Two-Stage Stochastic Algorithm for System Cost-Reliability and Maintenance Optimization Considering Uncertain Future Usage Scenarios

This paper develops a two-stage stochastic cost-reliability optimization model for multi-component systems subjected to uncertain stress exposure which leads to uncertain component and system degradation and wear. Optimization models are used for making decisions of system configurations to be applied in different and occasional diverse future usage scenarios. Component reliability and cost data can provide the necessary information of conditions and inputs to determine an optimum preventive maintenance policy. In this formulation, the system is exposed to distinct usage scenarios that are collectively represented as the future usage profile. This profile is described by a set of specific scenarios and their probability of occurrence or likelihood. Once these future usage profiles are known or can be estimated, the cost rate for prospective system configurations and associated maintenance can be modeled. The cost-reliability system design with maintenance modeling problem is defined as a two-stage stochastic cost-reliability programming problem with recourse. The decision variables for the first-stage are the selection of components and the number of components to be used in the system, where the cost rate objective function is constrained by component availability. The second stage variables are defined by the corrective and preventive maintenance plan in order to optimize the maintenance time interval for planned replacement of components in the system. A numerical example demonstrates that the system cost rate can be optimized by these decision variables. Keywords Two-stage stochastic problem, Preventive maintenance, Reliability 1. Introduction There have been significant researches related to optimization of system reliability based on long term forecasts of usage requirements and stresses. In any real application, the reliability of component and system reliability naturally change due to stresses and operating conditions. The stresses and operating conditions affecting component reliability then influence system reliability design. The new reliability analysis and optimization model involving cost minimization are evaluated based on variations of stress under changing circumstances. Preventive maintenance (PM) optimization is conducted simultaneously with the system reliability model. In the modeling, the system is exposed to distinct usages, which are represented as the future usage profile. In reliability optimization, redundancy allocation is a technique to increase system reliability for configuration design. Redundancy allocation problems with multiple choices of components are NP-hard. Several techniques have been studied to solve reliability optimization for redundancy allocation problems. Heuristics, meta-heuristics or exact algorithms have been used for solving redundancy allocation problems [1]. In addition, PM is a technique to reduce the number of system failures, with early failure detections the system cost is also optimized before the system fails. There are numerous techniques for optimization of PM policy. Maintenance optimization is formulated Chatwattanasiri, Coit, Feng to minimize costs of maintenance and/or to increase system availability [2,3]. Although many researches focus on PM policy, the best method to solve the PM optimization problem is still not always understood. The PM optimization for a two-unit cold standby redundant system was studied to minimize cost per unit time [4]. The optimization model of [5] is solved by Bender’s decomposition to schedule PM considering the total cost objective function. The PM scheduling models have also been formulated as mixed-integer linear programming models [5,6]. Cheng et al [7] presented optimization models to minimize system cost subject to reliability considering the PM time interval. The analytical model of PM was analyzed by considering the expected cost rate objective function and PM policy based on a Weibull distribution in [8]. They develop a mathematical formulation of PM period and the number of PM events. When uncertainty lies in component or system reliability for different scenarios, changes of stress in component operation, conditions, environmental vibration and/or temperature can cause different reliability characteristics for the components and system. Reliability optimization research has been developed for considering uncertainty using mathematical programming models. In general, stochastic programming (SP) with recourse to consider uncertainty has been studied. Dantzig [9], Birge and Louveaux [10] and Kall and Wallace [11] proposed the formulation of stochastic problems with recourse and reviewed the basic concepts, methods and applications. The mathematical program where parameters in the objective function or constraints are uncertain is considered as stochastic programming. When some parameters are random, then solutions and the optimal objective value of the problem are also random. The two-stage stochastic decision is for dealing with uncertainty. After the first-stage decision, a random event occurs affecting the outcome. Solving a recourse problem is to evaluate the second stage that compensates for any impact that might have been experienced from the first-stage decision. The two-stage stochastic programming model has been formulated in many standard forms. Two-stage stochastic integer programming and two-stage stochastic mixed integer programming have been developed [12]. A variety of applications can be formulated as two-stage stochastic integer programs in energy planning, manufacturing, logistics, etc. [13,14]. In this paper, maintenance costs and system reliability improvement for configuration designs are studied and the two-stage stochastic programming problem that relates system reliability optimization and PM is formulated. Many objective frameworks with randomness of future usages are proposed for the two-stage stochastic problem. The decision variable in first-stage is the number of components used in each system affecting system cost-rate and system reliability. The decision makers must make a decision at the current time from the best-known information. For the second-stage decision, corrective actions are designed based on realization of full information. The corrective actions represent the maintenance time interval for replacement of the components in the system. 2. Future usages Profile of Operating Conditions and Stresses Uncertainty in future usages can involve several factors of uncertain stress/load to components within the future usage profile, as shown in Figure 1. In some applications, system reliability estimation is often problematic due to unplanned variation, or changing operating stresses. More often, not only operating conditions and stresses but also environmental stresses such as temperature or vibration can be uncertain and dependent on usage conditions of components and systems in the future. Figure 1. Uncertainty in Future usages Component stress/load parameters are represented in the future usage scenarios as vectors, which describe how changing stresses in different future usage scenarios affect reliability functions. Uncertain stress factors are demonstrated in the form of a random vector U in the future usage profile. The model parameter Uk is a random variable of the k stress factor that the component or system experiences, e.g, temperature, humidity, voltage or Future Usage Environment Stresses Opera4ng Condi4ons Chatwattanasiri, Coit, Feng others. A random future usage vector is represented with c different operating usage and stress factors where each uncertain usage condition has different effects on each component, U = (U1, U2, ..., Uc ). The future usage profile defines possible future usage scenarios defined by future usage stress vectors. In practice, the possible future usage scenarios can be enumerated from prediction of how the system will be used and possible occurrences of each future usage. The future usage stress vector in each scenario l is determined from random future usage vector U. There are c different factors of future usage stresses which determine how stresses of different factors change in that future usage scenario. For example, the determined future usage vector for possible future scenario l is represented by vector ul, vector u2 for possible future scenario 2 and so on. Figure 2. Discrete scenarios in future usage profile In Figure 2, the possible future usage vectors in the future usage profile are defined as ul and each future usage scenario is associated with a probability pl. Let ul = (u1l, u2l,...,ucl) represent a determined vector of operating usage conditions and/or stresses. For example, for electronic components, temperature is a critical contributor to component failure (u1l), and for a given future usage scenario, the risk of component failure increases along with increasing temperatures. For mechanical components, mechanical loading (u2l) and stress (u3l) are important factors. The current operating condition and stress vector is u0 and it is known with certainty and given as u0 = 0. Each operating stress from future usage profile is defined by the usage vector ul. All stresses have been scaled from 0 to 1. For example, u2l is a second usage variable at future l. 3. System Model For this paper, a parallel system configuration is introduced for the stochastic reliability model, as shown in Figure 3. Since the number of identical components, x, is connected in parallel within a system, the system reliability is given in Eq. (1). R(x;t) =1− (1− r) (1) Figure 3. Parallel system For our proposed system reliability function, the future usage profile of operating conditions and stresses is considered for developing a system reliability model. The component reliability (ri) is no longer constant, but a random variable. The number of identical components, x, is connected and the system reliability is given as follows R(x,U;t) =1− (1− r(U;t)) (2) U is a random future usage vector, so r(U;t) and R(x,U;t) are also random variables. The system reliability equation is derived from the component reliabilities based on the future usage profile. p1 u0=0 Usage vector u1 Usage vector u2