Aspects of optimal insurance demand when there are uninsurable risks

The standard model of insurance demand is enlarged by the inclusion of uninsurable risks. When these risks are independent of the insurable risk standard methods can be applied to show the effects of uninsurable risks on the insurance premium and on optimal coinsurance. In case of dependence it is shown that expected utility theory can imply counterintuitive results for the behaviour of risk-averse insurance buyers. It is shown that expected local utility theory gives more intuitive results.     

Estimating the demand distributions of single-period items having frequent stockouts

Very often, the service level of a single-period newsboy-type product is set at such a low level that: (i) stockouts occur in the majority of the periods, and (ii) a large right-hand side of the empirical demand distribution is never observable. This paper reports a practical approach for estimating the periodic-demand distribution of such a product. The approach has three components: (i) using the non-parametric ‘product limit’ method to estimate the fractiles of the observable left-hand side of the empirical distribution; (ii) using a subjective approach and an ‘extrapolation of hourly sales’ approach to ‘fill in’ the missing right-hand side of the empirical distribution; (iii) fitting the estimates obtained in the preceding two components to a Tocher curve — which can handle the diversity of shapes of a realistic demand distribution and is also computationally very convenient for subsequent calculations for production/inventory decisions. The entire approach is shown to be simpler but more powerful than existing alternatives for the problem.     

Interval Newton/generalized bisection when there are singularities near roots

Interval Newton methods in conjunction with generalized bisection are important elemetns of algorithms which find theglobal optimum within a specified box X ⁿ of an objective function whose critical points are solutions to the system of nonlinear equationsF(X)=0with mathematical certainty, even in finite presision arithmetic. The overall efficiency of such a scheme depends on the power of the interval Newton method to reduce the widths of the coordinate intervals of the box. Thus, though the generalized bisection method will still converge in a box which contains a critical point at which the Jacobian matrix is singular, the process is much more costly in that case. Here, we propose modifications which make the generalized bisection method isolate singular solutions more efficiently. These modifications are based on an observation about the verification property of interval Newton methods and on techniques for detecting the singularity and removing the region containing it. The modifications assume no special structure forF. Additionally, one of the observations should also make the algorithm more efficient when finding nonsingular solutions. We present results of computational experiments.     

Stability in systems with aftereffect when there are singularities in the integral kernels

Systems with aftereffect are considered. The state of these systems is described by integrodifferential equations of the Volterra type, which depend on functionals in integral form and, in particular, on analytic functionals which are represented by Frechet series. The integral kernels can allow of singularities of Abel kernel singularities. The total stability (i.e. stability under persistent disturbances) is investigated, and the structure of the general solution is investigated in the neighbourhood of zero for an equation with a holomorphic non-linearity assuming asymptotic stability of the trivial solution of the linearized unperturbed equation. The conditions for instability are given in the critical case of a single zero root, which generalise results obtained previously.     

Forecasting of compound Erlang demand

Intermittent demand items dominate service and repair inventories in many industries and they are known to be the source of dramatic inefficiencies in the defence sector. However, research in forecasting such items has been limited. Previous work in this area has been developed upon the assumption of a Bernoulli or a Poisson demand arrival process. Nevertheless, intermittent demand patterns may often deviate from the memory-less assumption. In this work we extend analytically previous important results to model intermittent demand based on a compound Erlang process, and we provide a comprehensive categorisation scheme to be used for forecasting purposes. In a numerical investigation we assess the benefit of departing from the memory-less assumption and we provide insights into how the degree of determinism inherent in the process affects forecast accuracy. Operationalised suggestions are offered to managers and software manufacturers dealing with intermittent demand items.     

Optimal diffusion of a new technology when both demand and supply are nonstatic

The investment problem of a monopolized sector selling an innovated product is explored. Learning by doing is supposed to occur on the supply side, while learning by using is introduced to explain demand growth. Pontryagin's maximum principle is applied to the resulting optimal control problem, which includes supply capacity and cumulative output as state variables. The optimal investment policy turns out to be of a very simple form: all profit is retained and invested until capacity achieves its optimal size. In spite of this, the new technology price displays a variety of time patterns that heavily depend on the actual demand and cost conditions, as one would expect in the real world.The authors express their gratitude to Professor Sergio Rinaldi for helpful comments. This work was partially supported by Centro Teoria dei Sistemi, CNR, Milano, Italy.     

Forecasting newspaper demand with censored regression

Newspaper circulation has to be determined at the level of the individual retail outlets for each of the editions to be sold through such outlets. Traditional forecasting methods provide no insight into the impact of the service level: defined as the probability that no out-of-stock will occur. The service level results in out-of-stock situations, causing missed sales and oversupply or returns. In our application management sets a policy aiming at a 97% service level. The forecasting system developed provides estimates for excess deliveries and for the expected shortages. The results compare favourably to the traditional moving average approach previously employed by the publisher. Censored regression is a natural approach to the newspaper problem. It provides information on key policy variables and it is relatively simple to integrate into the distribution policy, with only small adaptations to the existing forecasting and distribution policy.     

