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In this paper, the authors prove that the Douglas space of second kind with a generalised form of special (α, β)-metric F, is conformally invariant.

For, the authors have used the notion of conformal transformation and Douglas space.

The authors found some results to show that the Douglas space of second kind with certain (α, β)-metrics such as Randers metric, first approximate Matsumoto metric along with some special (α, β)-metrics, is invariant under a conformal change.

The authors introduced Douglas space of second kind and established conditions under which it can be transformed to a Douglas space of second kind.

This paper investigates the temporal inflow profile that minimises the total cost of travel for a single route. The problem is formulated to consider the case in which the total demand to be serviced is fixed. The approach used here is a direct calculation of the first order variation of total system cost with respect to variations in the inflow profile. Two traffic models are considered; the bottleneck with deterministic queue and the kinematic wave model. For the bottleneck model a known solution is recovered. The wave model proves more difficult and after eliminating the possibility of a smooth inflow profile the restricted case of constant inflow is solved. As the space of possible profiles is finite dimensional in this case, the standard techniques of calculus apply. We establish a pair of equations that are satisfied simultaneously by the optimal inflow and time of first departure.

A nurse home care scheduling system is described. The objective is to provide medical care at patients’ homes using the fewest number of nurses possible to deliver the required care. The heuristic scheduling system is easy to implement as a computerized adaptive system. As such, it is easy to use on a daily basis and easy to update as new data related to completed treatment and new requests are obtained. A case study illustrates the advantages of implementing such a system.

Unfortunately, the occurrence of natural disasters is inevitable all over the world. Hence, this paper aims to analyze a scenario-based collaborative problem in a relief supply chain (RSC), where nongovernmental organizations can participate in relief activities with governmental organizations. This study focuses on location-allocation, inventory management and distribution planning under uncertain demand, budget, transportation and holding costs where government and private distribution centers receive relief items from suppliers then send them to affected areas. The performance of the proposed model is surveyed in a real case study in Dorud.

This paper develops a nonlinear mixed-integer programming model that seeks to maximize the coverage of demand points and minimize operating costs and traveled distance. The linear programming-metric technique and grasshopper optimization algorithm are applied to survey the model's applicability and efficiency.

This study compares noncollaborative and collaborative cases in terms of the number of applied distribution centers and RSC's goals, then demonstrates that the collaborative model not only improves the coverage of demand points but also minimizes cost and traveled distance. In fact, the presented approach helps governments efficiently surmount problems created after a disaster, notwithstanding existing uncertainties, by determining a strategic plan for collaboration with nongovernmental organizations for relief activities.

Relief strategies considered in previous research have not been sufficiently examined from the perspective of collaboration of governmental and nongovernmental organizations and provided an approach to develop the coverage of affected areas and reducing costs and traveled distance despite various uncertainties. Hence, the authors aim to manage RSCs better by offering a mathematical model whose performance has been proved in a real case study.

Considering the influence of environmental noise and modeling error during the process of the robotic automatic grinding aero-engine blade, this study aims to propose a method based on the extended state observer (ESO) to reduce the fluctuation of normal grinding force.

First, the measurement range of the six-dimensional force sensor is calibrated according to the maximum acceleration of end-effector and grinding force. Second, the gravity and zero drift compensation model is built to compensate for measurement error. Finally, the switching function is designed based on the difference between the expected grinding force and the actual feedback value. When the value of function stays within the switching band, a nonlinear active disturbance rejection control (ADRC) loop is applied. When the function value reaches outside the switching band, an ESO-based sliding mode control (SMC) loop is applied.

The simulated and experimental results show that the proposed control method has higher robustness compared with proportion-integral-derivative (PID), Fuzzy PID and ADRC.

The processing parameters of this paper are obtained based on the single-factor experiment without considering the correlation between these variables. A new control strategy is proposed, which is not only used to control the grinding force of blades but also promotes the development of industrial control.

ESO is used to observe environmental interference and modeling errors of the system for real-time compensation. The segment control method consisting of ESO-based SMC and ESO-based ADRC is designed to improve the robustness. The common application of the two parts realizes suppression of fluctuation of grinding force.

The traction cable is paralleled with the existing traction network of electrified railway through transverse connecting line to form the scheme of long distance power supply for the traction network. This paper aims to study the scheme composition and power supply distance (PSD) of the scheme.

Based on the structure of parallel traction network (referred to as “cable traction network (CTN)”), the power supply modes (PSMs) are divided into cable + direct PSM and cable + autotransformer (AT) PSM (including Japanese mode, French mode and new mode). Taking cable + Japanese AT PSM as an example, the scheme of long distance power supply for CTN under the PSMs of co-phase and out-of-phase power supply are designed. On the basis of establishing the equivalent circuit model and the chain circuit model of CTN, taking the train working voltage as the constraint condition, and based on the power flow calculation of multiple train loads, the calculation formula and process for determining the PSD of CTN are given. The impedance and PSD of CTN under the cable + AT PSM are simulated and analyzed, and a certain line is taken as an example to compare the scheme design.

Results show that the equivalent impedance of CTN under the cable + AT PSM is smaller, and the PSD is about 2.5 times of that under the AT PSM, which can effectively increase the PSD and the flexibility of external power supply location.

The research content can effectively improve the PSD of traction power supply system and has important reference value for the engineering application of the scheme.

This study aims to evaluate the melting process of the phase-change RT-35 material in a shell and tube heat exchanger saturated with a porous medium. Titanium porous media with isotropic and inhomogeneous structures are studied. The considered tubes in the shell and tube exchanger are made of copper with specific thicknesses. The phase-change material has a non-Newtonian behavior and follows the endorsed Carreau–Yasuda Model.

The enthalpy–porosity method is used for modeling of the melting process. The governing equations were transferred to their dimensionless forms. Finally, the equations are solved by applying the Galerkin finite element method.

The findings for different values of the relative permeability (K*) and permeability deviation angle (λ) are represented in the forms of charts, streamlines and constant temperature contours. The considerable effects of the relative permeability (K*) and deviation angle (λ) on the flow line patterns of the melting phase-change material are some of the significant achievements of this works.

This study was conducted using data from relevant research articles provided by reputable academic sources. The data included in this manuscript have not been published previously and are not under consideration by any other journal.