Optimal solutions for a dock assignment problem with trailer transportation

This paper presents a model for a dock assignment problem, where trailers need to be assigned to gates for a given period of time for loading or unloading activities. The parking lot is used as a buffer zone. Transportation between the parking lot and the gates is performed by additional resources called terminal tractors. The problem is modeled as a three-stage flexible flow shop, where the first and the third stage share the same identical parallel machines and the second stage consists of a different set of identical parallel machines. We examine multiple integer-programming formulations for the parallel-machine model in stage two and for the three-stage flow shop and we provide extensive computational results. Our goal is to explore the limits of the instance sizes that can be solved to guaranteed optimality within acceptable running times using integer programming.     

Integer linear programming formulations of multiple salesman problems and its variations

In this paper, we extend the classical multiple traveling salesman problem (mTSP) by imposing a minimal number of nodes that a traveler must visit as a side condition. We consider single and multidepot cases and propose integer linear programming formulations for both, with new bounding and subtour elimination constraints. We show that several variations of the multiple salesman problem can be modeled in a similar manner. Computational analysis shows that the solution of the multidepot mTSP with the proposed formulation is significantly superior to previous approaches.     

Queueing models for a flexible machining station Part II: The method of Coxian phases

A work station of a flexible manufacturing system (fms) is modeled as a multi-server queue with finite waiting room. The interarrival/service time distributions have squared coefficients of variations not less than 0.5 and are modeled as Coxian laws of two phases. A recursive scheme is developed to calculate the equilibrium queue length distribution. The model, together with the diffusion approximation model of Part I, can be used to aid the design of fms work stations.     

Seasonally perturbed prey–predator system with predator-dependent functional response

The effect of seasonality on the prey–predator model with predator-dependent trophic function is investigated analytically as well as numerically. The effect of periodic variations is considered on two different parameters of the system: the growth rate of prey and the death rate of the predators. The two parameters may not be in the same phase. The behavior of the system is simulated and bifurcation diagrams are obtained for different parameters. The results show that seasonality in two different parameters with or without phase difference can give rise to multiple attractors, including chaos, with variations in critical parameters.     

Facility location with increasing production costs

Facility location models determine the set of locations on a network that minimize the sum of the costs of investment, production, and distribution to meet a known set of demands. In this paper, we introduce a new type of facility location model, which combines aspects of the well-studied simple uncapacitated and capacitated facility location problems. Its distinctive feature is that unit production costs are modeled as increasing with scale of output. Such cost functions have practical value in handling cases in which capacity can be “stretched” by incurring some additional cost (e.g., by adding workers). Indeed, it is shown that average total costs are minimized at a point where average production costs are rising. Four different formulations for this problem are proposed. Using linear programming plus branch-and-bound as a solution method, the four formulations are tested and compared on a set of 216 problems with randomly generated data.     

A note on capacity models for network design

In network design problems,capacity constraints are modeled in three different ways depending on the application: directed, bidirected and undirected. In the literature, polyhedral investigations for strengthening mixed-integer formulations are done separately for each model. In this note, we examine the relationship between these models to provide a unifying approach and show that one can indeed translate valid inequalities from one to the others.     

Decoupling correlated and uncorrelated parametric uncertainty contributions for nonlinear models

For models with correlated parameters, the amount of uncertainty (generally measured by variance) in a model output contributed by a specific parameter encompasses two components: (1) the uncertainty contributed by the variations (used to represent uncertainty in the parameter) correlated with other parameters; and (2) the uncertainty contributed by the variations unique to the parameter of interest (i.e., uncorrelated variations or variations that cannot be explained by any other parameters in the model). A regression-based method has been proposed previously by Xu and Gertner (2008) [1] to decouple the correlated and uncorrelated contributions to uncertainties in model outputs by each parameter for linear models. Based on a modified version of the popular Fourier Amplitude Sensitivity Test (FAST), this paper develops a general approach for the quantification of the correlated and uncorrelated parametric uncertainty contributions in linear, nonlinear and non-monotonic models with linear or nonlinear dependence among parameters. The decoupling of correlated and uncorrelated contributions can help us determine if the uncertainty contributed by a specific parameter results from the uncertainty in itself or from its correlations with other parameters. Thus, this decoupling can be very useful in improving the understanding our modeled systems.     