The stability of an inverted pendulum when there are rapid random oscillations of the suspension point

The stability of the upper equilibrium position of a pendulum when the suspension point makes rapid random oscillations of small amplitude, is investigated. A class of random oscillations that make the system stable with unit probability for small friction is indicated. It is shown that, if there is no friction, there is no stability, which, as is well known, is not the case for harmonic oscillations of the suspension point. Some general results concerning the impossibility of stochastic stabilization of Hamiltonian systems are proved.     

On the convergence of the DFP algorithm for unconstrained optimization when there are only two variables

Let the DFP algorithm for unconstrained optimization be applied to an objective function that has continuous second derivatives and bounded level sets, where each line search finds the first local minimum. It is proved that the calculated gradients are not bounded away from zero if there are only two variables. The new feature of this work is that there is no need for the objective function to be convex. Received: June 16, 1999 / Accepted: December 24, 1999?Published online March 15, 2000     

Robust distribution planning for supplier-managed inventory agreements when demand rates and travel times are stationary

Supplier-managed inventory (SMI) is a partnering agreement between a supplier and his customers. Under this SMI agreement, inventory monitoring and ordering responsibilities are entirely transferred to the supplier. Subsequently, the supplier decides both the quantity and timing of his customer deliveries. The inventory routing problem is an underlying optimization model for SMI partnerships to cost-effectively coordinate and manage customer inventories and related replenishments logistics. This paper discusses the case where customer demand rates and travel times are stochastic but stationary, and proposes a version of the inventory routing optimization model that generates optimal robust distribution plans. The approach proposed to obtain and deploy these robust plans combines optimization and Monte Carlo simulation. Optimization is used to determine the robust distribution plan and simulation is used to fine-tune the plan's critical parameters such as replenishment cycle times and safety stock levels. Results of a simplified real-life case implementing the proposed optimization-simulation approach are shown and discussed in detail.     

A model in which there are Jech-Kunen trees but there are no Kurepa trees

By an ω₁-tree we mean a tree of power ω₁ and height ω₁. We call an ω₁-tree a Jech-Kunen tree if it hask-many branches for somek strictly between ω₁ and 2^ω1. In this paper we construct the models of CH plus 2^ω1 > ω₂, in which there are Jech-Kunen trees and there are no Kurepa trees. The research of the first author was partially supported by the Basic Research Fund, Israeli Academy of Science, Publ. No. 466.     

The contact problem when there are friction forces in the contact area for a three-component cylindrical base

The plane contact problem of elasticity theory on the interaction when there are friction forces in the contact area of an absolutely rigid cylinder (punch) with an internal surface of a cylindrical base, consisting of two circular cylindrical layers rigidly connected to one another and with an elastic space, is considered. The layers and space have different elastic constants. A vertical force and a counterclockwise torque, act on the punch, and the punch – base system is in a state of limiting equilibrium,. An exact integral equation of the first kind with a kernel represented in an explicit analytical form, is obtained for the first time for this problem using analytical calculation programs. The main properties of the kernel of the integral equation are investigated, and it is shown that the numerator and denominator of the kernel symbols can be represented in the form of polynomials in products of the powers of the moduli of the displacement of the layers and the half-space. A solution of the integral equation is constructed by the direct collocation method, which enables the solution of the problem to be obtained for practically any values of the initial parameters. The contact stress distributions, the dimensions of the contact area, the interconnection between the punch displacement and the forces and torques acting on it are calculated as a function of the geometrical and mechanical parameters of the layers and the space. The results of the calculations in special cases are compared with previously known results.     

Forecasting intermittent demand: a comparative study

Methods for forecasting intermittent demand are compared using a large data set from the UK Royal Air Force. Several important results are found. First, we show that the traditional per period forecast error measures are not appropriate for intermittent demand, even though they are consistently used in the literature. Second, by comparing the ability to approximate target service levels and stock holding implications, we show that Croston's method (and a variant) and Bootstrapping clearly outperform Moving Average and Single Exponential Smoothing. Third, we show that the performance of Croston and Bootstrapping can be significantly improved by taking into account that an order in a period is triggered by a demand in that period.     

Deterministic economic order quantity models with partial backlogging when demand and cost are fluctuating with time

Journal of the Operational Research Society - In today's time-based competition, the unit cost of a high-tech product declines significantly over its short life cycle while its demand...     

