Estimates of the region of attraction of continuous-time cascade systems

Xiao-Song Yang Department of Mathematics, Huazhong University of Science and Technology Wuhan, Hubei 430074, People's Republic of China Huimin Li Departments of Mathematics and Control and Engineering, Huazhong University of Science and Technology Wuhan, Hubei 430074, People's Republic of China Corresponding author. Email: yangxs{at}cqupt.edu.cn Received on May 1, 2006; Accepted on October 6, 2006 In this paper, we first present some new sufficient conditionsfor global asymptotic stability of continuous-time cascade systems.Then, we give some detailed results on the region of attractionof continuous-time cascade systems. Some examples are presentedto illustrate these results.     

Characterizing the frequency of orders received by a stockist

John E. Boylan Applied Social Sciences and Humanities, Buckinghamshire Chilterns University College, Buckinghamshire HP11 2JZ, UK F. R. Johnston Kerridge Computer Company, Charnham Park, Hungerford RG17 0YU, UK Email: estelle.shale{at}warwick.ac.uk Received on 26 January 2007. Accepted on 19 November 2007. It is regularly asserted that the frequency of orders receivedby a stockist can be represented as a Poisson process, but verylittle corroborative evidence has been published. This paperpresents some results supporting the assumption. An adaptationto the standard testing methodology is presented which overcomesthe complications arising from the necessity of sampling fromtime series data. The new approach is applied to a large rangeof stock-keeping units. Except for the extreme tail of the distribution,which is of little interest for most inventory applications,it is found that the frequency of orders is well representedby a Poisson process.     

Maximum principle for optimal control of sterilization of prepackaged food

Mi-Xia Wu College of Applied Sciences, Beijing University of Technology, Beijing 100022, People's Republic of China Corresponding author. Present address: 23 Paca PL, Rockville, MD 20852-1123, USA. E-mail: sunbing{at}amss.ac.cn, sunamss{at}gmail.com Received on November 13, 2006; Revision received December 8, 2006. Accepted on January 20, 2007 This paper is concerned with an optimal control problem of thesterilization of prepackaged food. The Dubovitskii–Milyutinapproach is adopted in investigation of the Pontryagin's maximumprinciple of the system. The necessary condition is presentedfor the problem with fixed final horizon and phase constraints.     

Pricing vulnerable European options with stochastic default barriers

CF Lo and KC Ku Institute of Theoretical Physics and Department of Physics, The Chinese University of Hong Kong, Shatin, Hong Kong, China Email: cho-hoi_hui{at}hkma.gov.hk Received on 31 July 2006. Accepted on 15 March 2007. This paper develops a valuation model of European options incorporatinga stochastic default barrier, which extends a constant defaultbarrier proposed in the Hull–White model. The defaultbarrier is considered as an option writer's liability. Closed-formsolutions of vulnerable European option values based on themodel are derived to study the impact of the stochastic defaultbarriers on option values. The numerical results show that negativecorrelation between the firm values and the stochastic defaultbarriers of option writers gives material reductions in optionvalues where the options are written by firms with leverageratios corresponding to BBB or BB ratings.     

Predicting overflow in an emergency department

G. B. Byrnes Centre for Molecular, Environmental, Genetic and Analytic Epidemiology, Department of Public Health, The University of Melbourne, Victoria, Australia C. A. Bain Directorate Office, Western and Central Melbourne Integrated Cancer Service, Victoria, Australia M. Fackrell Department of Mathematics and Statistics, The University of Melbourne, Victoria, Australia C. Brand Clinical Epidemiology and Health Service Evaluation Unit, Melbourne Health, Victoria, Australia D. A. Campbell Department of Medicine, Southern Clinical School, Monash University, Victoria, Australia P. G. Taylor Department of Mathematics and Statistics, The University of Melbourne, Victoria, Australia Email: l.au{at}ms.unimelb.edu.au Received on 9 October 2007. Accepted on 4 February 2008. Ambulance bypass occurs when the emergency department (ED) ofa hospital becomes so busy that ambulances are requested totake their patients elsewhere, except in life-threatening cases.It is a major concern for hospitals in Victoria, Australia,and throughout most of the western world, not only from thepoint of view of patient safety but also financially—hospitalslose substantial performance bonuses if they go on ambulancebypass too often in a given period. We show that the main causeof ambulance bypass is the inability to move patients from theED to a ward. In order to predict the onset of ambulance bypass,the ED is modelled as a queue for treatment followed by a queuefor a ward bed. The queues are assumed to behave as inhomogeneousPoisson arrival processes. We calculate the probability of reachingsome designated capacity C within time t, given the currenttime and number of patients waiting.     

