We consider two types of disruptions arising in the multi-depot vehicle scheduling; the delays and the extra trips. These disruptions may or may not occur during operations, and hence they need to be indirectly incorporated into the planned schedule by anticipating their likely occurence times. We present a unique recovery method to handle these potential disruptions. Our method is based on partially swapping two planned routes in such a way that the effect on the planned schedule is minimal, if these disruptions are actually realized. The mathematical programming model for the multi-depot vehicle scheduling problem, which incorporates these robustness considerations, possesses a special structure. This special structure causes the conventional column generation method fall short as the resulting problem grows also row-wise when columns are generated. We design an exact simultaneous column-and-row generation algorithm to find a valid lower-bound. The novel aspect of this algorithm is the pricing subproblem, which generates pairs of routes that form recovery solutions. Compromising on exactness, we modify this algorithm in order to enable it to solve practical-sized instances efficiently. This heuristic algorithm is shown to provide very tight bounds on the randomly generated instances in a short computation time.
In the present study, Poly(methyl methacrylate) (PMMA)/single-walled carbon nanotubes (SWCNT) composites were prepared by melt mixing to achieve suitable SWCNT dispersion and distribution and low electrical resistivity, whereby the SWCNT direct incorporation method was compared with masterbatch dilution. An electrical percolation threshold of 0.05–0.075 wt% was found, the lowest threshold value for melt-mixed PMMA/SWCNT composites reported so far. The influence of rotation speed and method of SWCNT incorporation into the PMMA matrix on the electrical properties and the SWCNT macro dispersion was investigated. It was found that increasing rotation speed improved macro dispersion and electrical conductivity. The results showed that electrically conductive composites with a low percolation threshold could be prepared by direct incorporation using high rotation speed. The masterbatch approach leads to higher resistivity values compared to the direct incorporation of SWCNTs. In addition, the thermal behavior and thermoelectric properties of PMMA/SWCNT composites were studied. The Seebeck coefficients vary from 35.8 µV/K to 53.4 µV/K for composites up to 5 wt% SWCNT.
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