Capacity allocation for demand of different customer-product-combinations with cancellations, no-shows, and overbooking when there is a sequential delivery of service

We consider a problem where different classes of customers can book different types of services in advance and the service company has to respond immediately to the booking request confirming or rejecting it. Due to the possibility of cancellations before the day of service, or no-shows at the day of service, overbooking the given capacity is a viable decision. The objective of the service company is to maximize profit made of class-type specific revenues, refunds for cancellations or no-shows as well as the cost of overtime. For the calculation of the latter, information of the underlying appointment schedule is required. Throughout the paper we will relate the problem to capacity allocation in radiology services. Drawing upon ideas from revenue management, overbooking, and appointment scheduling we model the problem as a Markov decision process in discrete time which due to proper aggregation can be optimally solved with an iterative stochastic dynamic programming approach. In an experimental study we successfully apply the approach to a real world problem with data from the radiology department of a hospital. Furthermore, we compare the optimal policy to four heuristic policies, of whom one is currently in use. We can show that the optimal policy significantly improves the currently used policy and that a nested booking limit type policy closely approximates the optimal policy and is thus recommended for use in practice.     

Asymptotic solution of contact problemsfor a relatively thick elastic layer when there are friction forces in the contact area

The plane contact problem of the theory of elasticity of the interaction between a punch, having a base in the form of a paraboloid,and a layer, taking Coulomb friction in the contact region into account, is considered. It is assumed that either the lower boundary of the layer is fixed or there are no normal displacements and shear stresses on it, and that normal and shear forces are acting on the punch. Here, the punch-layer system is in a condition of limit equilibrium, and the punch does not turn during the deformation of the layer. The case of quasi-statistics, when the punch moves evenly over the layer surface, can be considered similarly in a moving system of coordinates. The problem is investigated by the large-λ method (see [1–3], etc.), which is further developed here, namely, simple recurrence relations are derived for constructing any number of terms of the series expansion of the solution of the corresponding integral equation in negative powers of the dimensionless parameter λ related to the thickness of the layer.     

Forecasting energy demand using neural-network-based grey residual modification models

Grey forecasting models have taken an important role for forecasting energy demand, particularly the GM(1,1) model, because they are able to construct a forecasting model using a limited samples without statistical assumptions. To improve prediction accuracy of a GM(1,1) model, its predicted values are often adjusted by establishing a residual GM(1,1) model, which together form a grey residual modification model. Two main issues should be considered: the sign estimation for a predicted residual and the way the two models are constructed. Previous studies have concentrated on the former issue. However, since both models are usually established in the traditional manner, which is dependent on a specific parameter that is not easily determined, this paper focuses on the latter issue, incorporating the neural-network-based GM(1,1) model into a residual modification model to resolve the drawback. Prediction accuracies of the proposed neural-network-based prediction models were verified using real power and energy demand cases. Experimental results verify that the proposed prediction models perform well in comparison with original ones.     

Correlation when data are missing

Variable correlation is important for many operations research models. Manyinventory, revenue management, and queuing models presume uncorrelated demandbetween products, market segments, or time periods. The specific model applied,or the resulting policies of a model, can differ drastically depending onvariable correlation. Having missing data are a common problem for the realworld application of operations research models. This work is at the junction ofthe two topics of correlation and missing data. We propose a test ofindependence between two variables when data are missing. The typical method fordetermining correlation with missing data ignores all data pairs in which onepoint is missing. The test presented here incorporates all data. The test can beapplied when both variables are continuous, when both are discrete, or when onevariable is discrete and the other is continuous. The test makes no assumptionsabout the distribution of the two variables, and thus it can be used to extendapplication of non-parametric rank tests, such as Spearman's rankcorrelation, to the case where data are missing. An example is shown wherefailure to incorporate the incomplete data yields incorrect policies.     

Stochastic impulse control with regime switching for the optimal dividend policy when there are business cycles,taxes and fixed costs

We consider the problem of finding the optimal dividend policy for a company whose cash reserve follows a Brownian motion with drift and volatility modulated by an observable finite-state continuous-time Markov chain. The Markov chain represents the regime of the economy. We allow fixed costs and taxes associated with the dividend payments. This optimization problem generates a stochastic impulse control problem with regime switching. We solve this problem and obtain the first analytical solutions for the optimal dividend policy when there are simultaneously fixed costs, taxes and business cycles. Our results show that the optimal dividend policy depends strongly on the regime of the economy, on fixed costs and on taxes.     

Strategic new product pricing when demand obeys saturation effects

We analyze dynamic pricing strategies for new products over an infinite planning horizon in a duopolistic market. The sales dynamic is modelled as a linear demand function with saturation effects, marginal costs are assumed to be constant. The optimal pricing strategies are obtained as (degenerate) closed-loop Nash solutions. It is shown that the optimal dynamic prices are greater than the static ones. In the case of no discounting there is in addition to the constant solution also an equilibrium with monotonically increasing prices.