Routing Optimization with Generalized Consistency Requirements

The consistent vehicle routing problem (ConVRP) aims to design synchronized routes on multiple days to serve a group of customers while minimizing the total travel cost. It stipulates that customers should be visited at roughly the same time (time consistency) by several familiar drivers (driver consistency). This paper generalizes the ConVRP for any level of driver consistency and additionally addresses route consistency, which means that each driver can traverse at most a certain proportion of different arcs of routes on planning days, which guarantees route familiarity. To solve this problem, we develop two set partitioning-based formulations, one based on routes and the other based on schedules. We investigate valid lower bounds on the linear relaxations of both of the formulations that are used to derive a subset of columns (routes and schedules); within the subset are columns of an optimal solution for each formulation. We then solve the reduced problem of either one of the formulations to achieve an optimal solution. Numerical results show that our exact method can effectively solve most of the medium-sized ConVRP instances in the literature and can also solve some newly generated instances involving up to 50 customers. Our exact solutions explore some managerial findings with respect to the adoption of consistency measures in practice. First, maintaining reasonably high levels of consistency requirements does not necessarily always lead to a substantial increase in cost. Second, a high level of time consistency can potentially be guaranteed by adopting a high level of driver consistency. Third, maintaining high levels of time consistency and driver consistency may lead to lower levels of route consistency.

Transition study for asymmetric reflection between moving incident shock waves

The transition criteria seen from the ground frame are studied in this paper for asymmetrical reflection between shock waves moving at constant linear speed. To limit the size of the parameter space, these criteria are considered in detail for the reduced problem where the upper incident shock wave is moving and the lower one is steady, and a method is provided for extension to the general problem where both the upper and lower ones are unsteady. For the reduced problem, we observe that, in the shock angle plane, shock motion lowers or elevates the von Neumann condition in a global way depending on the direction of shock motion, and this change becomes less important for large shock angle. The effect of shock motion on the detachment condition, though small, displays non-monotonicity. The shock motion changes the transition criteria through altering the effective Mach number and shock angle, and these effects add for small shock angle and mutually cancel for large shock angle, so that shock motion has a less important effect for large shock angle. The role of the effective shock angle is not monotonic on the detachment condition, explaining the observed non-monotonicity for the role of shock motion on the detachment condition. Furthermore, it is found that the detachment condition has a wavefunction form that can be approximated as a hybrid of a sinusoidal function and a linear function of the shock angle.

A Hybrid Technique Based on a Genetic Algorithm for Fuzzy Multiobjective Problems in 5G, Internet of Things, and Mobile Edge Computing

Emerging commucation technologies, such as mobile edge computing (MEC), Internet of Things (IoT), and fifth-generation (5G) broadband cellular networks, have recently drawn a great deal of interest. Therefore, numerous multiobjective optimization problems (MOOP) associated with the aforementioned technologies have arisen, for example, energy consumption, cost-effective edge user allocation (EUA), and efficient scheduling. Accordingly, the formularization of these problems through fuzzy relation equations (FRE) should be taken into consideration as a capable approach to achieving an optimized solution. In this paper, a modified technique based on a genetic algorithm (GA) to solve MOOPs, which are formulated by fuzzy relation constraints with s -norm, is proposed. In this method, firstly, some techniques are utilized to reduce the size of the problem, so that the reduced problem can be solved easily. The proposed GA-based technique is then applied to solve the reduced problem locally. The most important advantage of this method is to solve a wide variety of MOOPs in the field of IoT, EC, and 5G. Furthermore, some numerical experiments are conducted to show the capability of the proposed technique. Not only does this method overcome the weaknesses of conventional methods owing to its potentials in the nonconvex feasible domain, but it also is useful to model complex systems.

Approximate Solution of a Reduced-Type Index- k Hessenberg Differential-Algebraic Control System

This study focuses on developing an efficient and easily implemented novel technique to solve the index- k Hessenberg differential-algebraic equation (DAE) with input control. The implicit function theorem is first applied to solve the algebraic constraints of having unknown state differential variables to form a reduced state-space representation of an ordinary differential (control) system defined on smooth manifold with consistent initial conditions. The variational formulation is then developed for the reduced problem. A solution of the reduced problem is proven to be the critical point of the variational formulation, and the critical points of the variational formulation are the solutions of the reduced problem on the manifold. The approximate analytical solution of the equivalent variational formulation is represented as a finite number of basis functions with unknown parameters on a suitable separable Hilbert setting solution space. The unknown coefficients of the solution are obtained by solving a linear algebraic system. The different index problems of linear Hessenberg differential-algebraic control systems are approximately solved using this approach with comparisons. The numerical results reveal the good efficiency and accuracy of the proposed method. This technique is applicable for a large number of applications like linear quadratic optima, control problems, and constrained mechanical systems.

