An implicit model for multi-activity shift scheduling problems

Abstract

We consider a multi-activity shift scheduling problem where the objective is to construct anonymous multi-activity shifts that respect union rules, satisfy the demand and minimize workforce costs. An implicit approach using adapted forward and backward constraints is proposed that integrates both the shift construction and the activity assignment problems. Our computational study shows that using the branch-and-bound procedure of CPLEX 12.6 on the proposed implicit model yields optimal solutions in relatively short times for environments including up to 2970 millions of explicit shifts. Our implicit model is compared to the grammar-based implicit model proposed by Côté et al. (Manag Sci 57(1):151–163, 2011b) on a large set of instances. The results prove that both implicit models have their strengths and weaknesses and are more or less efficient depending on the scheduling environment.

Appendices

Appendix A: The grammar used to adapt the Côté et al. (2011) model to instances in the Dahmen set

All the grammars used are presented in accordance with the notations adopted by Côté et al. (2011). For clarity, they are not stated in Chomsky normal form. For each production P,‘$\rightarrow _{S,[min,max]}$’ restricts the subsequences generated to have a position belonging to the set of periods S and a span between min and max periods. When S or [min, max] are not specified for a production, there are no restrictions on its position or span, respectively. The grammar used for the Dahmen set with no restrictions on the number of activities, denoted $\mathcal {G}$, is defined as follows:

$$\begin{aligned}&\mathcal{{G}} = (\Sigma =((a_j)_{j \in J=\lbrace 1, 2, \ldots , |A| \rbrace },b,r),\nonumber \\&N=(S, R, (Q_t, P_t, W_t, B_t)_{t \in T}, (J_j, T_j, A_j)_{j \in J}, P, S)\nonumber \\&\forall t \in T, \quad \forall j, k \in J: j \ne k\nonumber \\&S \rightarrow _{\lbrace 0 \rbrace ,[96,96]} R Q_t | Q_t R | R Q_t R \nonumber \\&Q_t \rightarrow _{S_t,[L_t^{min},L_t^{max}]} P_t B_t W_t \nonumber \\&B_t \rightarrow b^{LB_t}, R \rightarrow R r | r \nonumber \\&P_t \rightarrow _{[LAM_{t}^{min},LAM_{t}^{max}]} J_j T_j , W_t \rightarrow _{[LPM_{t}^{min},LPM_{t}^{max}]} J_j T_j \nonumber \\&P_t \rightarrow _{[LAM_{a_j,t}^{min},LAM_{a_j,t}^{max}]} J_j , W_t \rightarrow _{[LPM_{a_j,t}^{min},LPM_{a_j,t}^{max}]} J_j \nonumber \\&J_j \rightarrow _{[L_a^{min},L_a^{max}]} A_j, T_j \rightarrow J_{k} T_{k} | J_{k}, A_j \rightarrow A_j a_j | a_j, \nonumber \\&\text{ where } LAM_{a_j,t}^{min}=\max \lbrace LAM_{t}^{min}, L_{a_j}^{min} \rbrace \nonumber \\&LAM_{a_j,t}^{max}=\min \lbrace LAM_{t}^{max}, L_{a_j}^{max} \rbrace \nonumber \\&LPM_{a_j,t}^{min}=\max \lbrace LPM_{t}^{min}, L_{a_j}^{min} \rbrace \nonumber \\&LPM_{a_j,t}^{max}=\min \lbrace LPM_{t}^{max}, L_{a_j}^{max} \rbrace . \nonumber \end{aligned}$$

When restrictions on the number of activities are considered (that is, $NA_{am} , NA_{pm} = 2$), some of the productions above defining $\mathcal {G}$ are changed as follows:

$$\begin{aligned}&\mathcal{{G}}^{r} = (\Sigma =((a_j)_{j \in J=\lbrace 1, 2, \ldots , |A| \rbrace },b,r), N=(S, R, \nonumber \\&(Q_t, P_t, W_t, B_t)_{t \in T}, (J_j, A_j)_{j \in J}, (T_{j,k})_{j,k\in J: j \ne k}, P, S)\nonumber \\&\forall t \in T, \quad \forall j, k \in J: j \ne k\nonumber \\&P_t \rightarrow _{[LAM_{t}^{min},LAM_{t}^{max}]} J_j T_{j,k} \nonumber \\&W_t \rightarrow _{[LPM_{t}^{min},LPM_{t}^{max}]} J_j T_{j,k}\nonumber \\&T_{j,k} \rightarrow J_{k} T_{k,j} | J_{k}. \nonumber \end{aligned}$$

Appendix B: Additional results for Section 6

Table 7 Average, minimum and maximum number of explicit shifts for the generated instances

Table 8 Number of variables and constraints of model G for the Dahmen set with no restrictions on the number of activities

Table 9 Computational performance of models P and G for the instances in Dahmen set with no restrictions solved to optimality within an hour

Table 10 Computational performance of models $P^r$ and $G^r$ for the instances in Dahmen set solved to optimality by both models within an hour

Table 11 Model size of P and G for the Côté set