A hybrid method for the Probabilistic Maximal Covering Location–Allocation Problem

This paper presents a hybrid algorithm that combines a metaheuristic and an exact method to solve the Probabilistic Maximal Covering Location–Allocation Problem. A linear programming formulation for the problem presents variables that can be partitioned into location and allocation decisions. This model is solved to optimality for small- and medium-size instances. To tackle larger instances, a flexible adaptive large neighborhood search heuristic was developed to obtain location solutions, whereas the allocation subproblems are solved to optimality. An improvement procedure based on an integer programming method is also applied. Extensive computational experiments on benchmark instances from the literature confirm the efficiency of the proposed method. The exact approach found new best solutions for 19 instances, proving the optimality for 18 of them. The hybrid method performed consistently, finding the best known solutions for 94.5% of the instances and 17 new best solutions (15 of them optimal) for a larger dataset in one-third of the time of a state-of-the-art solver.     

Domain decomposition and model reduction for the numerical solution of PDE constrained optimization problems with localized optimization variables

We introduce a technique for the dimension reduction of a class of PDE constrained optimization problems governed by linear time dependent advection diffusion equations for which the optimization variables are related to spatially localized quantities. Our approach uses domain decomposition applied to the optimality system to isolate the subsystem that explicitly depends on the optimization variables from the remaining linear optimality subsystem. We apply balanced truncation model reduction to the linear optimality subsystem. The resulting coupled reduced optimality system can be interpreted as the optimality system of a reduced optimization problem. We derive estimates for the error between the solution of the original optimization problem and the solution of the reduced problem. The approach is demonstrated numerically on an optimal control problem and on a shape optimization problem.     

Solving a rich vehicle routing and inventory problem using column generation

The livestock collection problem (LCP) is a rich vehicle routing problem (VRP) extended with inventory constraints. The LCP is a complex planning problem taken from the meat industry, and the goal is to construct a set of vehicle routes to collect animals from farms for slaughter at a slaughterhouse. Several constraints dealing with animal welfare are added, some of these lead to a loading problem where the vehicle capacity depends on the loading sequence. In addition, global constraints to handle production and inventory at the slaughterhouse are needed. This paper presents an exact solution method for the LCP, based on column generation, that solves much larger instances to optimality than what has been done before. The algorithm presented here also solves a richer model that is closer to the underlying real-world problem than previously published work on exact methods for this problem is based on.     

Managing large fixed costs in vehicle routing and crew scheduling problems solved by column generation

We consider vehicle routing and crew scheduling problems that involve a lexicographic bi-level objective function (for instance, minimizing first the number of vehicles and second the operational costs) and can be solved by column generation where the subproblem is a resource constrained shortest path problem. Such problems are often modeled using a single-level objective function with a large fixed cost (weight) for ensuring the minimization of the primary objective. In this paper, we study the impact on the solution time of the fixed cost value. First, we present computational results which show that the solution time increases as the fixed cost value gets larger. Then, we develop an exact dynamic fixed cost procedure compatible with column generation that starts with a relatively small fixed cost value and increases it iteratively until optimality is reached. To prove optimality, a shortest path problem with resource constraints needs to be solved. Through a series of computational experiments on two types of problems, we show that this procedure can reduce solution times by up to 50% when compared to an approach relying on one very large fixed cost value.     

Spectral mixed Galerkin method for state constrained optimal control problem governed by the first bi-harmonic equation

An optimal control problem governed by the first bi-harmonic equation with the integral constraint for the state and its spectral approximations based on a mixed formulation are investigated. The optimality conditions of the exact and the discrete optimal control systems are derived. The a priori error estimates of high order spectral accuracy are obtained. Furthermore, a simple and efficient iterative algorithm is proposed to solve mixed discrete system. Some numerical examples are performed to verify the theoretical results.     