On time-dependent behavior of cross-ply laminated strips with viscoelastic interfaces

The two-dimensional problem of a simply supported laminated orthotropic strip with viscoelastic interfaces under static loading is studied. State-space formulations are developed based on the exact elasticity equations governing orthotropic media and the Kelvin–Voigt constitutive relation of interfaces. Since the response of the strip is time-dependent, the power series expansion technique is adopted to model the variations of elastic fields with time. Results show that the response of the laminated strip with viscoelastic interfaces changes remarkably with time, which is also significantly different from that of a plate with perfect interfaces or with viscous interfaces. Note that from the present analysis, the response for a laminated plate with spring-like interfaces or with viscous interfaces can be easily obtained because they are just two particular cases of the present Kelvin–Voigt model.     

The Organization of Interaction Between Oil Refineries

The paper deals with the appropriate form of interaction between two refineries with different demand patterns. This problem can be formulated as finding a decentralized solution of linear programming problems linked by buying and selling activities. The complete problem is first solved for central values of product demands and costs and revenues. The structure of the basis then determines the organization of the interaction in terms of which unit sets quantities and which prices, or whether centralized decisions should be made. If, for expected values of product demand and costs, the structure of the basis is the same then the related organization of trading can be used for day-to-day transactions. For a well-known oil refinery model it is found that, for fairly large demand variations, decentralized interaction is effective, but that the structure of the basis changes easily with crude price variations, and that simulations did not converge for these variations.     

Dynamic models of production with multiple operations and general processing times

Traditional integer programming model formulations for job-shops and flow-shops do not easily account for characteristics common to high-technology manufacturing such as high-volume semiconductor manufacturing. These characteristics are: (1) products (wafers) are processed by the same machine type more than once during the operation sequence, (2) many lots of similar type are run, and (3) there can be multiple machines of the same type. In this paper, we present two new integer programming formulations which easily account for these characteristics. The approach is based on restricting the allowed domain of events for the start of lot processing. The first model restricts production starts to the beginning of a planning time period. The second model uses a special time grid at each operation with width equal to the processing time, and allows starts to be scheduled at the grid points. In an example problem replicating a high-volume wafer fabrication process, it is shown that it is computationally practical to obtain solutions for the restricted start models where it is not computationally possible for the traditional integer programming model formulations.     

An Oligopolistic Investment Model of the Finnish Electricity Market

The investment problem faced by producers in deregulated electricity markets contains high uncertainties about the future. It can also be seen as a game, as only a small number of large players act in the market. A dynamic stochastic oligopoly model to describe the production and investment in such a situation is developed and applied to the Finnish electricity market. The demand growth rate is modeled as a stochastic variable. The strategies of the firms consist of investments and production levels for base and peak load periods. The firms have nuclear, hydro and thermal capacities, but are only allowed to invest in new thermal capacity. Using a so-called sample-path adapted open-loop information structure, the model contributes to the understanding of the dynamics of production, investment and market power in a medium time horizon. The solution method uses recent developments in variational inequality and mixed complementarity problem formulations.     

A probabilistic model of thermal explosion in polydisperse fuel spray

This work is concerned with an analysis of polydisperse spray droplets distribution on the thermal explosion processes. In many engineering applications it is usual to relate to the practical polydisperse spray as a monodisperse spray. The Sauter Mean Diameter (SMD) and its variations are frequently used for this purpose [13]. The SMD and its modifications depend only on “integral” characterization of polydisperse sprays and can be the same for very different types of polydisperse spray distributions.The current work presents a new, simplified model of the thermal explosion in a combustible gaseous mixture containing vaporizing fuel droplets of different radii (polydisperse). The polydispersity is modeled using a probability density function (PDF) that corresponds to the initial distribution of fuel droplets size. This approximation of polydisperse spray is more accurate than the traditional ‘parcel’ approximation and permits an analytical treatment of the simplified model. Since the system of the governing equations represents a multi-scale problem, the method of invariant (integral) manifolds is applied.An explicit expression of the critical condition for thermal explosion limit is derived analytically. Numerical simulations demonstrate an essential dependence of these thermal explosion conditions on the PDF type and represent a natural generalization of the thermal explosion conditions of the classical Semenov theory.     

Modeling the join curve between two co-axial carbon nanotubes

Making use of an applied mathematical model, we employ a calculus of variations technique to join two co-axial nanotubes. Due to the axial symmetry of the tubes, the three-dimensional problem can be reduced to a problem in two dimensions. The curvature squared for the join region is minimized for a prescribed join length and given tube radii. In this model, a certain non-dimensional parameter B arises, which approximately has the same numerical value when compared with the standard method for the joining between any two carbon nanotubes of different radii. This value occurs in consequence of adopting an angle of inclination of 9.594°, which occurs in the conventional method for joining two carbon nanotubes of different radii and which is necessary to accommodate a single pentagon. The simple calculus of variations model described here provides a general framework to connect nanotubes or other nanostructures.     