Robust H{infty} dynamic output feedback control for 2D linear parameter-varying systems

James Lam Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong Changhong Wang Space Control and Inertial Technology Research Center, Harbin Institute of Technology, Harbin 150001, People's Republic of China Corresponding author. Email: ligangwu{at}hit.edu.cn Email: james.lam{at}hku.hk Email: cwang{at}hit.edu.cn Received on June 28, 2006; Accepted on October 1, 2007 This paper is concerned with the problem of robust dynamicoutput feedback control for 2D discrete-time linear parameter-varyingsystems. Given a Fornasini–Marchesini local state-spacesystem with linear varying parameters, our attention is focussedon the design of full-order dynamic output feedback controller,which guarantees the closed-loop system to be asymptoticallystable and has a prescribed disturbance attenuation performance.A sufficient condition for the existence of a desired robustoutput feedback controller is established in terms of parameterizedlinear matrix inequalities, and the corresponding controllersynthesis is cast into a convex optimization problem which canbe efficiently handled by using standard numerical software.A numerical example is provided to illustrate the effectivenessof the proposed design method.     

Properties of the weighted logarithmic matrix norms

Mingzhu Liu Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China Email: ghu{at}hit.edu.cn, ghuca{at}yahoo.ca Email: mzliu{at}hit.edu.cn Received on January 31, 2006; Accepted on December 9, 2006 In this paper, we are concerned with the properties of the weightedlogarithmic matrix norms. A relation between the elliptic logarithmicmatrix norm and the weighted logarithmic matrix norm is given.Based on Lyapunov equations, two weighted logarithmic matrixnorms are constructed which are less than 1-logarithmic matrixnorm and -logarithmic matrix norm, respectively. Then, an iterativescheme is presented to obtain the logarithmically -efficientmatrix norm. Numerical examples are given to illustrate theresults.     

Estimation and decomposition of cost efficiency in the health care food service sector: an extended stochastic frontier approach

K. M. Matawie School of Computing and Mathematics, University of Western Sydney, Locked Bag 1797, Penrith South DC NSW 1797, Australia Email: a.assaf{at}uws.edu.au Corresponding author. Email: k.matawie{at}uws.edu.au Received on 11 February 2007. Accepted on 14 September 2007. This paper introduces and implements an extended stochasticfrontier model to estimate and decompose cost efficiency (CE)in the health care foodservice sector. The interesting featureof this model is that it allows differentiation between thethree different types of efficiency: technical efficiency (TE),allocative efficiency (AE) and their associated CE. A completeillustration of these efficiencies is presented using a cross-sectionalsample of 101 hospital foodservice operations. Results showedthat the model and all its parameter coefficients were plausible,significant and satisfy all theoretical requirements. It wasalso clear that the average efficiency scores were lower forTE (80%) than AE (88%), indicating that TE is the main sourceof CE and requires further attention.     

Demand forecasting adjustments for service-level achievement

John E. Boylan School of Business and Management, Buckinghamshire New University, Bucks, UK Email: a.syntetos{at}salford.ac.uk Received on 25 January 2007. Accepted on 3 October 2007. Demand forecasting and stock control are traditionally examinedas independent of each other. Even though this weakness hasbeen highlighted in the academic literature, little empiricalwork has been conducted on forecasting adjustments addressingthe interaction between forecasting and stock control. In thispaper, the relevant literature is critically reviewed. Subsequently,the empirical performance of some modifications and adjustments,on slow-moving items, is examined in detail. The data set consistsof the individual demand histories of 753 intermittent lineitems from the Royal Air Force (UK). Overall, the results indicatethat there is a scope for improving the performance of parametricstock control systems, and adjustments are indeed required inorder to account for the interaction between forecasting andstock control.     