Necessary condition for near optimal control of linear forward–backward stochastic differential equations

This paper investigates the near optimal control for a kind of linear stochastic control systems governed by the forward–backward stochastic differential equations, where both the drift and diffusion terms are allowed to depend on controls and the control domain is not assumed to be convex. In the previous work (Theorem 3.1) of the second and third authors, some problem of near optimal control with the control dependent diffusion is addressed and our current paper can be viewed as some direct response to it. The necessary condition of the near-optimality is established within the framework of optimality variational principle developed by Yong and obtained by the convergence technique to treat the optimal control of FBSDEs in unbounded control domains by Wu. Some new estimates are given here to handle the near optimality. In addition, an illustrating example is discussed as well.     

A fuzzy collaboration system for ubiquitous loading/unloading space recommendation in the logistics industry

A fuzzy collaboration system was designed for ubiquitous loading/unloading space recommendation in the logistics industry. The designed system allows drivers to share information regarding the availability of loading/unloading spaces without providing the exact number of available loading/unloading spaces, thus promoting drivers to use the system. To derive the exact number of loading/unloading spaces from the inexact information, a quadratic programming problem was formulated and solved. In addition, driver location and speed were modeled using fuzzy numbers to account for the uncertainty of their locations. Subsequently, fuzzy cross-referencing was used so that loading/unloading space information can be referenced from more than one location. The proposed methodology was applied to a small region in Seatwen District, Taichung City, Taiwan. The designed system reduced the average time required for a driver to locate a nearby loading/unloading space by 72%.     

A branch and bound algorithm for the response time variability problem

The response time variability problem (RTVP) is an NP-hard scheduling problem that has been studied intensively recently and has a wide range of real-world applications in mixed-model assembly lines, multithreaded computer systems, network environments and others. The RTVP arises whenever products, clients or jobs need to be sequenced in order to minimise the variability in the time between two successive points at which they receive the necessary resources. To date, the best exact method for solving this problem is a mixed integer linear programming (MILP) model, which solves to optimality most of instances with up to 40 units to be scheduled in a reasonable amount of time. The goal of this paper is to increase the size of the instances that can be solved to optimality. We have designed an algorithm based on the branch and bound (B&B) technique to take advantage of the particular features of the problem. Our computational experiments show that the B&B algorithm is able to solve larger instances with up to 55 units to optimality in a reasonable time.     

An exact method for the double TSP with multiple stacks

The double travelling salesman problem (TSP) with multiple stacks (DTSPMS) is a pickup and delivery problem in which all pickups must be completed before any deliveries can be made. The problem originates from a real‐life application where a 40‐foot container (configured as 11 rows of three columns) is used to transport 33 pallets from a set of pickup customers to a set of delivery customers. The pickups and deliveries are performed in two separate trips, where each trip starts and ends at a depot and visits a number of customers. The aim of the problem is to produce a packing plan for the pallets that minimizes the total transportation cost given that the container cannot be repacked at any stage. In this paper we present an exact solution method based on matching k‐best tours to each of the separate pickup and delivery TSPs. The approach is shown to outperform the only known previous exact method for this problem in that solutions can be obtained faster and previously unsolved instances containing as many as 18 customers can now be solved to optimality.     

Assessment of an EMG-based method for continuous estimates of low back compression during asymmetrical occupational tasks.

Variables, such as peak and accumulated moments and spine compression forces, have been shown to be risk factors for occupational low back pain. Estimates of these forces during prolonged, dynamic, asymmetric tasks using biomechanical models is complex and time-consuming. A simple technique for continuous measurement of these variables over a prolonged period is needed to measure the distribution of spinal loading during both sagittal plane lifts and complex asymmetrical jobs. The aim of this study was to determine whether a linear normalization of erector spinae EMG to spine compression force, called compression normalized EMG (CNEMG), could be used to estimate spinal loading for simulations of asymmetrical occupational tasks. The estimates of spine compression force obtained using the normalized EMG are presented in the form of an amplitude probability distribution function and are compared with estimates of a three-dimensional biomechanical model. The per cent time a worker spends above particular levels of spinal loading of interest, such as the NIOSH action limit for compression, are displayed. Five males performed simulated occupational tasks. The exposure time at a specific level of spine compression force for a combination of three tasks, estimated by CNEMG, was, on average, within 6.5% of the time calculated by the biomechanical model. However, if the task combination was dominated by an axial twisting moment, then the difference was, on average, 13.4%. The difference in magnitude of spine compression at a specific probability was, on average, 14.9% and when axial trunk twist dominated, 30.7%. It is concluded that CNEMG can estimate probability at a specific level of spine compression force when the task combination is characterized by a predominant extensor moment in the sagittal plane. Estimates of spine compression at a specific probability, and estimates obtained during task combinations dominated by an axial twisting moment, are poor.     

