Abstract
This paper presents a column-and-constraint generation algorithm for two-stage stochastic programming problems. A distinctive feature of the algorithm is that it does not assume fixed recourse and as a consequence the values and dimensions of the recourse matrix can be uncertain. The proposed algorithm contains multi-cut (partial) Benders decomposition and the deterministic equivalent model as special cases and can be used to trade-off computational speed and memory requirements. The algorithm outperforms multi-cut (partial) Benders decomposition in computational time and the deterministic equivalent model in memory requirements for a maintenance location routing problem. In addition, for instances with a large number of scenarios, the algorithm outperforms the deterministic equivalent model in both computational time and memory requirements. Furthermore, we present an adaptive relative tolerance for instances for which the solution time of the master problem is the bottleneck and the slave problems can be solved relatively efficiently. The adaptive relative tolerance is large in early iterations and converges to zero for the final iteration(s) of the algorithm. The combination of this relative adaptive tolerance with the proposed algorithm decreases the computational time of our instances even further.
Highlights
Two-stage stochastic programming problems are often solved by Benders decomposition (Benders 1962) (BD), known as the L-shaped decomposition method in the stochastic programming literature
partial Benders decomposition (PBD) includes the deterministic equivalent model (DEM) of a subset of the scenarios directly into the master problem resulting in a stronger master problem that can lead to faster convergence
We demonstrate that a column-and-constraint generation (C&CG) algorithm can be used for two-stage stochastic programming problems
Summary
Two-stage stochastic programming problems are often solved by Benders decomposition (Benders 1962) (BD), known as the L-shaped decomposition method in the stochastic programming literature (van Slyke and Wets 1969). Tönissen et al (2019) show that a C&CG algorithm outperforms BD and MCBD with two orders of magnitude for a two-stage robust maintenance location routing problem for rolling stock with discrete scenarios. For instances with a large number of scenarios, the C&CG algorithm outperforms the plain use of any state-of-the-art MIP solver in memory requirements and computational time We demonstrate this by performing computational experiments on a maintenance location routing problem. We showcase the algorithm on a maintenance location routing problem for rolling stock We show that this algorithm can be used to trade-off computational speed and memory requirements and that it can perform better than MC(P)BD or the DEM.
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