تقریب بهبود یافته برای عملکرد نوسازی و انتگرال آن در یک برنامه مدیریت موجودی دو پله ای

A simple but effective approximation is proposed to compute the renewal function, M(t)M(t), and its integral. The asymptotic approximation of the renewal function and its integral, which are widely used in decision makings involving a renewal process, may not perform well when tt is not large enough. To overcome the inaccuracy of the asymptotic approximation, we propose a modified approximation that computes the renewal function and its integral based on the probability distribution function of inter-renewal time when the distribution function is known or based on its mean and standard deviation when the distribution function is unknown. The proposed approximation provides closed form expressions, which are important in decision makings, for the renewal function and its integral for the entire range of tt rather than numerically computes them for given values of tt. Extensive numerical experiments on commonly used distributions are conducted and demonstrate better performance of the proposed approximations compared to the asymptotic approximation. The new approximations are further applied to a case study of a two-echelon inventory system and result in better solutions compared to the reported results based on the asymptotic approximation.

Let the random variable XnXn denote the intercurrence or inter-renewal time between the (n −1)th and n th events in a renewal process. Assume X1,X2,…X1,X2,… to be a sequence of nonnegative, independent random variables having a common probability distributionView the MathML sourceF(x)=P{Xk≤x},x≥0,k=1,2,…, andView the MathML sourceμ1=E(Xi),0<μ1<∞. DefineView the MathML sourceS0=0,Sn=∑i=1nXi,n=1,2,…, so that SnSn would be the time epoch at which the n th event occurs. For eachView the MathML sourcet≥0, N(t)N(t) is the largest integer n≥0n≥0 so thatView the MathML sourceSn≤t. The random variable of N(t)N(t) represents the number of events up to time tt and the renewal function M(t)M(t) is defined by Tijms (2003) as View the MathML sourceM(t)=E[N(t)],t≥0. Turn MathJax on Define the cumulative distribution function (c.d.f.) View the MathML sourceFn(t)=P{Sn≤t},t≥0,n=1,2,…,andF1(t)=F(t). It is implied that View the MathML sourceP{N(t)≥n}=Fn(t),n=1,2,…. Given an F(t)F(t), the renewal function M(t)M(t) satisfies the integral equation of View the MathML sourceM(t)=F(t)+∫0tM(t−x)dF(x). Turn MathJax on The integral equation has a unique solution of M(t)M(t), which is bounded on finite intervals under the assumption that F(t)F(t) is continuous inView the MathML sourcet, F(0)=0F(0)=0, and F(∞)=1F(∞)=1 (Cox, 1962). Let f(t)f(t) denote the corresponding probability density function (p.d.f.), if exists, forView the MathML sourceF(t). Then, View the MathML sourceM(t)=F(t)+∫0tM(t−x)f(x)dx. Turn MathJax on The renewal function View the MathML sourceM(t)and its integral View the MathML sourceI(t)=∫0tM(x)dx play an important role in decision makings involving the renewal process, such as inventory planning, supply chain planning, reliability and maintenance analysis (e.g., Bahrami et al., 2000, Barlow and Proschan, 1965, Sheikh and Younas, 1985 and Tijms, 1994). However, obtaining the renewal function, M(t)M(t), analytically is complicated and even impossible for most distribution functions. As an analytical method, the Laplace transform M(s)M(s) of the renewal function satisfies View the MathML sourceM(s)=f0(s)s(1−f0(s)), Turn MathJax on where f0(s)f0(s) is the Laplace transforms of the density function of the inter-renewal time, f(t)f(t) (From, 2001). It is usually difficult to obtain M(t)M(t) through the inversion of M(s)M(s) (Jaquette, 1972). We can obtain an exact computation of the renewal function, M(t)M(t) for all t≥0t≥0 analytically only for a few special cases of F(t)F(t) (Tijms, 2003), such as the exponential distribution. Furthermore, in many real-life applications the distribution for the inter-renewal time may not be known. Therefore, approximations of the renewal function have drawn much interest in the literature and result in various methods. The asymptotic expansion is very helpful in the approximation of the renewal function and its integral because of its simplicity (Spearman, 1989). The asymptotic approximation only requires the first several moments (the first two moments for the renewal function approximation and the first three for its integral) and does not need the exact distribution function for the inter-renewal times. Because it provides a closed-form, the asymptotic approximation has been widely applied to the optimization problems that