Design of input signals for parameter estimation

The optimum energy-constrained and time-constrained input signal is obtained for estimating the parameters of a system. The output is corrupted by nonstationary, nonwhite additive observation noise, and the observation time is finite. The reproducing kernel Hilbert space formulation is used to obtain the parameter estimates and the error covariance matrix in terms of the input. The performance index, assumed to be a function of the error covariance matrix, is minimized by a variational procedure. A necessary condition for optimality is that the input satisfy a nonlinear Fredholm equation. An example estimates the gain of a single time constant system where the observation noise has an exponential autocorrelation function. For broadband noise, the optimum input is a portion of a sinusoid. For a noise bandwidth narrower than the system bandwidth, the optimum input switches sign as rapidly as possible, but near-optimum performance can be obtained with a relatively high frequency sinusoidal input.     

Assessment of an EMG-based method for continuous estimates of low back compression during asymmetrical occupational tasks

Variables, such as peak and accumulated moments and spine compression forces, have been shown to be risk factors for occupational low back pain. Estimates of these forces during prolonged, dynamic, asymmetric tasks using biomechanical models is complex and time-consuming. A simple technique for continuous measurement of these variables over a prolonged period is needed to measure the distribution of spinal loading during both sagittal plane lifts and complex asymmetrical jobs. The aim of this study was to determine whether a linear normalization of erector spinae EMG to spine compression force, called compression normalized EMG (CNEMG), could be used to estimate spinal loading for simulations of asymmetrical occupational tasks. The estimates of spine compression force obtained using the normalized EMG are presented in the form of an amplitude probability distribution function and are compared with estimates of a three-dimensional biomechanical model. The per cent time a worker spends above particular levels of spinal loading of interest, such as the NIOSH action limit for compression, are displayed. Five males performed simulated occupational tasks. The exposure time at a specific level of spine compression force for a combination of three tasks, estimated by CNEMG, was, on average, within 6.5% of the time calculated by the biomechanical model. However, if the task combination was dominated by an axial twisting moment, then the difference was, on average, 13.4%. The difference in magnitude of spine compression at a specific probability was, on average, 14.9% and when axial trunk twist dominated, 30.7%. It is concluded that CNEMG can estimate probability at a specific level of spine compression force when the task combination is characterized by a predominant extensor moment in the sagittal plane. Estimates of spine compression at a specific probability, and estimates obtained during task combinations dominated by an axial twisting moment, are poor.     

ε-Optimality and duality for multiobjective fractional programming

Using the scalar ε-parametric approach, we establish the Karush-Kuhn-Tucker (which we call KKT) necessary and sufficient conditions for an ε-Pareto optimum of nondifferentiable multiobjective fractional objective functions subject to nondifferentiable convex inequality constraints, linear equality constraints, and abstract constraints. These optimality criteria are utilized as a basis for constructing one duality model with appropriate duality theorems. Subsequently, we employ scalar exact penalty function to transform the multiobjective fractional programming problem to an unconstrained problem. Under this case, we derive the KKT necessary and sufficient conditions without a constraint qualification for ε-Pareto optimality of multiobjective fractional programming.     

