Capacity expansion under a service level constraint for uncertain demand with lead times

For a service provider, stochastic demand growth along with expansion lead times and economies of scale may complicate a capacity planning problem. We consider a service provider who has to maintain certain minimum level of service and is interested in knowing the optimal timings and sizes of the future capacity expansions. This service level is defined in terms of unsatisfied demand over an expansion cycle. Under this service level constraint, the service provider wants to minimize the infinite time horizon cost of expansion. We assume a stationary policy where the timing and the sizes of the expansions are determined as fixed proportions of the capacity position, where the capacity position is the capacity that will be available when the current expansion is completed. We assume that the demand for the capacity follows a geometric Brownian motion (GBM) process. We discuss a method to check the GBM process fit for any data series representing the demand values and find that the data for electric utility consumption in the US, and the airline passenger enplanement data over a period of 15 years satisfy the assumptions of a GBM process. Using properties of the demand process, we can use financial option pricing theory to express the service level in terms of the decision variables. Particularly, we use the Up-and-Out partial barrier call option price expression to formulate the service level constraint. We use cutting plane algorithm to solve the optimization problem. Numerical optimization shows that it could be optimal to accumulate initial shortage before initiating the next capacity expansions for a low growth, low volatility demand and also when the expansion lead times are shorter. However, when the demand grows at a high rate or is more volatile, it is optimal to start the next expansion project before the demand reaches the current capacity position

Recent advances in directional statistics

Mainstream statistical methodology is generally applicable to data observed in Euclidean space. There are, however, numerous contexts of considerable scientific interest in which the natural supports for the data under consideration are Riemannian manifolds like the unit circle, torus, sphere and their extensions. Typically, such data can be represented using one or more directions, and directional statistics is the branch of statistics that deals with their analysis. In this paper we provide a review of the many recent developments in the field since the publication of Mardia and Jupp (1999), still the most comprehensive text on directional statistics. Many of those developments have been stimulated by interesting applications in fields as diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics, image analysis, text mining, environmetrics, and machine learning. We begin by considering developments for the exploratory analysis of directional data before progressing to distributional models, general approaches to inference, hypothesis testing, regression, nonparametric curve estimation, methods for dimension reduction, classification and clustering, and the modelling of time series, spatial and spatio-temporal data. An overview of currently available software for analysing directional data is also provided, and potential future developments discussed.Comment: 61 page

Capacity expansion under a service-level constraint for uncertain demand with lead times

For a service provider facing stochastic demand growth, expansion lead times and economies of scale complicate the expansion timing and sizing decisions. We formulate a model to minimize the infinite horizon expected discounted expansion cost under a service-level constraint. The service level is defined as the proportion of demand over an expansion cycle that is satisfied by available capacity. For demand that follows a geometric Brownian motion process, we impose a stationary policy under which expansions are triggered by a fixed ratio of demand to the capacity position, i.e., the capacity that will be available when any current expansion project is completed, and each expansion increases capacity by the same proportion. The risk of capacity shortage during a cycle is estimated analytically using the value of an up-and-out partial barrier call option. A cutting plane procedure identifies the optimal values of the two expansion policy parameters simultaneously. Numerical instances illustrate that if demand grows slowly with low volatility and the expansion lead times are short, then it is optimal to delay the start of expansion beyond when demand exceeds the capacity position. Delays in initiating expansions are coupled with larger expansion sizes

Brief Review on Formation Control of Swarm Robot

This paper presented review formation control ofswarm robot. Recently the problems formation control of swarmrobots has attracted much attention, and several formationcontrol schemes were proposed based on various strategies. Theformation control strategies to solved these problem on swarmrobots, with considering regulation concept in control theory.Swarm intelligence algorithms takes the full of advantages of thefeature of swarm robotics, and provides a great solution forproblem formation control on swarm robots

Mathematical Modelling and Methods for Load Balancing and Coordination of Multi-Robot Stations

The automotive industry is moving from mass production towards an individualized production, individualizing parts aims to improve product quality and to reduce costs and material waste. This thesis concerns aspects of load balancing and coordination of multi-robot stations in the automotive manufacturing industry, considering efficient algorithms required by an individualized production. The goal of the load balancing problem is to improve the equipment utilization. Several approaches for solving the load balancing problem are suggested along with details on mathematical tools and subroutines employed.Our contributions to the solution of the load balancing problem are fourfold. First, to circumvent robot coordination we construct disjoint robot programs, which require no coordination schemes, are flexible, admit competitive cycle times for several industrial instances, and may be preferred in an individualized production. Second, since solving the task assignment problem for generating the disjoint robot programs was found to be unreasonably time-consuming, we model it as a generalized unrelated parallel machine problem with set packing constraints and suggest a tailored Lagrangian-based branch-and-bound algorithm. Third, a continuous collision detection method needs to determine whether the sweeps of multiple moving robots are disjoint. We suggest using the maximum velocity of each robot along with distance computations at certain robot configurations to derive a function that provides lower bounds on the minimum distance between the sweeps. The lower bounding function is iteratively minimized and updated with new distance information; our method is substantially faster than previously developed methods. Fourth, to allow for load balancing of complex multi-robot stations we generalize the disjoint robot programs into sequences of such; for some instances this procedure provides a significant equipment utilization improvement in comparison with previous automated methods