Closed-form Optimal Solution Research Articles

Optimal feedback design for nonlinear stochastic systems using the pseudospectral method

The development of smart structures involves the use of various types of controllable damping devices by integrating them into structural systems for energy dissipation. An optimal feedback control law that can consider the nonlinearity embedded in the systems and the random characteristics of the external inputs is desirable to control such dissipation mechanism. This paper proposes an innovative numerical technique to obtain the optimal feedback control strategy of nonlinear stochastic systems. The systems under investigation are modeled as mechanical oscillators and a damping device. The numerical approach proposed to obtain the optimal control strategy involves solving a nonlinear partial differential equation—the Hamilton–Jacobi–Bellman equation. Because civil engineering structural systems may exhibit nonlinear hysteretic behavior under extreme loading conditions, the application of the obtained control strategy could provide an optimal feedback control law to reduce the system response under random excitations (such as earthquakes, wind load and sea waves). Several numerical examples are presented to verify optimality and demonstrate the efficacy of the proposed optimal control solution. First, a linear oscillator is used to verify that the obtained solution is indeed the optimal solution by comparing it to the closed-form optimal solution. Then the proposed method is applied to several nonlinear systems. In each nonlinear example, control optimality is demonstrated by comparing the system responses and control costs obtained using the proposed optimal controls with those obtained using corresponding linearized optimal controls.

No Refund or Full Refund: When Should a Fashion Brand Offer Full Refund Consumer Return Service for Mass Customization Products?

We analytically explore in this paper the consumer return policy under fashion mass customization (MC) program. To be specific, we model the stochastic fashion MC program with the consideration of consumer demand uncertainty. If a consumer return policy is implemented, we further consider return uncertainty. By modeling the optimization objective of the risk averse MC fashion brand via a mean-variance approach, we derive the closed-form optimal solution under each case. We then conduct both analytical and numerical sensitivity analyses. For the scenario with full refund and return, we reveal the analytical conditions under which the optimal retail price and the optimal number of options available for customization (called the “optimal modularity level”) vary monotonically with respect to the salvage value and the return service charge. For the scenario when there is no refund and return, we show that the optimal retail price and the optimal modularity level are decreasing in the MC fashion brand's degree of risk aversion, the demand uncertainty, and the price-demand sensitivity coefficient. In addition, our numerical analysis indicates that whether the risk averse MC fashion brand would prefer offering consumer return with full refund to no return depends heavily on the demand-return correlation (DRC) parameter.

Theoretical and Practical Solutions for High-Order Superdirectivity of Circular Sensor Arrays

An analytical and closed-form optimal solution expressed in elementary functions for superdirectivity of a circular sensor array is proposed in this paper. By utilizing the circulant property of the data covariance matrix for circular sensor arrays, such solutions are derived to accurately calculate the optimal beamforming vector, optimal beam pattern, corresponding directivity factor, and eigenbeams. Based on these solutions, it is proved that such array possesses several important properties which facilitate an eigenbeam decomposition and synthesis approach for practical implementations. Experiments conducted result in superdirective beams which contain zeroth- to fifth- and sixth-order eigenbeams. As a typical result, the superdirective beam which contains zeroth- to fifth-order eigenbeams has a 28 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">°</sup> beamwidth and a 12-dB directivity index with a small circular array radius of 0.267 wavelength.

How to make the replenishment and payment strategy under flexible two-part trade credit

In the existing inventory models concerning the two-part trade credit, a common assumption is that the retailer either pays for all the ordered items within a short permissible delay period and receives a cash discount or pays for all the ordered items within a long permissible delay period at the regular price. In this paper, this unrealistic assumption is relaxed. We assume that the retailer may pay any fraction of the purchase cost within the short permissible delay period and receives a cash discount and then the rest is paid within the long permissible delay period. A decision model is proposed for a retailer to determine the optimal ordering policy and payment plan. The closed-form optimal solution to the model is developed and analyzed. Numerical studies show that a retailer can obtain more benefits from the proposed payment plan than from the extreme payment plan in the existing literature.