involve the renewal process, such as inventory planning, reliability and maintenance planning (e.g., Cetinkaya et al., 2008). However, asymptotic expansions for M(t)M(t) are not accurate for small values of tt and may yield poor optimal solutions. This paper tries to address this drawback of asymptotic approximations by proposing a new approach of approximation. At the same time, our proposed approximations keep the positive features of asymptotic approximations such as simplicity, closed-form expression for optimization, and independence from the distributions of inter-renewal times. It is rather easy to compute M(t)M(t) numerically for a given value of tt (Jaquette, 1972) and a variety of approaches have been developed in the literature, such as cubic-splining algorithm by McConalogue and Pacheco (1981) to compute the renewal function by numerical convolution, the generating function algorithm by Giblin (1984), and power series expansion. The power series method is used for Weibull distribution in most studies (e.g., Wang and Pham, 1999 and Weiss, 1981) but can be extended to all distributions with a power series expansion (Smeitink and Dekker, 1990). Smith and Leadbetter (1963) found an iterative solution for the case in which the inter-renewal time follows a Weibull distribution. Another iterative solution method with the Weibull distributed inter-renewal time was given by White (1964). A numerical integration approach, which covers Weibull, Gamma, Lognormal, truncated Normal and inverse Normal distributions, was offered by Baxter et al. (1982). Garg and Kalagnanam (1998) proposed a Pade approximation approach, a class of rational polynomial approximants (Baker and Graves-Morris ,1996), to solve the renewal equation for the inverse Normal distribution. Their method uses Pade approximants to compute the renewal function near the origin and switches to the asymptotic values farther from the origin. They presented a polynomial switch-over function in terms of the coefficient of variation of the distribution, enabling one to determine a priori if the asymptotic value can be used instead of computing the Pade approximant. A shortcoming of their method is that it does not provide a compact closed-form until applying the numerical method of Xie (1989). Kaminskiy and Krivtsov (1997) used a Monte Carlo simulation, which provides a universal numerical solution to the renewal function equation, covering essentially any parametric or empirical distribution used to model time-to-failure distributions. A method called the RS-method was established by Xie (1989) for solving renewal-type integral equations based on direct numerical Riemann–Stieltjes integration. The RS-method is particularly useful when the probability density function has singularities. The numerical method of Xie (1989) was used as a starting point of a numerical approximation proposed by From (2001) that constructs a two-piece modified rational function with the second piece being a linear function of View the MathML sourcet. An approximation for the renewal function of a failure distribution with an increasing failure rate was proposed by Jiang (2010). Although all the methods that numerically compute the renewal function are generally accurate for the small values of tt but do not provide a closed-form expression that is useful for decision makings. Our proposed approximation not only is accurate in the range of small values of tt but also provides a closed-form expression to facilitate optimization.

In this paper we introduced a simple but effective approximation for the renewal function and its integral. The commonly used approximations of the renewal function and its integral based on asymptotic behaviors do not perform well for small values ofView the MathML sourcet. The main focus of this research is to improve the performance of the approximation when the values of tt are outside the typical asymptotic range. By plotting and studying the simulated values of renewal function and renewal function integral, we developed a three-piece function with two switch-over points that resembles the simulation results of renewal function and its integral. Intensive numerical studies show that our modified approximation outperforms the asymptotic approximation. The modified approximation is easy to implement, especially useful for decision makings. A comprehensive application case from the literature is used to demonstrate the applicability and performance of the proposed modified approximation. In the future, more applications will be investigated to test the effectiveness of the proposed modified approximations.