Deriving compact extended formulations via LP-based separation techniques

The best formulations for some combinatorial optimization problems are integer linear programming models with an exponential number of rows and/or columns, which are solved incrementally by generating missing rows and columns only when needed. As an alternative to row generation, some exponential formulations can be rewritten in a compact extended form, which have only a polynomial number of constraints and a polynomial, although larger, number of variables. As an alternative to column generation, there are compact extended formulations for the dual problems, which lead to compact equivalent primal formulations, again with only a polynomial number of constraints and variables. In this this paper we introduce a tool to derive compact extended formulations and survey many combinatorial optimization problems for which it can be applied. The tool is based on the possibility of formulating the separation procedure by an LP model. It can be seen as one further method to generate compact extended formulations besides other tools of geometric and combinatorial nature present in the literature.     

Multicriteria Semi-Obnoxious Network Location Problems (MSNLP) with Sum and Center Objectives

Locating a facility is often modeled as either the maxisum or the minisum problem, reflecting whether the facility is undesirable (obnoxious) or desirable. But many facilities are both desirable and undesirable at the same time, e.g., an airport. This can be modeled as a multicriteria network location problem, where some of the sum-objectives are maximized (push effect) and some of the sum-objectives are minimized (pull effect).We present a polynomial time algorithm for this model along with some basic theoretical results, and generalize the results also to incorporate maximin and minimax objectives. In fact, the method works for any piecewise linear objective functions. Finally, we present some computational results.     

带有升降气囊与压块的飞艇动力学建模

研究飞艇的动力学建模,将飞艇的机体视为浮力与重力相等的浸没刚体,且考虑了飞艇机体与升降气囊以及压块之间的动力学耦合作用．整体的动力学方程首先通过Newton-Euler定律和Kirchhoff方程推出．此外,应用Hamilton与Lagrange半直积约化理论,可以将动力学方程解释为Lie-Poisson系统或者Euler-Poincaré系统．这两种动力学描述在基于能量的控制设计中有着重要作用．     

Two improved formulations for the minimum latency problem

In network problems, latency is associated with the metric used to evaluate the length of the path from a root vertex to each vertex in the network. In this work we are dealing with two applications or variations of the minimum latency problem known as the repairman problem and the deliveryman problem. We have developed two integer formulations for the minimum latency problem and compared them with other two formulations from the literature for the time-dependent traveling salesman problem. The present work highlights the similarities and differences between the different formulations. In addition, we discuss the convenience of including a set of constraints in order to reduce the computation time needed to reach the optimal solution. We have carried out extensive computational experimentation on asymmetrical instances, since they provide the characteristics of the deliveryman and repairman problems in a better way.     

The Fshape Framework for the Variability Analysis of Functional Shapes

This article introduces a full mathematical and numerical framework for treating functional shapes (or fshapes) following the landmarks of shape spaces and shape analysis. Functional shapes can be described as signal functions supported on varying geometrical supports. Analyzing variability of fshapes’ ensembles requires the modeling and quantification of joint variations in geometry and signal, which have been treated separately in previous approaches. Instead, building on the ideas of shape spaces for purely geometrical objects, we propose the extended concept of fshape bundles and define Riemannian metrics for fshape metamorphoses to model geometric-functional transformations within these bundles. We also generalize previous works on data attachment terms based on the notion of varifolds and demonstrate the utility of these distances. Based on these, we propose variational formulations of the atlas estimation problem on populations of fshapes and prove existence of solutions for the different models. The second part of the article examines thoroughly the numerical implementation of the tangential simplified metamorphosis model by detailing discrete expressions for the metrics and gradients and proposing an optimization scheme for the atlas estimation problem. We present a few results of the methodology on a synthetic dataset as well as on a population of retinal membranes with thickness maps.     

The maximum capture problem with random utilities: Problem formulation and algorithms

A model for the optimal location of new facilities in a competitive market is introduced under the hypothesis that customers' behavior can be modeled by random utility functions. It means that the company, that wished to locate, uses a random utility model to forecast the market share of a location. Therefore the company cannot forecast the behavior of every customer in a deterministic fashion, but is able to embed him by a probability distribution. Three formulations are proposed to compute upper bounds of the objective function and compared in a numerical simulation. A branch and bound method is developed and tested on examples with up to 50 potential locations, and a Variable Neighborhood Search heuristic is proposed to solve larger instances.