Employee substitutability as a tool to improve the robustness in personnel scheduling

Abstract

Organisations usually construct personnel rosters under the assumption of a deterministic operating environment. In the short term, however, organisations operate in a stochastic environment as operational variability arises. This variability leads to the occurrence of unexpected events such as employee absenteeism and/or a demand for personnel that is higher or lower than expected. In order to deal with these uncertainties, organisations need to adopt proactive and reactive scheduling strategies to protect the personnel roster and to respond to this operational variability, respectively. In this paper, we discuss a proactive approach that exploits the concept of employee substitutability to improve the flexibility of a personnel shift roster to respond to schedule disruptions. We propose a pre-emptive programming approach to construct a medium-term personnel shift roster that maximises the employee substitutability value. Moreover, we assess different proactive strategies to introduce robustness with respect to the definition and formulation of employee substitutability and different reactive strategies that impact the decision freedom for schedule recovery. The robustness of the generated personnel shift rosters is evaluated using a three-step methodology of roster construction, daily simulation and optimisation, and evaluation.

Appendix: Operational allocation model

Notation

Sets

G :

set of skills (index m)

N :

set of employees (index i)

S :

set of shifts (index j)

General parameters

d :

day under consideration in the operational planning horizon

$b_{im}$ :

1 if employee i possesses skill m, 0 otherwise

$c^w_{imdj}$ :

wage cost of assigning an employee i to shift j, day d and skill m

$c^{wu}_{mdj}$ :

shortage cost for shift j, day d and skill m

$p_{idj}$ :

preference penalty cost if an employee i receives a shift assignment j on day d

$l_j$ :

duration of shift j

$\kappa ^{\alpha }_{idj}$ :

1 if employee i is allowed to receive an assignment during shift j on day d, 0 otherwise

$\kappa ^{f}_{id}$ :

the total number of hours employee i has to receive on day d

Simulation parameters

$a_{id}$ :

1 if employee i is available on day d, 0 otherwise

$R^{'w}_{mdj}$ :

simulated staffing requirements for shift j, day d and skill m

Roster change parameters

$x^{'w}_{imdj}$ :

1 if employee i received a shift assignment j for skill m on day d in the baseline personnel shift roster, 0 otherwise

$c^{w\delta }_{imdj}$ :

roster change cost for assigning an employee i to shift j, day d and skill m with  $c^{w\delta }_{imdj}$ > 0   if $x^{'w}_{imdj} = 0$ $c^{w\delta }_{imdj} = 0$ otherwise

$c^{v}_{id}$ :

duty cancellation cost for employee i on day d with  $c^{v}_{id}$ > 0   if $\sum _{m\in G}\sum _{j\in S}x^{'w}_{imdj}$ = 1 and $a_{id}$ = 1    $c^{v}_{id}$ =  0 otherwise

Variables

$x^w_{imdj}$ :

1 if employee i receives a shift assignment j for skill m on day d, 0 otherwise

$x^{v}_{id}$ :

1 if employee i receives a day off on day d, 0 otherwise

$x^{wu}_{mdj}$ :

the shortage of employees for shift j, day d and skill m

Mathematical formulation

$$\begin{aligned}&\min \quad \sum _{i\in N} \sum _{m\in G} \sum _{j\in S} \left( c^w_{imdj} + c^{w\delta }_{imdj} + p_{idj}\right) x^w_{imdj} + \sum _{i\in N} c^v_{id} x^v_{id} \nonumber \\&\quad + \sum _{m\in G} \sum _{j\in S} c^{wu}_{mdj}x^{wu}_{mdj} \end{aligned}$$

(24)

$$\begin{aligned}&\sum _{i\in N} b_{im}x^w_{imdj} + x^{wu}_{mdj} \ge R^{'w}_{mdj} \qquad \forall m\in G, \forall j\in S \end{aligned}$$

(25)

$$\begin{aligned}&\sum _{m\in G} \sum _{j\in S}x^w_{imdj} \le a_{id} \qquad \forall i\in N \end{aligned}$$

(26)

$$\begin{aligned}&\sum _{m\in G} \sum _{j\in S}x^w_{imdj} + x^v_{id} = 1 \qquad \forall i\in N \end{aligned}$$

(27)

$$\begin{aligned}&\sum _{m\in G} x^w_{imdj} \le \kappa ^{\alpha }_{idj} \qquad \forall i\in N, \forall j\in S \end{aligned}$$

(28)

$$\begin{aligned}&\sum _{m\in G} \sum _{j\in S} l_j x^w_{imdj} \ge \kappa ^{f}_{id}a_{id} \qquad \forall i\in N \end{aligned}$$

(29)

$$\begin{aligned}&x^w_{imdj} \in \{0,1\} \qquad \forall i\in N, \forall m\in G, \forall j\in S\nonumber \\&x^v_{id} \in \{0,1\} \qquad \forall i\in N\\&x^{wu}_{mdj} \ge 0 \qquad \forall m\in G, \forall j\in S\nonumber \end{aligned}$$

(30)

The objective function (Eq. 24) minimises the wage cost, roster change cost, preference penalty cost, cancellation cost and the cost for understaffing. The objective function weights are as follows, i.e.

• Every employee has a wage cost ($c^{w}_{imdj}$) of 10$\times 1.2^{\sum _{m\in G}b_{im} - 1}$.

• The roster change cost ($c^{w\delta }_{imdj}$) depends on the chosen scenario (cf. Table 6).

• Every employee has a preference penalty cost ($p_{idj}$) that is randomly generated in the range of 1 to 5.

• The duty cancellation cost ($c^v_{id}$) is 5.

• The shortage cost ($c^{wu}_{mdj}$) is fixed at 20.

Constraint (25) imposes the staffing requirements and every employee can only receive a shift assignment if (s)he is available (Eq. 26). Constraint (27) ensures that every employee receives either a shift assignment or a day off. The shifts assigned to the employees need to satisfy the time-related constraints (Eqs. 4–7). The satisfaction of the minimum rest period (Eq. 4), maximum number of hours that can be assigned (Eq. 5) and maximum consecutive working assignments (Eq. 7) is ensured through the definition of $\kappa ^{\alpha }_{idj}$ in constraint (28). Finally, constraint (29) ensures that every employee works a minimum number of hours over the complete planning period (Eq. 6). Note that if an employee is unavailable on a working day ($a_{id}$=0), we adapt the minimum number of hours for this employee such that the employee does not have to catch up this duty. We define the integrality conditions in Eq. (30).