On a market equilibrium theorem with an infinite number of commodities

A direct proof is given of the market equilibrium theorem of Gale, Nikaido and Debreu for an infinite-dimensional commodity space. The theorem is closely related to a recent result of Aliprantis and Brown, but allows for excess demand correspondences rather than excess demand functions.     

Oil Price Shocks and Long Run Price and Import Demand Behavior

The effect which the oil price time series has on the long run properties of Vector AutoRegressive (VAR) models for price levels and import demand is investigated. As the oil price variable is assumed to be weakly exogenous for the long run parameters, a cointegration testing procedure allowing for weakly exogenous variables is developed using a LU decomposition of the long run multiplier matrix. The likelihood based cointegration test statistics, Wald, Likelihood Ratio and Lagrange Multiplier, are constructed and their limiting distributions derived. Using these tests, we find that incorporating the oil price in a model for the domestic or import price level of seven industrialized countries decreases the long run memory of the inflation rate. Second, we find that the results for import demand can be classified with respect to the oil importing or exporting status of the specific country. The result for Japan is typical as its import price is not influenced by GNP in the long run, which is the case for all other countries.     

考虑保质期的非即时易腐品最优订购与定价决策

针对有保质期约束的非即时易腐品,零售商可以在产品非腐败阶段和腐败阶段进行差异化定价来调整市场需求。即零售商可以调整产品非腐败阶段的定价策略,使得非即时易腐品在进入腐败阶段之前全部售出;或者在产品腐败阶段降低零售价格以刺激市场需求。因此,在需求依赖价格的假设下,本文研究了零售商关于有保质期约束的非即时易腐品的最优订购与定价决策。通过分析零售商单位时间利润函数的理论性质,得到了零售商的最优订购与定价策略。基于线性和指数型两种需求函数形式进行数值算例,可以发现针对保质期较长或者非腐败阶段较长的易腐品,零售商会延长订货周期来增加销售收入。特别是在指数型需求函数情境下,当市场需求与零售价格高度相关时,零售商更有意愿降价来刺激销量,从而使得易腐品在非腐败阶段内销售完毕。     

On the effect of centralisation on expected profits in a multi-location Newsboy problem

This paper is a multi-location, single-period, single-product inventory problem with an opportunity for centralisation. The decentralised model in which a separate inventory is kept at every location is compared to the centralised system in which the demands are satisfied from one central warehouse. The two systems are then compared when excess demands are penalised and lost, and in the more general case where a portion or all of the excess demand at a location may be reallocated among other locations with the remaining inventory. The revenue, salvage, cost, and penalty functions at each location are assumed to be identical. Expected profits are evaluated and conditions under which the centralised model outperforms the decentralised one are derived.     

On the propagation of a multi-dimensional shock of arbitrary strength

In this paper the kinematics of a curved shock of arbitrary strength has been discussed using the theory of generalised functions. This is the extension of Moslov’s work where he has considered isentropic flow even across the shock. The condition for a nontrivial jump in the flow variables gives the shock manifold equation (sme). An equation for the rate of change of shock strength along the shock rays (defined as the characteristics of the sme) has been obtained. This exact result is then compared with the approximate result of shock dynamics derived by Whitham. The comparison shows that the approximate equations of shock dynamics deviate considerably from the exact equations derived here. In the last section we have derived the conservation form of our shock dynamic equations. These conservation forms would be very useful in numerical computations as it would allow us to derive difference schemes for which it would not be necessary to fit the shock-shock explicitly.     

Adomian decomposition method is a special case of Lyapunov’s artificial small parameter method

The Adomian decomposition method and Lyapunov’s artificial small parameter method are two popular analytic methods for solving nonlinear differential equations. In this short paper, we show that the Adomian decomposition method can be derived from the more general Lyapunov’s artificial small parameter method.     

