Optimal myopic policies and index policies for stochastic scheduling problems

Stochastic scheduling problems are considered by using discounted dynamic programming. Both, maximizing pure rewards and minimizing linear holding costs are treated in one common Markov decision problem. A sufficient condition for the optimality of the myopic policy for finite and infinite horizon is given. For the infinite horizon case we show the optimality of an index policy and give a sufficient condition for the index policy to be myopic. Moreover, the relation between the two sufficient conditions is discussed.     

Application of renewal theory to call handover counting and dynamic location management in cellular mobile networks

Mobility management in wireless cellular networks is one of the main issues for resource optimization. It is aimed to keep track of Mobile Stations (MSs) in the different Location Areas (LAs) or Registration Areas (RAs) for an efficient call delivery. The optimization issues of these location strategies look for a minimization of the generated signaling traffic. We describe the three basic strategies for location management: distance-based, time-based and movement-based, and their corresponding optimization cost. We emphasize that counting the number of wireless cell crossings or handovers occurring in the call duration time or during inter-call times is a fundamental issue for mobility management analysis. We present the main approaches in the literature to deal with these problems with a special emphasis to renewal theory to model the probabilistic structure of these optimization problems.     

Optimal maintenance of a production-inventory system with idle periods

In this paper we consider a production-inventory system in which an input generating installation supplies a buffer with a raw material and a production unit pulls the raw material from the buffer with constant rate. The installation deteriorates in time and the problem of its optimal preventive maintenance is considered. It is assumed that the installation after the completion of its maintenance remains idle until the buffer is evacuated. Under a suitable cost structure it is shown that the average-cost optimal policy for fixed buffer content is of control-limit type, i.e. it prescribes a preventive maintenance of the installation if and only if its degree of deterioration is greater than or equal to a critical level. Using the usual regenerative argument, the average cost of a control-limit policy is computed exactly and then, the optimal control-limit policy is determined. Furthermore, the stationary probabilities of the system under the optimal policy are computed.     

状态相依效用下的超额损失再保险——投资策略

假设保险公司的盈余过程和金融市场的资产价格过程均由可观测的连续时间马尔科夫链所调节, 以最大化终端财富的状态相依的期望指数效用为目标, 研究了保险公司的超额损失再保险-投资问题. 运用动态规划方法, 得到最优再保险-投资策略的解析解以及最优值函数的半解析式. 最后, 通过数值例子, 分析了模型各参数对最优值函数和最优策略的影响.     

Approximate Optimal Solutions and Nonlinear Lagrangian Functions*

There is an increasing interest in the study of optimality conditions of approximate solutions for nonlinear optimization problems. In this paper, relationships between approximate optimal values and approximate roots of a nonlinear function are explored via a nonlinear Lagrangian function. Almost approximate optimal solutions are investigated by means of nonlinear Lagrangian functions.     

Modeling Optimal Allocation of Deficit Irrigation: Application to Crop Production in Saudi Arabia

We develop an integrated dynamic programming—linear programming (LP) model to solve for optimal land exploitation for a given crop. The model applies deficit irrigation in order to increase the irrigated area at the expense of reducing the crop yield per unit area. The dynamic program guarantees that deficit irrigation is considered only when it is economically efficient. Moreover, it provides the best irrigation level for each growth stage of the crop, accounting for the varying impact of water stress overtime. The LP provides the best tradeoff between expanding the irrigated area and decreasing water share per hectare. The model objective is to maximize the total expected crop yield. The model is particularly applicable for regions suffering from irrigation water scarcity, such as Saudi Arabia. The implementation was made for crops in Al-Jouf Region, north of Saudi Arabia     

Functional Optimal Estimation Problems and Their Solution by Nonlinear Approximation Schemes

The design of state estimators for nonlinear dynamic systems affected by disturbances is addressed in a functional optimization framework. The estimator contains an innovation function that has to be chosen within a suitably defined class of functions in such a way to minimize a cost functional given by the worst-case ratio of the ℒ _p norms of the estimation error and the disturbances. Since this entails an infinite-dimensional optimization problem that under general hypotheses cannot be solved analytically, an approximate solution is sought by minimizing the cost functional over linear combinations of simple “basis functions,” represented by computational units with adjustable parameters. The selection of the parameters is made by solving a constrained nonlinear programming problem, where the constraints are given by pointwise conditions that ensure the well-definiteness of the functional and the existence of a solution. Penalty terms are introduced in the cost function to account for constraints imposed on points that result from sampling the sets to which the trajectories of the state and of the estimation error belong. To ensure an efficient covering of the sets, low-discrepancy sampling techniques are exploited that generate samples deterministically spread in a uniform way, without leaving regions of the space undersampled. Work supported by a PRIN grant from the Italian Ministry of University and Research (Project “New Techniques for the Identification and Adaptive Control of Industrial Systems”) and by the EU and the Regione Liguria trough the Regional Programs of Innovative Action of the European Regional Development Fund.     