An inventory model for stock with advertising sensitive demand

K. S. Chaudhuri Department of Mathematics, Jadavpur University, Kolkata-700 032, West Bengal, India Corresponding author. Email: shib_sankar{at}yahoo.com Email: k_s_chaudhuri{at}yahoo.com The effect of advertising media and sales effort on future demandof merchandize is considered. In an oligopolistic marketingsystem, it is quite natural to boost sales by using advertisingand sales effort to earn more money from a business sector.Determination of demand and costs due to advertising and saleseffort is quite difficult. The approach in this paper is toconcentrate on investment for the purpose of advertising andincreasing sales effort in order to maximize profit. The paperextends an inventory model over finite and infinite time horizonswhere demand is influenced by the level of stock, advertisingmedia and sales effort. The associated profit maximization problemis solved by the ‘Euler–Lagrange method’.The model is illustrated with a numerical example.     

Statistical solution to the capacity problem in direct-sequence code-division multiple-access communication systems

Tohru Kohda Department of Computer Science and Communication Engineering, Kyushu University, Fukuoka 812-8581, Japan Minoru Tabata Department of Mathematical Sciences, Osaka Prefecture University, Osaka 599-8531, Japan Email: eshima{at}med.oita-u.ac.jp Received on September 21, 2005; Accepted on May 2, 2006 Capacity in the direct-sequence code-division multiple-access(DS/CDMA) communication system was considered according to thecode acquisition performance with the conventional serial-searchmethod, because code acquisition needs a difficult operation.Since capacity in DS/CDMA systems is defined by the maximumnumber of users that can simultaneously transmit their signalswith the same carrier frequencies, assuring a larger capacityis very important in DS/CDMA systems with respect to the economyof frequency source. This paper reconsiders the capacity problemthrough a statistical approach to code acquisition. First, aDS/CDMA system model is reviewed. Second, properties of thecounting method for code acquisition are discussed. It is provedthat the method can acquire the target user's code under anynumber of interference users that simultaneously transmit theirsignals, and that the method can guarantee a considerable precisionin code acquisition. Third, an observation time necessary forcode acquisition is given from a statistical discussion, andthe original counting method is modified into a simpler one.It is concluded that the capacity of the DS/CDMA communicationsystem can be set by the ‘bit error-rate-based capacityor signal-to-noise ratio-based capacity’, i.e. ‘post-acquisition-basedcriterion’, rather than the ‘acquisition-based capacity’.     

A calibration algorithm for simulation-based pricing models

Don L. McLeish Department of Statistics and Actuarial Science, University of Waterloo, Waterloo,Ontario N2L 3G1, Canada Corresponding author. Email: mreesor{at}uwo.ca Email: dlmcleis{at}math.uwaterloo.ca Received on 20 February 2007. Accepted on 15 March 2007. Derivative pricing models require calibration to market conditionsin order to determine quantities such as hedging positions andthe prices of other instruments. For stochastic models and/orcomplex derivatives whose prices are not of an analytic form,prices must be computed via simulation and the calibration ismore difficult. A method to facilitate the calibration of simulation-basedpricing models is proposed. The algorithm uses a statisticallydesigned experiment to select the points at which simulationsare performed. The method is quite general as it is independentof the stochastic model for the underlying and allows for differentobjective functions that can incorporate information such asopen interest and volume. Furthermore, market prices from European-and/or American-style derivatives covering a range of strikeprices and maturities can be handled by this technique. Examplesshow the procedure is successful at calibrating well-known assetpricing models to both simulated and market data.     