Joint demand and capacity management in a restaurant system

This paper studies the demand and capacity management problem in a restaurant system. A queueing-based optimization model with underlying quasi birth-and-death process and state-dependent functions is developed to address the dynamic and nonlinearity difficulties. In particular, our model explicitly captures the demand changes with respect to the system congestion state on a near real-time dynamic basis. With this framework, we empirically examine the relative performance of commonly used strategies for the case of a local restaurant. The study shows that a strategy that balances service quality and cost yields maximum profit. The result indicates that the traditional view of the conflict between service quality and cost can be overcome by using an interdisciplinary perspective of marketing and operations. Both perspectives should be embraced in academic research and industrial practice in capacity planning decisions for services.     

Introducing Undergraduates to the Spirit of OR whilst Imparting Substantive Skills

This paper presents analytical expressions for the moments of job flow time in a job shop by computing the moments of a random number of random variables assuming that the job shop satisfies the sufficient conditions of the Jackson decomposition principle. The mathematical equivalence of the proposed method to the Laplace transform method is shown and then demonstrated in a simple example. Then, expressions for the moments of job flow time in more realistic (and more complex) job shop settings are derived by computing the moments of a random number of random variables, provided that expressions for the moments of job flow time are available for the corresponding single stage shop. It is shown that the analytical expressions developed are exact as long as the assumptions of the Jackson decomposition principle hold. Any violation of these assumptions makes the derived expressions approximations, and their accuracy should be tested by simulation.     

Decomposition in multi-item inventory control

This paper considers a Lagrangian decomposition approach to a stochastic demand multi-item inventory control problem with a single resource constraint. The work is a generalization of existing decomposition methods.Three decomposition methods are proposed, and bounds on the loss of optimality for each are given in terms of the Lagrange multiplier used. One method allows the calculation of the complete decision rule in advance of the realization of the states, but is expected to perform worse than the other two methods. The second and third method allow the determination of decisions as an optimization problem as the states are realized. Since, in any problem with many states, only a small proportion will actually be realized even in a large time-horizon problem, there may be some advantage in taking this approach.     

A Geometric Study of the Split Decomposition

This paper sheds new light on split decomposition theory and T-theory from the viewpoint of convex analysis and polyhedral geometry. By regarding finite metrics as discrete concave functions, Bandelt-Dress' split decomposition can be derived as a special case of more general decomposition of polyhedral/discrete concave functions introduced in this paper. It is shown that the combinatorics of splits discussed in connection with the split decomposition corresponds to the geometric properties of a hyperplane arrangement and a point configuration. Using our approach, the split decomposition of metrics can be naturally extended to distance functions, which may violate the triangle inequality, using partial split distances.     

A first order characterization of generalized monotonicity

It is shown that pseudomonotone and quasimonotone maps can be characterized by a first order property provided they are regular. This result extends the well known characterization of nonvanishing generalized monotone maps to an essentially larger class. The paper supplements a recent contribution by Crouzeix and Ferland (1996) and solves a related open problem concerning homogeneous excess demand functions which occur in general equilibrium theory. Received: April 10, 1998 / Accepted: February 29, 2000?Published online April 20, 2000     

Stochastic multi-site capacity planning of TFT-LCD manufacturing using expected shadow-price based decomposition

This paper presents a stochastic optimization model and efficient decomposition algorithm for multi-site capacity planning under the uncertainty of the TFT-LCD industry. The objective of the stochastic capacity planning is to determine a robust capacity allocation and expansion policy hedged against demand uncertainties because the demand forecasts faced by TFT-LCD manufacturers are usually inaccurate and vary rapidly over time. A two-stage scenario-based stochastic mixed integer programming model that extends the deterministic multi-site capacity planning model proposed by Chen et al. (2010) [1] is developed to discuss the multi-site capacity planning problem in the face of uncertain demands. In addition a three-step methodology is proposed to generate discrete demand scenarios within the stochastic optimization model by approximating the stochastic continuous demand process fitted from the historical data. An expected shadow-price based decomposition, a novel algorithm for the stage decomposition approach, is developed to obtain a near-optimal solution efficiently through iterative procedures and parallel computing. Preliminary computational study shows that the proposed decomposition algorithm successfully addresses the large-scale stochastic capacity planning model in terms of solution quality and computation time. The proposed algorithm also outperforms the plain use of the CPLEX MIP solver as the problem size becomes larger and the number of demand scenarios increases.     