A Closed-form Formula for the Fair Allocation of Gains in Cooperative N-person Games

Abstract This paper provides a closed-form optimal solution to the multi-objective model of the fair allocation of gains obtained by cooperation among all players. The optimality of the proposed solution is first proved. Then, the properties of the proposed solution are investigated. At the end, a numerical example in inventory control environment is given to demonstrate the application and to compare the results with the ones of an existing method.

MMSE-Based MIMO Cooperative Relaying Systems: Closed-Form Designs and Outage Behavior

In this paper, we investigate minimum mean squared error (MMSE) based amplify-and-forward cooperative multiple antenna relaying systems where a non-negligible direct link exists between the source and the destination. First, we provide a new design strategy for optimizing the relay amplifying matrix. Instead of conventional optimal design approaches resorting to an iterative gradient method, we propose a near optimal closed-form solution which provides an insight. As relay systems with a direct link incur a non-convex problem in general, we exploit the decomposable property of the error covariance matrix and a relaxation technique imposing a structural constraint on the problem. Next, we study the error performance limit of the proposed scheme using diversity-multiplexing tradeoff analysis, which leads to several interesting observations on MMSE-based cooperative relaying systems. Finally, through numerical simulations, we confirm that the proposed solution shows the performance very close to the optimum with much reduced complexity and the analysis closely matches with simulation results.

Closed-Form Optimal Power Allocation for Weighted Rate Sum Maximization in Gaussian Broadcast Channel

This paper is motivated by optimal adaptive resource allocation policies at the base station in wireless communication systems, and we particularly study the maximization of the weighted rate sum in Gaussian broadcast channel. The problem of finding the optimal power allocation in Gaussian broadcast channel is non-convex and intractable to solve using traditional methods with low complexity. In this paper, we derive a closed-form optimal solution of the weighted rate sum maximization in Gaussian broadcast channel. The derived closed-form optimal solution can give the optimal power allocation for the weighted rate sum maximization one time and does not need to determine the water level and iterative process. Thus, it is very easy to be implemented with an adaptive power allocation in the real communication systems.

PORTFOLIO SELECTION WITH REGIME-SWITCHING: DYNAMIC PROGRAMMING APPROACHES

I study an optimal consumption and portfolio selection problem with regime-switching using a dynamic programming method. With constant relative risk aversion (CRRA) utility I obtain optimal solutions in closed-form.

Maximizing Gathered Samples in Wireless Sensor Networks with Slepian-Wolf Coding

We consider an energy constrained wireless sensor network, with arbitrary number of nodes, where source nodes utilize Slepian-Wolf (SW) coding before transmission to a joint decoder. We investigate optimal and near-optimal SW coding rates, transmit powers, and transmit durations that maximize the number of collected samples during the network lifetime, subject to channel capacity, SW rate region, and residual energy constraints. We find optimal (near-optimal) closed-form solutions in the absence (presence) of an energy constraint at the joint decoder. We take into account the energy consumption of SW encoding and decoding and communication circuitry. Numerical results demonstrate the effectiveness of the proposed optimization, especially when the joint decoder is not energy constrained.

A Characterization of the Optimal Management of Heterogeneous Environmental Assets under Uncertainty

The application herein involves the optimal management of renewable and nonrenewable resources within the context of a stochastic model of optimal control. By characterizing the two dimensional Bellman solution, three rules with respect to resource management are established. Within the context of coastal development, this analysis may help to explain why renewable resources may become increasingly vulnerable to random external shocks as nonrenewable resources are depleted. Although existence of an optimal closed form solution to the multi-sector Bellman model remains an open mathematical question, this analysis offers a characterization which can be applied to other scenarios in economics or finance in which two assets following stochastic processes interact.

