The instant system availability of a repairable system with the renewal equation was studied. The starting point monotonicity of was proved and the upper bound of is also derived. It was found that the interval of instant system availability monotonically decreases. In addition, we provide examples that validate the analytically derived properties of based on the Lognormal, Gamma and Weibull distributions and the results show that the value of T is slightly smaller than its value defined in Theorem 2. The procedure of using a bathtub as application for this article is also discussed.

In this paper, we introduce the Heston-Hull-White (the hybrid) model in the pension fund management. The optimal investment and benefit payments policies for the DC pension fund with an income drawdown option are presented explicitly. In this study, the pension fund manager is allowed to invest the fund wealth in riskless and risky assets. The risk asset price dynamics evolve according to the Hybrid Stochastic Volatility (Heston-Hull-White) model. The goal of the pension fund manager is to maintain the standard of lives for the pension fund members after retirement. We derived the HJB equation and established the closed-form solution for the optimal control policies using stochastic dynamic programming principle. To illustrate the economic implications of the model, we provide a numerical example and simulations. The results reveal that the optimal investment and benefit payments strategies convergence successful with time t. The risk asset price volatility and interest rate mean reversion are inversely proportional to investment policy. The interest rate volatility and risk preference are both directly proportional to investment and benefit payments. The pension fund managers are advised to control the investment and the benefit payments policies to achieve the goals of the pension fund members.

In this article we develop a model for determining the appropriate level of inspection sampling for any manufacturing process. The model is useful for manufacturers, who naturally are concerned with profits and therefore with minimizing the cost of production. The model design aims to reduce total manufacturing cost and has general applicability to various manufacturing operations. The model considers the interests of consumers, who wish to minimize the cost of production while simultaneously ensuring the final product is of high quality. The cost parameters for production, the acceptance test, and admissible strategy are applied in the model. The cost components are formulated along with the minimization of the expected cost, and we used the repairable systems to guarantee the maintenance and sustainability of the economic system. We also discuss the assumptions and their appropriateness, as well as the application of the model to burn-in of system components.

Abstract-This paper analyzes optimal investment strategies for a DC pension fund under the Hull-White interest rate model. Under this model, the pension fund manager can invest capital in the bank account, stock index, and real estates. The dynamics of the interest rate follows the Hull-White interest rate model, and a drifted Brownian motion drives the financial market. The pension fund manager aims to maximize the expected terminal utility of wealth in a complete market setting under constant relative risk aversion (CRRA). This paper derives the Hamilton-JacobiBellman (HJB) equation associated with the control problem using a dynamic programming technique. We obtain the explicit solution for optimal investment policies by solving the HJB equation and optimal value function. The results show that the optimal proportion invested in risky assets is higher in stock than in a real estate.

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