Optimal pricing and production in an inventory model

This paper deals with the problem of simultaneously determining the optimal price policy and production rate over a given planning horizon. For nonlinear demand functions and convex inventory and shortage cost functions the optimal solution paths are derived by using optimal control theory. The treatment of linear nonsmooth cost functions requires the use of a generalized maximum principle. The solution method is a phase portrait analysis providing insight into the optimal pricing and production policies as well as the resulting inventory paths. Moreover, it is shown that in the case of nonsmooth piecewise linear cost functions the equilibrium is approached within finite time although the model is nonlinear in the control variables. Finally it is illustrated that exogenous fluctuations in the demand rate (seasonal demand pattern) amount to cyclical optimal solutions.     

超额市场需求函数

本文通过研究超额需求函数与多项式的关系得：（１）对于（ｎ—１）个关于ｐ１,ｐ２…,ｐｎ－１的多项式Ｅ。（ｐ１,ｐ２,…,ｐｎ－１）,ｓ＝１,２,…,ｎ—１,若满足条件：则Ｅｓ,ｓ＝１,２,…,ｎ—１均为一个经济的超额需求函数．（２）对无穷级数ｇ（ｐ）,也存在含有两个消费者与无穷种商品的经济,使得其超额需求函数恰好为ｇ（ｐ）．     

Balanced and Unbalanced Components of Moist Atmospheric Flows with Phase Changes

Atmospheric variables(temperature, velocity, etc.) are often decomposed into balanced and unbalanced components that represent low-frequency and high-frequency waves, respectively. Such decompositions can be defined, for instance, in terms of eigenmodes of a linear operator. Traditionally these decompositions ignore phase changes of water since phase changes create a piecewise-linear operator that differs in different phases(cloudy versus non-cloudy). Here we investigate the following question: How can a balanced–unbalanced decomposition be performed in the presence of phase changes? A method is described here motivated by the case of small Froude and Rossby numbers,in which case the asymptotic limit yields precipitating quasi-geostrophic equations with phase changes. Facilitated by its zero-frequency eigenvalue, the balanced component can be found by potential vorticity(PV) inversion, by solving an elliptic partial differential equation(PDE), which includes Heaviside discontinuities due to phase changes. The method is also compared with two simpler methods: one which neglects phase changes, and one which simply treats the raw pressure data as a streamfunction. Tests are shown for both synthetic, idealized data and data from Weather Research and Forecasting(WRF) model simulations. In comparisons, the phase-change method and no-phase-change method produce substantial differences within cloudy regions, of approximately 5K in potential temperature, due to the presence of clouds and phase changes in the data. A theoretical justification is also derived in the form of a elliptic PDE for the differences in the two streamfunctions.     

Pricing policy in dynamic markets and a generalization of the Ramsey rule

Pricing policy in a regulated monopoly industry is usually based on maximizing welfare or some other measure of utility level of return on investment. Previously, the Ramsey pricing policy which states that the percentage deviation of quasi-optimal price from marginal cost for each product must be inversely proportional to its price elasticity of demand, has been developed for a static market. The Ramsey framework assumes instantaneous demand response to price changes; empirical evidence suggests demand changes occur dynamically through time.In this paper an optimum pricing rule for a profit maximizing firm based on a general time varying demand model in a dynamic market is obtained assuming a single price change at the beginning of the planning period. A dynamic market equivalent of the well known inverse elasticity law of the static market is developed. Defining the concept of average price elasticity for dynamic markets we show that the inverse elasticity law of static markets takes an inequality form in dynamic markets. For demand functions which decrease, increase or are constant with time the optimum price markups are greater than, less than, or equal to the inverse of the average price elasticity, respectively.The results are then generalized to the case of a constrained welfare maximizing firm. This leads to the development of a dynamic market generalization of the well known Ramsey pricing rule. A simple rule for making quantitative arguments about the relative size of the optimum price in static and dynamic markets is also derived.This work was completed when the author was with Bell Laboratories, USA.     

