On the P-formulation and the Split-Fraction-Formulation for the Generalized Pooling Problem

The generalized pooling problem (GPP) is a NP-hard problem for which the solution time for securing a global optimal solution heavily depends on the strength of the problem formulation. The existing GPP formulations use either quality variables (P-formulation and the variants) or split-fraction variables (SF-formulation and the variants) to model the material balance at the pools. This paper is the first attempt to develop theoretical results for comparing the strength of P-formulation and SF-formulation. It is found that, an enhanced version of P-formulation, called P-formulation, is at least as strong as SF-formulation under mild conditions. Furthermore, P-formulation becomes identical to P-formulation when the pooling network comprises only mixers and splitters. With additional conditions that are often satisfied at the root node, P-formulation is proved to be as least as strong as SF-formulation. The theoretical results are verified by the computational study of 23 problem instances.

Neural Approximate Dynamic Programming for On-Demand Ride-Pooling

On-demand ride-pooling (e.g., UberPool, LyftLine, GrabShare) has recently become popular because of its ability to lower costs for passengers while simultaneously increasing revenue for drivers and aggregation companies (e.g., Uber). Unlike in Taxi on Demand (ToD) services – where a vehicle is assigned one passenger at a time – in on-demand ride-pooling, each vehicle must simultaneously serve multiple passengers with heterogeneous origin and destination pairs without violating any quality constraints. To ensure near real-time response, existing solutions to the real-time ride-pooling problem are myopic in that they optimise the objective (e.g., maximise the number of passengers served) for the current time step without considering the effect such an assignment could have on assignments in future time steps. However, considering the future effects of an assignment that also has to consider what combinations of passenger requests can be assigned to vehicles adds a layer of combinatorial complexity to the already challenging problem of considering future effects in the ToD case.A popular approach that addresses the limitations of myopic assignments in ToD problems is Approximate Dynamic Programming (ADP). Existing ADP methods for ToD can only handle Linear Program (LP) based assignments, however, as the value update relies on dual values from the LP. The assignment problem in ride pooling requires an Integer Linear Program (ILP) that has bad LP relaxations. Therefore, our key technical contribution is in providing a general ADP method that can learn from the ILP based assignment found in ride-pooling. Additionally, we handle the extra combinatorial complexity from combinations of passenger requests by using a Neural Network based approximate value function and show a connection to Deep Reinforcement Learning that allows us to learn this value-function with increased stability and sample-efficiency. We show that our approach easily outperforms leading approaches for on-demand ride-pooling on a real-world dataset by up to 16%, a significant improvement in city-scale transportation problems.

1DCNN Fault Diagnosis Based on Cubic Spline Interpolation Pooling

The conventional pooling method for processing one-dimensional vibration signals may lead to certain issues, such as weakening and loss of feature information. The present study proposes the cubic spline interpolation pooling method. The method is appropriate for processing one-dimensional signals. The proposed method can transform the pooling problem into a linear fitting problem, use the cubic spline interpolation method with outstanding fitting effects, and calculate the fitting function of the input signals. Moreover, the values of the interpolation points are sequentially taken as the feature value output. Furthermore, the network using the conventional pooling method and the pooling network model proposed in the present study are compared, tested, and analyzed on the constructed simulation signals and the measured bearing dataset. It is concluded that the proposed pooling method can reduce the data dimension while improving the network feature extraction capability and is more appropriate for pooling one-dimensional signals.