Globally Optimizing QAOA Circuit Depth for Constrained Optimization Problems

We develop a global variable substitution method that reduces n-variable monomials in combinatorial optimization problems to equivalent instances with monomials in fewer variables. We apply this technique to 3-SAT and analyze the optimal quantum unitary circuit depth needed to solve the reduced problem using the quantum approximate optimization algorithm. For benchmark 3-SAT problems, we find that the upper bound of the unitary circuit depth is smaller when the problem is formulated as a product and uses the substitution method to decompose gates than when the problem is written in the linear formulation, which requires no decomposition.

Numerical analytic method for solving a large class of generalized non-homogeneous variable coefficients KdV problems based on lie symmetries

In this study, we use Lie symmetry method to reduce a generalized KdV equation with initial and boundary conditions with non-homogeneous variable coefficients to initial value problem. In particular, we concentrate on the cases that the reduced IVP cannot be solved analytically. We also compare the approximated solutions of IVP using numerical methods with IBVP, and note that they are more efficient than doing the same procedure for IBVP. In fact, it is shown that reducing IBVP to IVP and solving the reduced problem numerically will lead to more accurate solutions.

Reduction method for solving Boundary Value Problem on Graph to it's internal edge: طريقة التخفيض لحل مسألة قيم حدية على البيان إلى ضلع داخلي منه

The main aim to this research is to reduce the boundary value problem for fourth differential equation on geometric graph with cycles to a problem on a internal edge provided that the right hand side of the differential equation is identically zero on some subgraph of the original graph, and in this research we find the sign of some coefficients in the boundary conditions of the reduced problem and relationship between these coefficients. That's helping us to prove existence and uniqueness for a boundary value problem resulting from this reduction. In order to reach our desired goal, we study the reduction method of boundary value problem for fourth differential equation on tree geometric graph(no cycles), finally we can say that our research help us to study green function on edge(interval) instead of complex sty ding on geometric graph(with cycles).

Numerical algorithms without saturation for the Schroedinger equation of hydrogen atom

Математически проблема сводится к задаче на собственные значения для оператора Лапласа во всем пространстве с кулоновским потенциалом. Для численного решения этой задачи применяется новый математический аппарат, разработанный автором. Инверсией относительно единичной сферы задача сводится к проблеме собственных значений в проколотом в центре единичном шаре. Граничное условие в бесконечности (нулевое) переходит в центр шара. В шаре можно исключить периодическую переменную $\varphi$ и построить дискретизацию, наследующую свойство разделения переменных дифференциального оператора ($h$-матрица). По $\varphi$ выбиралось 11 точек. Клетки $\Lambda_0$, $\Lambda_1$, $\Lambda_2$, $\Lambda_3$, $\Lambda_4$ и $\Lambda_5$ в $h$-матрице соответствуют линиям Lyman, Balmer, Paschen, Brackett, Pfund и Humphreys. Из рассмотрения, представленных расчетов видим, что $\alpha$-линия Lyman определена с точностью $5.43\%$. Таким образом, совпадение результатов расчетов с теоретическими значениями удовлетворительное. Mathematically, the problem under consideration is reduced to the eigenvalue problem for the Laplace operator in the entire space with the Coulomb potential. The new mathematical apparatus developed by the author is applied to the numerical solution of the reduced problem. This problem is reduced to the eigenvalue problem in the unit ball punctured at the center after inversion with respect to the unit sphere. The null boundary condition at infinity is transformed to the condition at the center of the unit sphere. In the sphere it is possible to split off the periodic variable $\varphi$ and to construct the discretization inheriting the property of the separation of variables of the differential operator (the $h$-matrix). Eleven points is chosen based on the values of $\varphi$. The blocks $\Lambda_0$, $\Lambda_1$, $\Lambda_2$, $\Lambda_3$, $\Lambda_4$, and $\Lambda_5$ of the $h$-matrix correspond to the Lyman, Balmer, Paschen, Brackett, Pfund, and Humphreys lines. From the obtained numerical results, it follows that the Lyman-alpha line is determined with the accuracy equal to 5.43\%. Thus, the coincidence of the numerical results with the theoretical values is satisfactory.