General theory of domain decomposition: Indirect methods

According to a general theory of domain decomposition methods (DDM), recently proposed by Herrera, DDM may be classified into two broad categories: direct and indirect (or Trefftz‐Herrera methods). This article is devoted to formulate systematically indirect methods and apply them to differential equations in several dimensions. They have interest since they subsume some of the best‐known formulations of domain decomposition methods, such as those based on the application of Steklov‐Poincaré operators. Trefftz‐Herrera approach is based on a special kind of Green's formulas applicable to discontinuous functions, and one of their essential features is the use of weighting functions which yield information, about the sought solution, at the internal boundary of the domain decomposition exclusively. A special class of Sobolev spaces is introduced in which boundary value problems with prescribed jumps at the internal boundary are formulated. Green's formulas applicable in such Sobolev spaces, which contain discontinuous functions, are established and from them the general framework for indirect methods is derived. Guidelines for the construction of the special kind of test functions are then supplied and, as an illustration, the method is applied to elliptic problems in several dimensions. A nonstandard method of collocation is derived in this manner, which possesses significant advantages over more standard procedures. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 296–322, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10008     

Moment Sets of Bell–Shaped Distributions: Extreme Points,Extremal Decomposition and Chebysheff Inequalities

The paper deals with sets of distributions which are given by moment conditions and convex constraints on derivatives of their cumulative distribution functions. A general albeit simple method for the study of their extremal structure, extremal decomposition and topological or measure theoretical properties is developed. Its power is demonstrated by the application to bell–shaped distributions. Extreme points of their moment sets are characterized completely (thus filling a gap in the previous theory) and inequalities of Chebysheff type are derived by means of general integral representation theorems.     

The Dynamic Interaction of Speculation and Diversification

A discrete time model of a financial market is developed, in which heterogeneous interacting groups of agents allocate their wealth between two risky assets and a riskless asset. In each period each group formulates its demand for the risky assets and the risk‐free asset according to myopic mean‐variance maximizazion. The market consists of two types of agents: fundamentalists, who hold an estimate of the fundamental values of the risky assets and whose demand for each asset is a function of the deviation of the current price from the fundamental, and chartists, a group basing their trading decisions on an analysis of past returns. The time evolution of the prices is modelled by assuming the existence of a market maker, who sets excess demand of each asset to zero at the end of each trading period by taking an offsetting long or short position, and who announces the next period prices as functions of the excess demand for each asset and with a view to long‐run market stability. The model is reduced to a seven‐dimensional nonlinear discrete‐time dynamical system, that describes the time evolution of prices and agents' beliefs about expected returns, variances and correlation. The unique steady state of the model is determined and the local asymptotic stability of the equilibrium is analysed, as a function of the key parameters that characterize agents' behaviour. In particular it is shown that when chartists update their expectations sufficiently fast, then the stability of the equilibrium is lost through a supercritical Neimark–Hopf bifurcation, and self‐sustained price fluctuations along an attracting limit cycle appear in one or both markets. Global analysis is also performed, by using numerical techniques, in order to understand the role played by the chartists' behaviour in the transition to a regime characterized by irregular oscillatory motion and coexistence of attractors. It is also shown how changes occurring in one market may affect the price dynamics of the alternative risky asset, as a consequence of the dynamic updating of agents' portfolios.     

考虑空间效应的多支撑管线随机地震响应分析

提出了一种能考虑地震动空间变化效应的多支撑管线随机地震响应分析的解析方法.证明了多点地震作用下结构的平稳随机响应分析可转化为求解支座简谐运动时的确定性响应,直接给出了含有待定系数的简谐响应的形式,并通过边界条件和连续性条件建立待定系数的求解方程.与拟静位移分解法相比,该方法不用计算结构的振型以及拟静位移分量,完全是基于解析推导,因此在计算效率方面优势明显.数值算例中,采用该方法和拟静位移分解法计算了一个6跨管线在空间多点地震作用下的随机响应,对比验证了方法的正确性和高效性.