A mixed integer programming approach to multi-skilled workforce scheduling

Abstract

The potential benefits of using human resources efficiently in the service sector constitute an incentive for decision makers in this industry to intelligently manage the work shifts of their employees, especially those dealing directly with customers. In the long term, they should attempt to find the right balance between employing as few labor resources as possible and keeping a high level of service. In the short run (e.g., 1 week), however, contracted staff levels cannot be adjusted, and management efforts thus focus on the efficient assignment of shifts and activities to each employee. This article proposes a mixed integer program model that solves the short-term multi-skilled workforce tour scheduling problem, enabling decision makers to simultaneously design workers’ shifts and days off, assign activities to shifts and assign those to employees so as to maximize and balance coverage of a firm’s demand for on-duty staff across multiple activities. Our model is simple enough to be solved with a commercial MIP solver calibrated by default without recurring to complex methodologies, such as extended reformulations and exact and/or heuristic column generation subroutines. A wide computational testing over 1000 randomly generated instances suggests that the model’s solution times are compatible with daily use and that multi-skilling is a significant source of labor flexibility to improve coverage of labor requirements, in particular when such requirements are negatively correlated and part-time workers are a scarce resource.

Appendix

1.1 Validation of statistical model

We explore the following a linear regression of $\Delta ^\mathrm{COV}$ as a function of the variables $p^\mathrm{PH}$, $p^\mathrm{FT}$ and $\sigma ^\mathrm{LRP}$

$$\begin{aligned} \Delta ^\mathrm{COV}_i&= \lambda _0 + \lambda _1 p^\mathrm{PH}_i +\lambda _2 p^\mathrm{FT}_i+ \lambda _3 \sigma ^\mathrm{LRP}_i + \lambda _4 p^\mathrm{PH}p^\mathrm{FT}_i\nonumber \\&\quad +\lambda _5p^\mathrm{FT}\sigma ^\mathrm{LRP}_i+\lambda _6p^\mathrm{PH}\sigma ^\mathrm{LRP}_i + \epsilon _i, \quad \forall i\in \{1,\ldots ,1000\}, \end{aligned}$$

(23)

where $\lambda$ is our vector of parameters and $\epsilon$ is the regression error term. The results of this estimation are presented in Table 6. We note that many parameters have poor t-tests including all variables related to $p^\mathrm{PH}$. If we remove one by one parameters with t-tests smaller than 1.96 and recalibrate each time, then we end up with a model without the effect of $p^\mathrm{PH}$. This means that there would be no statistical reason to include any of the above regressors related to this variable in the model. Also, the intercept is negative and this invalidates this model, since it could predict negative values of coverage gains.

Model (23) could be hiding non-linear effects and motivates us to look at a category-based regression model. First, we defined categories (or value ranges) for variables $p^\mathrm{FT}$ and $\sigma ^\mathrm{LRP}$ to group observations with similar values in $\Delta ^\mathrm{COV}$. These categories are given in Table 7. Then, we can calibrate the following regression model:

$$\begin{aligned} \Delta ^\mathrm{COV}_i= \sum _{k \in \mathrm{Categories}}\theta _k \mathbb {I}_{\{i \text { in category } k\}}+\epsilon '_i, \quad \forall i\in \{1,\ldots ,1000\}, \end{aligned}$$

(24)

where $\mathbb {I}_{A}$ is an indicator function equal to 1 if the boolean condition $A\in \{TRUE, FALSE\}$ is TRUE, and $\theta$ is the vector of parameters to estimate. The calibration is given in Table 8 and it has statistically significant t-tests at a 95 % confidence level for each parameter estimated. It also has increasing values over $p^\mathrm{FT}$ and $\sigma ^\mathrm{LRP}$ as expected. There is no statistical evidence to reject that $\theta _{A2}=\theta _{A3}$ and $\theta _{C3}=\theta _{C4}$, so we merge these classes. The new calibration is given in Table 4.

Table 6 Calibration of model (23)

Table 7 Definition of categories

Table 8 Parameter estimation for model (24)