Two new algorithms for solving optimization problems with one linear objective function and finitely many constraints of fuzzy relation inequalities

This paper studies the optimization model of a linear objective function subject to a system of fuzzy relation inequalities (FRI) with the max-Einstein composition operator. If its feasible domain is non-empty, then we show that its feasible solution set is completely determined by a maximum solution and a finite number of minimal solutions. Also, an efficient algorithm is proposed to solve the model based on the structure of FRI path, the concept of partial solution, and the branch-and-bound approach. The algorithm finds an optimal solution of the model without explicitly generating all the minimal solutions. Some sufficient conditions are given that under them, some of the optimal components of the model are directly determined. Some procedures are presented to reduce the search domain of an optimal solution of the original problem based on the conditions. Then the reduced domain is decomposed (if possible) into several sub-domains with smaller dimensions that finding the components of the optimal solution in each sub-domain is very easy. In order to obtain an optimal solution of the original problem, we propose another more efficient algorithm which combines the first algorithm, these procedures, and the decomposition method. Furthermore, sufficient conditions are suggested that under them, the problem has a unique optimal solution. Also, a comparison between the recently proposed algorithm and the known ones will be made.     

A new algorithm for minimizing a linear objective function subject to a system of fuzzy relation equations with max-product composition

In this paper, we study a problem of minimizing a linear objective function subject to a system of fuzzy relation equations with max-product composition. First, we present a standard form of an original model. Then by using the concept of chained-set suite, we educe optimal solutions of the standard form, with which we can get optimal solutions of the original model. On basis of these results, finally we get an algorithm for the studied problem, which is simple and rapid for application.     

A branch-and-price algorithm for the two-stage guillotine cutting stock problem

We investigate the two-stage guillotine two-dimensional cutting stock problem. This problem commonly arises in the industry when small rectangular items need to be cut out of large stock sheets. We propose an integer programming formulation that extends the well-known Gilmore and Gomory model by explicitly considering solutions that are obtained by both slitting some stock sheets down their widths and others down their heights. To solve this model, we propose an exact branch-and-price algorithm. To the best of our knowledge, this is the first contribution with regard to obtaining integer optimal solutions to Gilmore and Gomory model. Extensive results, on a set of real-world problems, indicate that the proposed algorithm delivers optimal solutions for instances with up to 809 items and that the hybrid cutting strategy often yields improved solutions. Furthermore, our computational study reveals that the proposed modelling and algorithmic strategy outperforms a recently proposed arc-flow model-based solution strategy.     

Bus driver duty optimization using an integer programming and evolutionary hybrid algorithm

This paper describes the details of a successful application where an integer programming and evolutionary hybrid algorithm was used to solve a bus driver duty optimization problem. The task is NP-hard, therefore theoretically optimal solutions can only be calculated for very small problem instances. Our aim is to obtain solutions of good quality within reasonable time limits. We first applied an integer programming approach to a set partitioning problem. The model was solved with a column generation algorithm in a branch and bound scheme. In order to solve larger real-life problems, we have combined the integer programming method with a greedy 1+1 steady state evolutionary algorithm. The resulting hybrid algorithm was capable of providing near-optimal solutions within reasonable timescales to larger instances of the bus driver scheduling problem. We present the results and running times of our algorithm in detail, as well as possible directions of future improvements.     

最小费用流的最优参数配置问题

本文提出并讨论了最小费用流的反问题：如何在有限的投资条件下，最有效地扩充容量参数，达到一个予定的流值。建立了反问题的数学模型，给出了最优参数配置的算法。     

Optimal buying at the global minimum in a regime switching model

This paper addresses the problem of buying an asset at its expected globally minimal price, to that end, we model it as an optimal stopping problem with regime switching driven by a continuous-time Markov chain. We characterize the optimal stopping time by optimizing the value functions and writing them as solutions of a system of integral equations. Finally we develop a stochastic recursive algorithm for numerical implementation.     

Lökal optimale lösungen der kombinierten strom- und spannungsoptimlerung auf netzwerken im anwendungsfall der abaufplanung

A mathematical problem is defined, which is a combination between the classical network flow problem with the classical potential problem. As a branch of application is presented a model for the quantitative and temporal control of resource flows. Because the problem is NP-hard, the following text deals with the obtainment of only local optimal solutions. It is demonstrated, that it is possible to transform the known algorithm of network optimization. The sketched algorithm of solution is demonstrated by an example.     