Optimal regulation policies for a multipurpose reservoir with seasonal input and return function

Optimal regulation policies for a multipurpose reservoir are characterized analytically for the case where the input and return functions are subject to seasonal variations and the return function is piecewise linear, concave, and separable in reservoir volume and discharge. Efficient upper and lower bounds on optimal policies are derived for two important special cases. Necessary and sufficient conditions are obtained for a finite-time horizon optimal policy to be also optimal for the infinite-time horizon.This research was supported in part by the Office of Water Resources Research, Grant No. A-043-CAL, through the Water Resources Center, University of California.     

Locating and staffing service centers under service level constraints

Many firms experience demand from geographically dispersed customers. This demand is satisfied by mobile servers that travel to the site of the customer. To achieve this in a cost-effective manner, the firm needs to decide where to locate its service centers, which customer regions to assign to the centers and the staffing level   at each center so that customers experience a defined level of service at minimum cost. To determine adequate staffing levels, we approximate a service center and the customer regions assigned to it as an M/G/s$M/G/s$ queueing system. Based on this queueing model, we explore properties of two different staffing level functions. The queueing model is embedded in a large-scale integer program. Using the concept of column generation, we develop an algorithm that can efficiently solve moderate-sized problems.     

Optimal risk control and dividend policies under excess of loss reinsurance

We study the optimal reinsurance policy and dividend distribution of an insurance company under excess of loss reinsurance. The objective of the insurer is to maximize the expected discounted dividends. We suppose that in the absence of dividend distribution, the reserve process of the insurance company follows a compound Poisson process. We first prove existence and uniqueness results for this optimization problem by using singular stochastic control methods and the theory of viscosity solutions. We then compute the optimal strategy of reinsurance, the optimal dividend strategy and the value function by solving the associated integro-differential Hamilton–Jacobi–Bellman Variational Inequality numerically.     

Optimal dynamic advertising policies in the presence of continuously distributed time lags

The Nerlove-Arrow model of optimal dynamic advertising policies is generalized by incorporating a continuously distributed lag between advertising expenditures and increases in the stock of goodwill. This leads to a control problem where the equation of motion is given by an integro-differential equation. The transitory and steady-state properties of the optimal policies are examined, both for a general lag function and for a gamma distributed lag. The dependence of the steady-state solution on the parameters of the gamma distribution is also investigated. An example is given using specific demand and cost functions.     

Optimal core acquisition and remanufacturing policies under uncertain core quality fractions

Cores acquired by a remanufacturer are typically highly variable in quality. Even if the expected fractions of the various quality levels are known, then the exact fractions when acquiring cores are still uncertain. Our model incorporates this uncertainty in determining optimal acquisition decisions by considering multiple quality classes and a multinomial quality distribution for an acquired lot. We derive optimal acquisition and remanufacturing policies for both deterministic and uncertain demand. For deterministic demand, we derive a simple closed-form expression for the total expected cost. In a numerical experiment, we highlight the effect of uncertainty in quality fractions on the optimal number of acquired cores and show that the cost error of ignoring uncertainty can be significant. For uncertain demand, we derive optimal newsboy-type solutions for the optimal remanufacture-up-to levels and an approximate expression for the total expected cost given the number of acquired cores. In a further numerical experiment, we explore the effects of demand uncertainty on the optimal acquisition and remanufacturing decisions, and on the total expected cost.     

Optimal control of a dam under seasonal electricity prices

This study concerns the optimal control of a hydroelectric dam under seasonal electricity prices: high in winter, low in summer. The goal is to maximize the expected discounted infinite-horizon return through a policy that determines the amount of electricity to be produced in each time period, depending on the water level at the beginning of the period under consideration. The prices are assumed to be deterministic, and the flows into the reservoir are seasonal, stochastic, but independent from one period to another. The electric power generated is proportional to the amount of water flow through the turbines. There exist seepage and evaporation losses.It is shown that in the simplest price structure, the optimal policy is entirely determined by a single critical water level in each period of time, at which one starts producing. An example shows that the discretization of the reservoir levels can destroy this property. A method is proposed to avoid this difficulty. Another way of defining a policy is through goal-levels. This approach is shown to give higher returns than the standard approach.     