Combining an alternative multi-delivery policy into economic production lot size problem with partial rework

This paper combines an alternative multi-delivery policy into an imperfect economic production quantity (EPQ) model with partial rework, with the purpose of reducing supplier’s stock holding cost. We extend the problem examined by Chiu et al. [Chiu, Y.-S. P, Chiu, S. W., Li, C.-Y., & Ting, C.-K. (2009). Incorporating multi–delivery policy and quality assurance into economic production lot size problem. Journal of Scientific and Industrial Research, 68(6), 505–512] by considering an n+1 delivery policy in lieu of n multi-delivery plan for the specific EPQ model with partial rework. Under such a policy, an initial delivery of perfect (finished) items is distributed to customer for satisfying product demand during manufacturer’s regular production time and rework time. At the end of rework, fixed quantity n installments of the finished products are delivered to customer at a fixed interval of time. As a result, a closed-form optimal replenishment batch size solution to the problem is obtained. A numerical example with analysis and comparison is provided to show practical usage of the proposed model and demonstrate its significant savings in stock holding cost.

Retailing policy for perishable item sold from two bins with mixed issuing policy

This paper deals with a retailer who sells single perishable product in two bins. The fresh items are sold at a list price in the primary bin and the unsold items that have reached a certain allowed age are transferred to the secondary bin to be sold at a discount price. It is assumed that the demand is affected by inventory level, selling price, product freshness, and demand leakage caused by the price difference of two bins. Also, products are sold under mixed issuing policy which is defined as a weighted sum of Last-In-First-Out and First-In First-Out issuing policy. With the objective of maximizing the retailer's profit, we develop mathematical models for the following two cases: (1) opening primary shop only and (2) opening both primary shop and secondary shop. Noting that the objective function is too complicated to possess a closed-form optimal solution, a solution procedure is developed based on Tabu search. The validity of the model is shown by solving an example problem and the sensitivity study is performed on the system parameters.

Data Assimilation: A Fully Nonlinear Approach to Ensemble Formation Using Indistinguishable States

Abstract Operational forecasting with simulation models involves the melding of observations and model dynamics to determine a set of initial conditions for each forecast. The Kalman filter (KF) provides the optimal closed-form solution to a general linear stochastic (perfect model) case, while the target of the problem has not even been defined in the case of imperfect models. Data assimilation in a nonlinear, perfect-model scenario is considered. It is shown that a new fully nonlinear approach based upon the indistinguishable states (IS) systematically outperforms the ensemble Kalman filter (EnKF). The IS provides an ensemble of initial conditions, consistent with (i) the model dynamics, (ii) the observational noise model, and (iii) the particular observations over a window. It is argued that this is the relevant limit to consider in data assimilation, when the desire is to place high probability density in the vicinity of the target state. The advantages of the IS approach come in part from its ability to provide attractor-balanced ensembles near any attracting manifold the system may evolve on. The use of an EnKF, provides a computationally cheaper alternative that place points in the general vicinity of the target. A low (i.e., 2) dimensional example is used to provide easily visualized evidence for these claims, which are then tested in a higher (i.e., 12) dimensional system. Inasmuch as the IS approach is shown to outperform the EnKF systematically in these perfect-model experiments, it provides an interesting alternative approach when informative ensembles are desired.

Cooperative FSO Systems: Performance Analysis and Optimal Power Allocation

In this paper, we investigate the cooperative diversity technique as a candidate solution for combating turbulence-induced fading over free-space optical (FSO) links. In particular, we propose a novel cooperation strategy that is suitable for quantum-limited FSO systems with any number of relays and we derive closed-form expressions for the error performance of this strategy. In scenarios where the channel-state-information (CSI) is available at the different nodes, we propose an optimal power allocation strategy that satisfies the Karush-Kuhn-Tucker (KKT) conditions and that further boosts the performance of FSO networks. It turned out that this closed-form optimal solution corresponds to transmitting the entire optical power along the “strongest link” between the source and the destination nodes. A simple procedure is proposed for selecting this link and for distributing the power among its different hops.