Analysis of supply contracts from a supplier's perspective

Hexin Wang and Khairy A. H. Kobbacy Centre for Operational Research and Applied Statistics, University of Salford, Salford, M5 4WT, UK Email: w.wang{at}salford.ac.uk Received on 9 May 2006. Accepted on 22 December 2006. Incentive structure and demand uncertainty may cause supplychains to operate at a low efficiency. Therefore, many supplycontracts are employed in practice to improve the performanceof supply chains, i.e. to benefit all members involved in thechain. Supply chain contracts provide mechanisms to change theincentive structures of the supply chain members so that theirdecisions can improve the supply chain efficiency, while alsoprotect their own interests. It is important to understand theimpacts of supply contracts and their differences from a supplier'sperspective, since it is often the supplier who initiates asupply contract. This paper reports on a comprehensive analysisof supply contracts from a supplier's perspective. Six commonlyused supply contracts are analysed and the contract parametersare optimized to maximize the supplier's expected profit withconsideration to improve the retailer's profit. This case hasnot been thoroughly investigated in literature to date. Therisk-sharing mechanism and the division of the increased profitbetween the retailer and supplier for some of the contractsare also investigated in detail.     

A new algorithm for finding numerical solutions of optimal feedback control

Bing Sun Department of Mathematics, Beijing Institute of Technology, Beijing 100081, People's Republic of China and School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa Email: bzguo{at}iss.ac.cn Received on March 15, 2007; Revision received October 17, 2007. A new algorithm for finding numerical solutions of optimal feedbackcontrol based on dynamic programming is developed. The algorithmis based on two observations: (1) the value function of theoptimal control problem considered is the viscosity solutionof the associated Hamilton–Jacobi–Bellman (HJB)equation and (2) the appearance of the gradient of the valuefunction in the HJB equation is in the form of directional derivative.The algorithm proposes a discretization method for seeking optimalcontrol–trajectory pairs based on a finite-differencescheme in time through solving the HJB equation and state equation.We apply the algorithm to a simple optimal control problem,which can be solved analytically. The consistence of the numericalsolution obtained to its analytical counterpart indicates theeffectiveness of the algorithm.     

Forecasting item-level demands: an analytical evaluation of top-down versus bottom-up forecasting in a production-planning framework

Rajesh Piplani School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798, Republic of Singapore Email: vish{at}pmail.ntu.edu.sg Received on 12 January 2007. Accepted on 19 November 2007. We compare the performance of top–down (TD) and bottom–up(BU) strategies for forecasting the demand of an item that belongsto a product family. The TD strategy forecasts the sum of theitem demands and distributes it to the individual item basedupon the historical demand proportion of each item in the family.The BU strategy forecasts each item demand individually usingthe historical demand data for the particular item. All theitem demands, which may be correlated with each other, are assumedto follow a first-order univariate moving average process. Asis common in a production-planning environment, the forecastingunder both strategies is carried out using the exponential smoothingtechnique. We show that the performance of the two forecastingstrategies is nearly identical, regardless of the coefficientof correlation between the item demands, the items' proportionin the family and the coefficient of the serial correlationterm of the demand process. We further investigate the relativeperformance of the two strategies when a fixed (rather thanthe optimal) smoothing constant is used for forecasting thedemand under both strategies.     

Decentralized dynamic pole assignment with low-order compensators

N. Karcanias Control Engineering Centre, School of Engineering and Mathematical Sciences, City University, Northampton Square London EC1V OHB, UK Email: n.karcanias{at}city.ac.uk Received on June 14, 2006; Accepted on October 2, 2006 The problem of arbitrary pole placement via dynamic decentralizedoutput feedback is studied for minimal systems described bya proper transfer function matrix P(s) R^{m x p}(s) (m = m_i andp = p_i), with McMillan degree n. The family of controllersto be used includes those decentralized controllers with channelswhose ith channel has maximum observability index at most d_i.The method presented here is based on asymptotic linearizationaround a decentralized degenerate compensator of the pole placementmap related to the problem. It is shown that the method worksgenerically when m⁺p > n, where m⁺ = min{d_i(p_i + m_i –1) + m_i}, i = 1, ..., , and the smallest d_i of the compensatorof the ith channel is the integral part of (n – pm_i)/p(p_i+ m_i – 1).     