Optimal budgetary and monetary policies under uncertainty: A stochastic control approach

In this paper, we determine optimal budgetary and monetary policies for Austria using a small macroeconometric model. We use a Keynesian model of the Austrian economy, called FINPOL1, estimated by ordinary least squares, which relates the main objective variables of Austrian economic policies, such as the growth rate of real gross domestic product, the rate of unemployment, the rate of inflation, the balance of payments, and the ratio of the federal budget deficit to GDP, to fiscal and monetary policy instruments, namely expenditures and revenues of the federal budget and money supply. Optimal fiscal and monetary policies are calculated for the model under a quadratic objective function using the algorithm OPTCON for the optimum control of nonlinear stochastic dynamic systems. Several control experiments are performed in order to assess the influence of different kinds of uncertainty on optimal budgetary and monetary policies. Apart from deterministic optimization runs, different assumptions about parameter uncertainties are introduced; the results of these different stochastic optimum control experiments are compared and interpreted.     

Optimal research and development expenditures under an incremental tax incentive scheme

The effect of an incremental tax incentive on optimal R and D spending of a profit maximizing firm is investigated. A dynamic programming approach permits the complete characterization of an optimal policy for a one-period-memory piecewise linear incentive scheme, and efficient computation of an ?-optimal policy in the more general case.     

Optimal ordering policies for perishable multi-item under stock-dependent demand and two-level trade credit

In this paper, a multi-item inventory model for perishable items is developed, where the demand rates of the items are stock dependent, two-level trade credit is adopted and the restriction of inventory capacity is also considered. The major objective is to determine the optimal cycle time and order quantities such that the total profit is maximized. The existence and uniqueness of the optimal cycle is discussed by Lagrange approach, and line search algorithms are developed to find the optimal solution of the model. Furthermore, numerical examples are given to illustrate the methods. The sensitivity of the solution to changes in the values of different parameters is also discussed.     

Data and queueing analysis of a Japanese air-traffic flow

Congestion is a major cause of inefficiency in air transportation. A question is whether delays during the arrival phase of a flight can be absorbed more fuel-efficiently in the future. In this context, we analyze Japan’s flow strategy empirically and use queueing techniques in order to gain insight into the generation of the observed delays. Based on this, we derive a rule to balance congestion delays more efficiently between ground and en-route. Whether fuel efficiency can be further improved or not will depend on the willingness to review the concept of runway pressure.     

Optimal policies for inventory systems with two types of product sharing common hardware platforms: Single period and finite horizon

In this paper, we consider an inventory system whose products share a common hardware platform but are differentiated by two types of software. Choice of different software results in different installation cost and different selling price of the whole product. Product with different software also faces different customer demand. We investigate the optimal proportion of an order to be installed with software 1 or 2, that maximizes expected profit in the single and multiple period scenarios, respectively. The optimal policy is analytically obtained and proved to be an order-up-to policy in each scenario. Our investigation reveals that whether to replenish, and how much to replenish each product depend not only on its own initial inventory level, and system parameters, but also the initial inventory level of the other product. We perform numerical experiments using the optimal policies we have derived in the paper.     

Optimal Covering of Plane Domains by Circles Via Hyperbolic Smoothing

We consider the problem of optimally covering plane domains by a given number of circles. The mathematical modeling of this problem leads to a min–max–min formulation which, in addition to its intrinsic multi-level nature, has the significant characteristic of being non-differentiable. In order to overcome these difficulties, we have developed a smoothing strategy using a special class C^∞ smoothing function. The final solution is obtained by solving a sequence of differentiable subproblems which gradually approach the original problem. The use of this technique, called Hyperbolic Smoothing, allows the main difficulties presented by the original problem to be overcome. A simplified algorithm containing only the essential of the method is presented. For the purpose of illustrating both the actual working and the potentialities of the method, a set of computational results is presented.     

Optimal investment and reinsurance policies in insurance markets under the effect of inside information

In this paper, we study the problem of optimal investment and proportional reinsurance coverage in the presence of inside information. To be more precise, we consider two firms: an insurer and a reinsurer who are both allowed to invest their surplus in a Black–Scholes‐type financial market. The insurer faces a claims process that is modeled by a Brownian motion with drift and has the possibility to reduce the risk involved with this process by purchasing proportional reinsurance coverage. Moreover, the insurer has some extra information at her disposal concerning the future realizations of her claims process, available from the beginning of the trading interval and hidden from the reinsurer, thus introducing in this way inside information aspects to our model. The optimal investment and proportional reinsurance decision for both firms is determined by the solution of suitable expected utility maximization problems, taking into account explicitly their different information sets. The solution of these problems also determines the reinsurance premia via a partial equilibrium approach. Copyright © 2011 John Wiley & Sons, Ltd.