \sum_{w=1}^{sw} \sum_{k=1}^{5} b^{rec}_{kw} + 3, \,\,\, i \in rn \end{array}$$

(11)

1.1.5 A.1.5 Scheduling rules

$$\sum\limits_{j=1}^{3}X_{ijl} \leq 1, \quad i \in I, ~ l=1,...,7sw $$

(12)

$$X_{i3l} + X_{i,1,l+1} \leq 1,\quad i \in I, ~ l=1,...,7sw-2 $$

(13)

$$ X_{i3l} + X_{i,2,l+1} \leq 1,\quad i \in I, ~ l=1,...,7sw-2 $$

(14)

$$ X_{i3l} + X_{i,1,l+2} \leq 1,\quad i \in I, ~ l=1,...,7sw-2 $$

(15)

$$ X_{i11} = 0, \quad i \in lsan $$

(16)

$$ X_{ijl} = 0, \quad i \in lsun, ~ (j,l) \in \{(1,1),(2,1),(1,2)\} $$

(17)

$$ et_i := pt_i \big(36,2n_i + 37(1-n_i) \big), \quad i \in I^a $$

$$ et_i := pt_i \big(36,2n_i + 38.15(1-n_i) \big),\quad i \in I^t $$

$$\begin{array}{lll} lbt^{w}et_i &\leq& \sum\limits_{k=1}^{7} \sum\limits_{j=1}^{3}t_{ijk}X_{i,j,k+7(w-1)} + AT_{iw} \leq ubt^{w}et_i, \\ i &\in& I, ~ w=1,...,sw \end{array}$$

(18)

$$\begin{array}{lll} lbt^{tot}et_isw &\leq& \sum\limits_{w=1}^{sw} \Bigg( \sum\limits_{k=1}^{7} \sum\limits_{j=1}^{3}t_{ijk}X_{i,j,k+7(w-1)} + AT_{iw} \Bigg)\\ &\leq& ubt^{tot}et_isw, \quad i \in I \end{array}$$

(19)

$$\begin{array}{lll} &&{\kern-6pt}\sum\limits_{ll=l}^{l+5} \sum\limits_{j=1}^{3} (X_{i,j,ll} + AP_{i,j,ll}) \leq 5, \quad i \in I, \\ &&{\kern-6pt} l \in \{1,\ldots,7sw\!-\!5\} \!\setminus \!\{l : \exists ~ c'_{i,1,ll} \Leftrightarrow \texttt{\footnotesize PURPLE}, ~ ll\!=\!l,...,l\!+\!5 \} \end{array}$$

(20)

$$\begin{array}{lll} && {\kern-8pt} \sum\limits_{k=1}^{7} \sum\limits_{j=1}^{3} t_{ijk}X_{i,j,k+7(w-1)}\! +\! AT_{iw}\! -\! et_i\! + \!TNS_{i,w-1} \!=\! TNS_{iw}, \\ && i \in I, ~ w=1,...,sw \textrm{ with } TNS_{i0} = ts_i \textrm{ and } \\ && TNS_{iw} \in [-20,20] \end{array}$$

(21)

$$\begin{array}{lll} &&{\kern-6pt} X_{i,j,k+7(w-1)} + R_{ikw} \leq 1, \quad i \in rn, ~ j=1,\ldots,3 ,\\ &&{\kern1pt} k=1,\ldots,5, ~ w=1,\ldots,sw \end{array}$$

(22)

$$\begin{array}{lll} & &{\kern-6pt} X_{i,3,k-1+7(w-1)} + R_{ikw} \leq 1,~i \in rn, ~ (k,w) \in \{(k,w): \\ & & k=1,\ldots,5;~w=1,\ldots,sw; ~ k=w \neq 1 \} \end{array}$$

(23)

$$\begin{array}{lll} & &{\kern-6pt} X_{i,3,k-2+7(w-1)} + R_{ikw} \leq 1, \quad i \in rn, (k,w) \in \{(k,w): \\ & & k=1,\ldots,5; ~ w=1,\ldots,sw; ~ k=w \neq 1; \\ && k=w+1\neq2 \} \end{array}$$

(24)

$$ R_{i11} = 0, \quad i \in lsan $$

(25)

$$ R_{ik1} = 0, \quad i \in lsun, ~ k=1,2 $$

(26)

$$ RT_{iw} = \left\{ \begin{array}{ll} t^{rec} \displaystyle \sum\limits_{k=1}^{5} R_{ikw}, & \quad i \in rn, ~ w=1,...,sw\\ 0, & \quad i \in I \setminus rn, ~ w=1,...,sw \end{array} \right. $$