Stochastic comparisons for allocations of policy limits and deductibles with applications

In this paper, we study the problem of comparing losses of a policyholder who has an increasing utility function when the form of coverage is policy limit and deductible. The total retained losses of a policyholder ∑ i = 1 n ( X i − l i ) + are ordered in the usual stochastic order sense when X i ( i = 1 , … , n ) are ordered with respect to the likelihood ratio order. The parallel results for the case of deductibles are obtained in the same way. It is shown that the ordering of the losses are related to the characteristics (log-concavity or log-convexity) of distributions of the risks. As an application of the comparison results, the optimal problems of allocations of policy limits and deductibles are studied in usual stochastic order sense and the closed-form optimal solutions are obtained in some special cases.

Optimal Guidance Solution for Impact Angle Control Based on Linearization about Collision Triangle

AbstractIn this paper, an optimal guidance problem is investigated where specific missile-target geometry at the time of the intercept of, or a terminal impact angle with respect to, a maneuvering target is prescribed. The kinematics is linearised to obtain a closed-form optimal solution and the proposed approach addresses the question of how to accomplish the linearization in a right manner. We use, as the basis of linearization, the perfect (or zero effort) collision triangle for the impact angle control problem which varies depending on the value of the prescribed impact angle, and solve a linear optimal guidance problem. The performance of the resultant guidance law is illustrated by performing some numerical simulations and the selection of a guidance variable that effectively capture the distance between the missile and the zero effort collision triangle is discussed.

The economic lot size of the integrated vendor–buyer inventory system derived without derivatives: A simple derivation

In this paper, we first complement an inappropriate mathematical error on the total cost in two previous published research papers in inventory field. Then an arithmetic–geometric inequality method is proposed not only to simplify the algebraic method of completing prefect squares, but also to complement their shortcomings. In addition, we also provide a closed-form optimal solution to the integrated vendor–buyer inventory system without using complex derivatives. Finally, the proposed method seems to be easy-to-understand and simple-to-apply by individuals who are not familiar with differential calculus.

Solving fi nite production rate model with scrap and multiple deliveries

This paper examines fi nite production rate (FPR) model with scrap and multiple deliveries. Classic FPR model assumes continuous issuing policy for satisfying demand and perfect quality production for all items produced. However, in real life vendor-buyer integrated system, multiple shipment policy is practically used and generation of defective items during a production run is inevitable. Mathematical modeling and analysis is used, renewal reward theorem is employed to cope with the variable production cycle length, and the integrated long-run average cost per unit time is derived. A closed-form optimal batch size solution to the problem is obtained. A numerical example demonstrates its practical usage

Newsvendor games: convex optimization of centralized inventory operations

A finite set of outlets with randomly fluctuating demands bands together to reduce costs by buying, storing and distributing their inventory jointly. This is termed inventory centralization and is a type of risk pooling. The expected centralization cost can be lowered even further, without disrupting the demand behavior at individual outlets, by inducing the outlets to correlate their individual demands. Given that the outlets’ demands are normally distributed, the lowering of the centralized cost corresponds to a semidefinite optimization problem. This paper establishes a closed-form optimal solution of the semidefinite program and a fair allocation of the centralized cost at optimality.

A Solution of the Dichromatic Model for Multispectral Photometric Invariance

In this paper, we address the problem of photometric invariance in multispectral imaging making use of an optimisation approach based upon the dichromatic model. In this manner, we cast the problem of recovering the spectra of the illuminant, the surface reflectance and the shading and specular factors in a structural optimisation setting. Making use of the additional information provided by multispectral imaging and the structure of image patches, we recover the dichromatic parameters of the scene. To do this, we formulate a target cost function combining the dichromatic error and the smoothness priors for the surfaces under study. The dichromatic parameters are recovered through minimising this cost function in a coordinate descent manner. The algorithm is quite general in nature, admitting the enforcement of smoothness constraints and extending in a straightforward manner to trichromatic settings. Moreover, the objective function is convex with respect to the subset of variables to be optimised in each alternating step of the minimisation strategy. This gives rise to an optimal closed-form solution for each of the iterations in our algorithm. We illustrate the effectiveness of our method for purposes of illuminant spectrum recovery, skin recognition, material clustering and specularity removal. We also compare our results to a number of alternatives.