Advancement of Knowledge

To address the integration of the discontinuous tour scheduling problem with the MASSP, we proposed different modeling approaches based on decomposition methods and formal languages. While the use of formal languages (context-free grammars) allows to define the work rules for the composition of shifts and the allocation of work activities and breaks to shifts in an efficient way, the use of decomposition methods (CG and BD) allows to decompose the problem into smaller ones, easier to solve.

In Chapter 4 we introduced two B&P algorithms to solve the personalized MATSP in a discontinuous environment. Two formulations were presented: a Daily-based formulation and a Tour-based formulation in which columns correspond to shifts and tours, respectively. In the pricing subproblems for the Daily-based formulation, columns (shifts) are modeled using context-free grammars. In the Tour-based formulation, columns (tours) are composed with an exact two-phase procedure. In the first phase, daily shifts are modeled with context- free grammars. In the second phase, the daily shifts are assembled into tours by using a label setting algorithm for the resource-constrained shortest-path problem. The master problem of both formulations is based on a generalized set partitioning problem. We formally showed that the Tour-based formulation was stronger in terms of its LP relaxation lower bound when compared with the Daily-based formulation.

We tested and compared the two approaches on randomly generated instances for the mono- activity and multi-activity tour scheduling problem over a one week planning horizon. Two methods were tested to find integer solutions. A heuristic approach in which the MIP version of the problem including only the set of generated columns at the root node was solved by a state-of-the-art B&B code and an exact approach corresponding to a B&P where the branching rule was designed to preserve the structure of the pricing subproblems. The

Daily-based formulation exhibited better solution times than the Tour-based formulation for the heuristic approach. On the other hand, the Tour-based formulation had a better performance in the exact approach, being able to find integer solutions for all the instances with an optimality gap lower than 1%.

The Tour-based formulation was also tested on a mono-activity problem presented in Brunner and Bard [17]. The experiments suggested that the solution times and quality of our formu- lation are comparable with the solution times and quality reported by Brunner and Bard [17].

Since scalability issues arise as the number of employees becomes large, in Chapter 5, we presented an approach that combines BD and CG to solve the anonymous MATSP in a discontinuous environment. We took advantage of the special block structure of the problem to decompose it into smaller problems, easier to solve. Therefore, the model consists of a master problem and a set of Benders daily subproblems. The master problem links the tours with daily shift shells, while the Benders subproblems assign work activities and breaks to daily shifts. Due to its structure, the master problem is solved by CG. Additionally, the Benders subproblems are modeled with context-free grammars to implicitly tackle all the work rules for the composition of shifts and to efficiently allocate work activities and breaks to them. Although Benders primal subproblems do not possess the integrality property, we showed that adopting an alternative algorithmic strategy that combines the generation of integer Benders cuts with classical Benders cuts, guarantees the convergence of the method, to an optimal solution to the original problem.

The combined BD and CG approach was tested on real-world instances and randomly generated instances for the discontinuous anonymous MATSP (one-week planning horizon) and for the anonymous MASSP (one-day planning horizon). Results on weekly instances suggest that our method is able to find high quality integer solutions for instances dealing with up to ten work activities. When compared with a B&P approach, our method exhibited faster solution times and provided better upper bounds for the most difficult instances. Regarding the MASSP, the BD approach was able to solve to optimality instances with up to thirty work activities. When the method was compared with the grammar-based integer programming approach presented in Côté et al. [30], our method presented competitive and often better solution times.

Because alternative personnel scheduling approaches that include demand uncertainty can lead to significant reductions in labor costs, in Chapter 6 we presented a two-stage stochastic programming approach to solve the anonymous MATSP when demand is uncertain. The

method is an extension of the combined BD and CG approach presented in Chapter 5. Therefore, the first-stage problem is in charge of the decisions corresponding to the allocation of employees to weekly tours and to daily shifts, while the second-stage problems are in charge of the decisions related to the allocation of work activities and breaks to shifts and to the undercovering and overcovering of employee requirements (recourse actions). Since the number of tours becomes large with an increase in shift and tour flexibility, the first- stage problem was solved with a heuristic CG method. On the other hand, the second-stage problems are modeled with the grammar-based integer programming model presented in Côté et al. [30].

A multi-cut L-shaped method was implemented as a solution approach. The method was evaluated on instances with different demand profiles. The results suggest that the perfor- mance of the method depends on the distribution of the demand, as well as on the numbers of scenarios and work activities included. Specifically, the multi-cut L-shaped method exhib- ited a better performance, in terms of CPU time, when evaluated on instances with a level demand behavior than when evaluated on instances with a unimodal behavior. However, we observed that the use of the stochastic model has a larger impact on instances with unimodal demand behavior since it prevents the occurrence of additional staffing costs when compared with the expected value problem. On the contrary, since the value of the stochastic solution is close to zero, a deterministic approach based on the expected demand often appears to be sufficient for instances with a level demand behavior.