Return difference bode diagram for optimal system design

This paper is concerned with a graphical method for the determination of bandwidth and optimal feedback gains for linear systems employing a quadratic form performing index. Using a fundamental frequency-domain condition of optimality developed by R. E. Kalman [1], the gain and bandwidth estimates are found from a logarithmic amplitude vs. frequency plot of the optimal system return difference. The method can be extended to systems with multiple outputs. An attempt is made to provide some insight into the connection between classical and optimal design, and into the price of optimality in terms of bandwidth.     

Convergence and Optimality of the Adaptive Nonconforming Linear Element Method for the Stokes Problem

In this paper, we analyze the convergence and optimality of a standard adaptive nonconforming linear element method for the Stokes problem. After establishing a special quasi-orthogonality property for both the velocity and the pressure in this saddle point problem, we introduce a new prolongation operator to carry through the discrete reliability analysis for the error estimator. We then use a specially defined interpolation operator to prove that, up to oscillation, the error can be bounded by the approximation error within a properly defined nonlinear approximate class. Finally, by introducing a new parameter-dependent error estimator, we prove the convergence and optimality estimates.     

Worst-case Traffic for Oblivious Routing Functions

This paper presents an algorithm to find a worst-case trafficpattern for any oblivious routing algorithm on an arbitrary interconnectionnetwork topology. The linearity of channel loading offered by obliviousrouting algorithms enables the problem to be mapped to a bipartitemaximum-weight matching, which can be solved in polynomial time forrouting functions with a polynomial number of paths. Finding exact worstcaseperformance was previously intractable, and we demonstrate an examplecase where traditional characterization techniques overestimate thethroughput of a particular routing algorithm by 47%.     

On large deviations for Poisson stochastic integrals

We obtain asymptotically exact estimates for large deviations of Poisson stochastic integrals. We also find a region where such an integral can be approximated by the corresponding Gaussian random variable. In of all these results, we obtain nonasymptotic estimates for remainder terms.     

Maximum-throughput mapping of SDFGs on multi-core SoC platforms

Data-Flow models are attracting renewed attention because they lend themselves to efficient mapping on multi-core architectures. The key problem of finding a maximum-throughput allocation and scheduling of Synchronous Data-Flow graphs (SDFGs) onto a multi-core architecture is NP-hard and has been traditionally solved by means of heuristic (incomplete) algorithms with no guarantee of global optimality. In this paper we propose an exact (complete) algorithm for the computation of a maximum-throughput mapping of applications specified as SDFG onto multi-core architectures. This is, to the best of our knowledge, the first complete algorithm for generic SDF graphs, including those with loops and a finite iteration bound. Our approach is based on Constraint Programming, it guarantees optimality and can handle realistic instances in terms of size and complexity. Extensive experiments on a large number of SDFGs demonstrate that our approach is effective and robust.     

Finding the pseudogradient of the objective function in procedures for estimation of interframe deformations of images

In the recursive estimate of parameters of interframe geometric deformations of digital images, calculating the pseudogradient from a local sample and current estimates of deformation parameters to be measured involves finding derivatives in terms of finite differences. The maximal convergence rate of estimates of the deformation parameters can be achieved when using the most informative readings of images in a local sample. Under this approach, there exist optimal values of increments in the parameters and basic axes, depending on the mismatch between estimates and correlation properties of the image. The performance of the procedures may be increased if the increments are optimized at each iteration. The potential accuracy of estimation of the parameters, as given by the Cramer-Rao inequality, serves as an optimality criterion when calculating the pseudogradient of the objective function. It is shown that the condition of maximum of information can be reduced to the condition of maximization of the ratio of the expectation of the pseudogradient to its mean square deviation. Application of optimized values of increments enables one to reduce considerably (up to several times) the computational cost with the same accuracy of estimation.     

Control problem for pricing in a commercial organization

A mathematical business optimization model related to product warehousing and sales is stated as a discrete linear quadratic optimal control problem. Necessary optimality conditions obtained according to Dubovitsky and Milyutin formalism are used to construct an algorithm yielding the exact solution to the problem.