Replacement Policy for Repairable System under Various Failure Types with Finite Time Horizon

The repairable system suffers various type of failure and each failure type has different repair cost. Assume the failure process of the system as Non-homogenous Poisson Process (NHPP). The system is replaced after it experienced a predetermined number of minimal repairs. Considering finite time horizon, the paper proposes a replacement model for the system. It firstly proves that the failure process of each type of failure also follows NHPP. Then it develops a model to estimate the total cost which covers minimal repair cost for each type of failure and system replacement cost. To obtain the numerical solution, the paper introduces a numerical approach to approximate renewal function and a nonlinear programming model is developed. A numerical example is presented eventually. 1 Notations λ(t) Failure rate of NHPP. Pi(t) Probability of ith failure type occurred at instantaneous time t. λi(t) Failure rate of NHPP for ith failure type. N Number of system failure. X1 Arrival time conditioned on one failure occurred during given time interval. f(s) Failure distribution density of arrival time conditioned on one failure occurred. N(s) Cumulative number of failures occurred before time s. N(s, t) Cumulative number of failures. occurred between time s and t. Λ(t) Mean cumulative number of failure occurred before time t. ni Number of type i failure occurred. Fi(t) Cumulative distribution function of type i failure. Ri(t) Survival distribution function of type i failure. m Number of types of failure. Ci Minimal repair cost for type i failure. Cr Replacement cost. C(t) Total maintenance cost to time t. Cmin(t) Total minimal repair cost to time t. A(t) The non convolution part of the renewal function. L(T) Cost per time unit with time horizon T. T Time horizon. Fi Value of cumulative distribution function at ith time step. Ai Value of A(t) at ith time step. 2 Introduction Most of the optimum replacement models are based on the reward renewal process with infinite time horizon. These models can obtain the analytical solution to optimum replacement time. Whereas in practice, the life length of system or the time horizon considered is finite, the models based on infinite time horizon may not be accurate. Jack (1991) presented a comparison between finite time horizon model and infinite time horizon model suitable for replacement decision and demonstrates that the cost per time unit based on finite time horizon would be 2.92% less than its corresponding cost per unit based on infinite time horizon. Castro and Alfa (2004) developed a model considering a lifetime for single unit system using age replacement policy. Other example is presented by Hartman and Murphy (2006). In some cases, the replacement models based on finite time horizon is more realistic. The paper proposes a replacement model based on finite time horizon. Assume the system subjects to NHPP[1], i.e. the system is the same as old after repair, and is suffering various types of failure. The system is replaced after it experienced a predetermined number of minimal repairs. The number of minimal repair that the system can tolerate differs at failure type. The paper proposes a methodology to determine the optimum number of minimal repairs before replacement. 3 Preliminary theory Assume the failure of the system follows NHPP.When system failure occurs, the probability of the ith type of failure is Pi(t). Ross (1996)proved the ith type of failure also follows NHPP when there are two types of failures. i.e. when parent event follows NHPP, their two child events are still NHPP. The paper considers the number of failure types(Child events) more than 2. Similar to the approach of proving for the two types of failure by Ross (1996), this Section proves that the th type of failure also follows NHPP when the number of types of failures is more than 2. 3.1 Probability of occurrence of type i failure Condition on a failure occurs during interval [0,t], the probability of failure occurs before time s is P {X1 < s |N(t) = 1} = P {X1 < s,N(t) = 1} N(t) = 1 (1) Equation(1) is rewritten to P {X1 < s |N(t) = 1} = P {N(s) = 1, N(s, t) = 0} N(t) = 1 (2) Due to independent increment of NHPP, the disjoint interval of NHPP is thus independent, then P {N(s) = 1, N(s, t) = 0} = P {N(s) = 1}P {N(s, t) = 0} (3) Hence, P {X1 < s |N(t) = 1} = Λ(s)e−Λ(s).e−Λ(s,t) Λ(t)e−Λ(t) = Λ(s) Λ(t) (4) Therefore, the density of failure distribution conditioned on one failure occurred during interval [0,t] is f(s) = λ(s) Λ(t) (5) Therefore, the probability of the ith type of failure occur during interval [0,t] is