Optimal allocation and backup of computer resources under asymmetric information and incentive incompatibility

This paper presents the optimal allocation and backup of computing resources in a multidivisional firm in the presence of asymmetric information and incentive incompatibility. A game-theoretic model is developed and transformed to a linear programming problem. The solution to this linear programming problem enables the corporate headquarters to design a resource allocation scheme such that the revelation principle prevails and all divisions tell the truth. To cope with the combinatorial explosion of complexity caused by the resource constraint, a greedy-type algorithm and an averaged version of the original linear programming problem are developed to provide the upper and lower bounds. The greedy-type algorithm generates exact solutions for a wide range of instances. The lower bounds coincide with the exact solutions for the cases where the computer resource is either scarce or abundant. The averaged-version resource allocation model with slight modifications solves the optimal computer backup capacity problem. It determines how much back up capacity the firm should purchase when the firm's computer breaks down.     

Numerical computation of singular control problems with application to optimal heating and cooling by solar energy

The method presented here is an extension of the multiple shooting algorithm in order to handle multipoint boundary-value problems and problems of optimal control in the special situation of singular controls or constraints on the state variables. This generalization allows a direct treatment of (nonlinear) conditions at switching points. As an example a model of optimal heating and cooling by solar energy is considered. The model is given in the form of an optimal control problem with three control functions appearing linearly and a first order constraint on the state variables. Numerical solutions of this problem by multiple shooting techniques are presented.     

Half thresholding eigenvalue algorithm for semidefinite matrix completion

The semidefinite matrix completion(SMC) problem is to recover a low-rank positive semidefinite matrix from a small subset of its entries. It is well known but NP-hard in general. We first show that under some cases, SMC problem and S1/2relaxation model share a unique solution. Then we prove that the global optimal solutions of S1/2regularization model are fixed points of a symmetric matrix half thresholding operator. We give an iterative scheme for solving S1/2regularization model and state convergence analysis of the iterative sequence.Through the optimal regularization parameter setting together with truncation techniques, we develop an HTE algorithm for S1/2regularization model, and numerical experiments confirm the efficiency and robustness of the proposed algorithm.     

Modeling and solution of the joint quay crane and truck scheduling problem

This paper addresses the joint quay crane and truck scheduling problem at a container terminal, considering the coordination of the two types of equipment to reduce their idle time between performing two successive tasks. For the unidirectional flow problem with only inbound containers, in which trucks go back to quayside without carrying outbound containers, a mixed-integer linear programming model is formulated to minimize the makespan. Several valid inequalities and a property of the optimal solutions for the problem are derived, and two lower bounds are obtained. An improved Particle Swarm Optimization (PSO) algorithm is then developed to solve this problem, in which a new velocity updating strategy is incorporated to improve the solution quality. For small sized problems, we have compared the solutions of the proposed PSO with the optimal solutions obtained by solving the model using the CPLEX software. The solutions of the proposed PSO for large sized problems are compared to the two lower bounds because CPLEX could not solve the problem optimally in reasonable time. For the more general situation considering both inbound and outbound containers, trucks may go back to quayside with outbound containers. The model is extended to handle this problem with bidirectional flow. Experiment shows that the improved PSO proposed in this paper is efficient to solve the joint quay crane and truck scheduling problem.     

A meta-heuristic approach for solving the Urban Network Design Problem

This paper proposes an optimisation model and a meta-heuristic algorithm for solving the urban network design problem. The problem consists in optimising the layout of an urban road network by designing directions of existing roads and signal settings at intersections. A non-linear constrained optimisation model for solving this problem is formulated, adopting a bi-level approach in order to reduce the complexity of solution methods and the computation times. A Scatter Search algorithm based on a random descent method is proposed and tested on a real dimension network. Initial results show that the proposed approach allows local optimal solutions to be obtained in reasonable computation times.     

A hybrid imperialist competitive algorithm for minimizing makespan in a multi-processor open shop

In this paper a new mixed-integer linear programming (MILP) model is proposed for the multi-processor open shop scheduling (MPOS) problems to minimize the makespan with considering independent setup time and sequence dependent removal time. A hybrid imperialist competitive algorithm (ICA) with genetic algorithm (GA) is presented to solve this problem. The parameters of the proposed algorithm are tuned by response surface methodology (RSM). The performance of the algorithm to solve small, medium and large sized instances of the problem is evaluated by introducing two performance metrics. The quality of obtained solutions is compared with that of the optimal solutions for small sized instances and with the lower bounds for medium sized instances. Also some computational results are presented for large sized instances.     