An unfitted finite-element method for elliptic and parabolic interface problems

Bhupen Deka Department of Mathematics, Assam University, Silchar-788011, India ^{A finite-element discretization, independent of the locationof the interface, is proposed and analysed for linear ellipticand parabolic interface problems. We establish error estimatesof optimal order in the H1-norm and almost optimal order inthe L2-norm for elliptic interface problems. An extension toparabolic interface problems is also discussed and an optimalerror estimate in the L2(0, T;H1())-norm and an almost optimalorder estimate in the L2(0, T;L2())-norm are derived for thespatially discrete scheme. A fully discrete scheme based onthe backward Euler method is analysed and an optimal order errorestimate in the L2(0, T;H1())-norm is derived. The interfacesare assumed to be of arbitrary shape and smooth for our purpose.}     

Optimality conditions for n x n infinite-order parabolic coupled systems with control constraints and general performance index

W. Kotarski Institute of Informatics, Silesian University, Bedzinska 60, 41-200 Sosnowiec, Poland Email: bahaa_gm{at}hotmail.com Email: kotarski{at}gate.math.us.edu.pl Received on March 14, 2006; Accepted on December 20, 2006 A distributed control problem for n x n parabolic coupled systemsinvolving operators with infinite order is considered. The performanceindex is more general than the quadratic one and has an integralform. Constraints on controls are imposed. Making use of theDubovitskii–Milyutin theorem, the necessary and sufficientconditions of optimality are derived for the Dirichlet problem.Yet, the problem considered here is more general than the problemsin El-Saify & Bahaa (2002, Optimal control for n x n hyperbolicsystems involving operators of infinite order. Math. Slovaca,52, 409–424), El-Zahaby (2002, Optimal control of systemsgoverned by infinite order operators. Proceeding (Abstracts)of the International Conference of Mathematics (Trends and Developments)of the Egyptian Mathematical Society, Cairo, Egypt, 28–31December 2002. J. Egypt. Math. Soc. (submitted)), Gali &El-Saify (1983, Control of system governed by infinite orderequation of hyperbolic type. Proceeding of the InternationalConference on Functional-Differential Systems and Related Topics,vol. III. Poland, pp. 99–103), Gali et al. (1983, Distributedcontrol of a system governed by Dirichlet and Neumann problemsfor elliptic equations of infinite order. Proceeding of theInternational Conference on Functional-Differential Systemsand Related Topics, vol. III. Poland, pp. 83–87) and Kotarskiet al. (200b, Optimal control problem for a hyperbolic systemwith mixed control-state constraints involving operator of infiniteorder. Int. J. Pure Appl. Math., 1, 241–254).     

Adaptive frame methods for elliptic operator equations: the steepest descent approach

Massimo Fornasier^¶ Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università "La Sapienza" in Roma, Via Antonio Scarpa, 16/B, I-00161 Roma, Italy Rob Stevenson^|| Department of Mathematics, Utrecht University, PO Box 80.010, NL-3508 TA Utrecht, The Netherlands ^{This paper is concerned with the development of adaptive numericalmethods for elliptic operator equations. We are particularlyinterested in discretization schemes based on wavelet frames.We show that by using three basic subroutines an implementable,convergent scheme can be derived, which, moreover, has optimalcomputational complexity. The scheme is based on adaptive steepestdescent iterations. We illustrate our findings by numericalresults for the computation of solutions of the Poisson equationwith limited Sobolev smoothness on intervals in 1D and L-shapeddomains in 2D.}     

Sparse convolution quadrature for time domain boundary integral formulations of the wave equation

S. A. Sauter Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland ^{Many important physical applications are governed by the waveequation. The formulation as time domain boundary integral equationsinvolves retarded potentials. For the numerical solution ofthis problem, we employ the convolution quadrature method forthe discretization in time and the Galerkin boundary elementmethod for the space discretization. We introduce a simple apriori cut-off strategy where small entries of the system matricesare replaced by zero. The threshold for the cut-off is determinedby an a priori analysis which will be developed in this paper.This analysis will also allow to estimate the effect of additionalperturbations such as panel clustering and numerical integrationon the overall discretization error. This method reduces thestorage complexity for time domain integral equations from O(M2N)to O(M2N logM), where N denotes the number of time steps andM is the dimension of the boundary element space.}