(27)

$$ RP_{i,j,k+7(w-1)} = \left\{ \!\!\begin{array}{ll} R_{ikw}, & i \in rn, ~ j=1, ~ k=1,\ldots,4, \\ & {\kern6pt}w=1,\ldots,sw \\ 0, & \textrm{otherwise} \end{array} \right. $$

(28)

$$\begin{array}{rll} AP_{ijl} &=& MP_{ijl} + RP_{ijl}, \quad i \in I, ~ j=1,\ldots,3, \\ l&=&1,\ldots,7sw \end{array}$$

(29)

$$ RP_{ijl} + MP_{ijl} \leq 1, \quad i \in I, ~ j=1,\ldots,3, ~ l=1,\ldots,7sw $$

(30)

$$ AT_{iw} = RT_{iw} + MT_{iw} \quad i \in I, ~ w=1,\ldots,sw $$

(31)

1.1.6 A.1.6 Quality aspects

$$ en := \sum\limits_{l=1}^{7sw}b^{tot}_{3l} \Big/ \sum\limits_{i \in I}n_i \nonumber $$

$$\begin{array}{lll} \sum\limits_{l=1}^{7sw}X_{i3l} &-& en + ns_i = NNS_i, \quad i \in I: ~ n_i=1, \\ &&{\kern-43pt} \textrm{ with } NNS_i \in [-2,2] \end{array}$$

(32)

$$ ew := \sum\limits_{w=1}^{sw} \Bigg( \sum\limits_{j=2}^{3} b^{tot}_{j,5+7(w-1)} + \sum\limits_{k=6}^{7} \sum\limits_{j=1}^{3} b^{tot}_{j,k+7(w-1)} \Bigg) \Big/ |I| $$

$$\begin{array}{lll} & & {\kern-8pt} \sum\limits_{w=1}^{sw} \Bigg( \sum\limits_{j=2}^{3} X_{i,j,5+7(w-1)} + \sum\limits_{k=6}^{7} \sum\limits_{j=1}^{3} X_{i,j,k+7(w-1)} \Bigg) - ew \\ & & + ws_i = WNS_i, \quad i \in I \textrm{ with } WNS_i \in [-2,2] \end{array}$$

(33)

$$\begin{array}{lll} &&{\kern-7pt} W_{iw} = \sum\limits_{j=1}^{2}X_{i,j,6+7(w-1)} = \sum\limits_{j=1}^{2}X_{i,j,7+7(w-1)}, \\ &&i \in I, ~ w=1,\ldots,sw \end{array}$$

(34)

$$ W_{iw} \leq X_{i,2,5+7(w-1)} + X_{i,1,7w+1}, \quad i \in I, ~ w=1,\ldots,sw-1 $$

(35)

$$ \sum\limits_{i \in I^a}NM_i = \sum\limits_{i \in I^t}NM_i= 2 $$

(36)

$$ W_{i,sw} \leq X_{i,2,5+7(sw-1)} + NM_i, \quad i \in I $$

(37)

$$ X_{i11} = 1, \quad i \in lw $$

(38)

1.1.7 A.1.7 Auxiliary constraints

$$\begin{array}{lll} &&{\kern-7pt} X_{ijl}+RP_{ijl}=1, \quad i \in I, ~j =1,2,3, ~l=1,\ldots,7sw \\ &&\textrm{such that } c'_{ijl} \Leftrightarrow \texttt{\footnotesize BLUE} \end{array}$$

(39)

$$\begin{array}{lll} &&{\kern-7pt} X_{ijl}+RP_{ijl}=0, \quad i \in I, ~j =1,2,3, ~l=1,\ldots,7sw \\ &&\textrm{such that } c'_{ijl} \Leftrightarrow \texttt{\footnotesize RED} \end{array}$$

(40)

$$\begin{array}{lll} & & {\kern-7pt} c_{ijl} := \big(p \cdot sw \cdot c'_{ijl} \big) \Big/ \sum\limits_{l=1}^{7sw} \sum\limits_{j=1}^{3}|c'_{ijl}|, \\ && i \in I, ~ j=1,\ldots,3, \, l=1,\ldots,7sw, \\ && \textrm{such that } c'_{ijl} \Leftrightarrow \texttt{\footnotesize GREEN, WHITE,} \textrm{ or } \texttt{\footnotesize YELLOW} \end{array}$$

$$ P_i = \sum\limits_{l=1}^{7sw} \sum\limits_{j=1}^{3} \Big( 2c_{ijl} \big( X_{ijl}+AP_{ijl} \big) - c_{ijl} \Big), \quad i \in I $$

(41)

$$ Q^a <= P_i, \quad i \in I^a $$

(42)

$$ Q^t <= P_i, \quad i \in I^t $$

(43)