A penalty function method for solving inverse optimal value problem

In order to consider the inverse optimal value problem under more general conditions, we transform the inverse optimal value problem into a corresponding nonlinear bilevel programming problem equivalently. Using the Kuhn–Tucker optimality condition of the lower level problem, we transform the nonlinear bilevel programming into a normal nonlinear programming. The complementary and slackness condition of the lower level problem is appended to the upper level objective with a penalty. Then we give via an exact penalty method an existence theorem of solutions and propose an algorithm for the inverse optimal value problem, also analysis the convergence of the proposed algorithm. The numerical result shows that the algorithm can solve a wider class of inverse optimal value problem.     

A branch and bound algorithm for scheduling trains in a railway network

The paper studies a train scheduling problem faced by railway infrastructure managers during real-time traffic control. When train operations are perturbed, a new conflict-free timetable of feasible arrival and departure times needs to be re-computed, such that the deviation from the original one is minimized. The problem can be viewed as a huge job shop scheduling problem with no-store constraints. We make use of a careful estimation of time separation among trains, and model the scheduling problem with an alternative graph formulation. We develop a branch and bound algorithm which includes implication rules enabling to speed up the computation. An experimental study, based on a bottleneck area of the Dutch rail network, shows that a truncated version of the algorithm provides proven optimal or near optimal solutions within short time limits.     

Conjugate gradient techniques for the optimal control evolution dam problem

This paper considers the numerical simulation of optimal control evolution dam problem by using conjugate gradient method.The paper considers the free boundary value problem related to time dependent fluid flow in a homogeneous earth rectangular dam.The dam is taken to be sufficiently long that the flow is considered to be two dimensional.On the left and right walls of the dam there is a reservoir of fluid at a level dependent on time.This problem can be transformed into a variational inequality on a fixed domain.The numerical techniques we use are based on a linear finite element method to approximate the state equations and a conjugate gradient algorithm to solve the discrete optimal control problem.This algorithm is based on Armijo's rule in the unconstrained optimization theory.The convergence of the discrete optimal solutions to the continuous optimal solutions,and the convergence of the conjugate gradient algorithm are proved.A numerical example is given to determine the location of the minimum surface     

Local search with a generalized neighborhood in the optimization problem for pseudo-Boolean functions

In the optimization problem for pseudo-Boolean functions we consider a local search algorithm with a generalized neighborhood. This neighborhood is constructed for a locally optimal solution and includes nearby locally optimal solutions. We present some results of simulations for pseudo-Boolean functions whose optimization is equivalent to the problems of facility location, set covering, and competitive facility location. The goal of these experiments is to obtain a comparative estimate for the locally optimal solutions found by the standard local search algorithm and the local search algorithm using a generalized neighborhood.     

The economic lot scheduling problem under extended basic period and power-of-two policy

The economic lot scheduling problem schedules the production of several different products on a single machine over an infinite planning horizon. In this paper, a nonlinear integer programming model is used to determine the optimal solution under the extended basic period and power-of-two policy. A small-step search algorithm is presented to find a solution which approaches optimal when the step size approaches zero, where a divide-and-conquer procedure is introduced to speed up the search. Further a faster heuristic algorithm is proposed which finds the same solutions in almost all the randomly generated sample cases.     

单体型装配问题及其算法

单核苷酸多态性(SNP)单体型装配问题就是从给定的来自某人染色体的SNP片段中去除错误,重构出尽可能与原来片段一致的单体型.这个问题有几个不同的模型最少片段去除(MFR)问题,最少SNP去除(MSR)问题以及最少错误纠正(MEC)问题.前两个问题的复杂性与算法已有一些学者研究过.第三个问题已被证明是NP完全问题,但这个问题的实际算法还没有.该文对MEC问题给出了一个分支定界算法,这个算法能得到问题的全局最优解.通过这个算法对实际数据的计算说明了MEC模型的合理性,即在一定条件下,通过修正最少的错误重构出的单体型确实是真实的单体型.由于分支定界算法对这样一个NP完全问题不能在可接受的时间内解规模较大的问题,文中又给出了求解MEC问题的两个基于动态聚类的算法,以便对规模较大的问题在可接受的时间内得到近似最优解.数值实际表明这两个算法很快,很有效.这两个算法总能得到与分支定界找到的全局最优解很接近的近似最优解.鉴于MEC问题是NP完全的,这两个算法是有效的、实际的算法.