ISSN : 2234-6473 (Online)

**Industrial Engineering & Management Systems Vol.12 No.1 pp.41-45**

DOI : https://doi.org/10.7232/iems.2013.12.1.041

# Some New Results on Uncertain Age Replacement Policy

### Abstract

- 1. INTRODUCTION
- 2. PRELIMINARIES
- 3. NEW THEOREMS ON UNCERTAIN AGE REPLACEMENT POLICY
- 4. CONCLUSION
- ACKNOWLEDGMENTS

### 1. INTRODUCTION

Among all kinds of maintenance policies, our major concerns are preventive maintenance and corrective maintenance. When corrective maintenance is carried out, the accidental failure of parts in a still operating system such as a flying aircraft, a running computer, a machine of manufacturing product, etc., is costly and dangerous, and it even brings bad reputation to enterprises. Therefore, preventive maintenance is needed to effectively avoid many breakdowns of parts in advance. However, preventive maintenance will generate too much expenses when it is carried out too frequently. So the best solution is a combination of these two kinds of maintenance. As a type of the combination maintenance, age replacement policy has been researched for a long time.

Barlow and Hunter (1960) firstly defined an age replacement policy as one in which a part is replaced at predetermined time T or at failure, whichever occurs first. They studied the basic replacement policies under the common assumption that a system after repair is “as good as new”. However, this assumption is not always reasonable in practice. On account of this fact, Brown and Proschan (1983) discussed an imperfect replacement model. Extensively, Tilquin and Cleroux (1975) proposed periodic replacement policy with adjustment costs. After that, Lin (1988) introduced a geometrical process to study two kinds of replacement policy for a simple repairable system based on the working age T and the number of failures N, respectively, and the explicit expressions of the average cost rate under these two kinds of policy are determined. Furthermore, Zhang (1994) considered a bivariate replacement policy (T, N) under which a unit is replaced at the working age T or at the time of the Nth failure, whichever occurs first.

In those studies, the ages of parts were considered as random variables, and the failure distribution function based on the probability theory is applicable only when the obtained probability is close enough to the real frequency. However, few or even no observed data is available in some situation. Therefore, we have to invite some experts to evaluate their belief degree. Since humans tend to overweight unlikely events (Kahneman and Tversky, 1979), the belief degree may have much larger deviation than the frequency, and the probability theory is no longer valid. In this case, we have a choice to use the uncertainty theory to deal with this type of human uncertainty. The uncertainty theory was founded by Liu (2007), and refined by Liu (2010) based on normality, duality, subadditivity, and product axioms. Nowadays the uncertainty theory has become a branch of axiomatic mathematics for modeling human uncertainty and has been applied to many aspects.

In this paper, by dealing with the parts’ lifetimes as uncertain variables, the optimal time of age replacement policy will be discussed when the lifetimes follow different uncertain distributions. The rest of this paper is organized as follows: Section 2 recalls some basic concepts and properties about the uncertain theory which will be used throughout the paper. In Section 3, the optimal replacement time of the age replacement policy is discussed when the lifetimes of the parts follow uncertain linear, zigzag and lognormal distribution. Finally, concluding remarks are given in Section 4.

### 2. PRELIMINARIES

In this section, we will introduce useful definitions and theorems of uncertainty theory that are applied throughout this paper.

Let Γ be a nonempty set. A collection L of E[1/ξ_{1}] is called a σ -algebra (a) if Γ∈ L ;(b) if Λ∈ L, then Λ^{c} ∈L ; and (c) if Λ_{1}, Λ_{2}, …∈L, then Λ_{1} ∪Λ_{2} ∪…∈L. Each element Λ in the σ -algebra L is called an event. Uncertain measure is a function from L to [0, 1]. In order to present an axiomatic definition of uncertain measure, it is necessary to assign to each event Λ a number M{Λ} which indicates the belief degree that the event Λ will occur. In order to ensure that the number M{Λ} has certain mathematical properties, Liu (2007) proposed the following three axioms:

**Axiom 1** (Normality Axiom). M{Γ} =1 for the universal set Γ.

**Axiom 2** (Duality Axiom). M{Λ} + M{Λ^{c}} =1 for any event Λ .

**Axiom 3** (Subadditivity Axiom). For every countable sequence of events Λ_{1}, Λ_{2}, … we have

Besides, in order to provide the operational law, Liu (2013) defined the product uncertain measure on the product σ -algebre L as follows,

**Axiom 4** (Product Axiom). Let (Γ_{k}, L_{k}, M_{k}) be uncertainty spaces for k = 1, 2,L Then the product uncertain measure M is an uncertain measure satisfying

where Λ_{k} are arbitrarily chosen events from L_{k} for k = 1, 2, …, respectively.

**Definition 1** (Liu, 2007). The set function M is called an uncertain measure if it satisfies the normality, duality, and subadditivity axioms.

**Definition 2** (Liu, 2007). An uncertain variable is a measurable function ξ from the uncertainty space (Γ, L, M) to the set of real numbers, i.e., for any Borel set B of real numbers, the set

is an event.

**Definition 3** (Liu, 2007). The uncertainty distribution Φ of an uncertain variable ξ is defined by

for any real number x.

**Definition 4** (Liu, 2009). The uncertain variables ξ_{1}, ξ_{2}, …, ξ_{m} are said to be independent if

for any Borel sets B_{1}, B_{2}, …, B_{m} of real numbers.

**Theorem 1** (Liu, 2010). Let ξ_{1}, ξ_{2}, …, ξ_{n} be independent uncertain variables with continuous uncertainty distributions Φ_{1}, Φ_{2}, …, Φ_{n}, respectively. If the function f(x_{1}, x_{2}, …, x_{n}) is strictly increasing with respect to x_{1}, x_{2}, …, x_{m} and strictly decreasing with respect to x_{m+1}, x_{m+2}, …, x_{n}, then

is an uncertain variable with uncertainty distribution

**Corollary 1** (Liu, 2013). Let ξ_{1}, ξ_{2}, …, ξ_{n} be iid uncertain variables.

Then ξ_{1 }+ ξ_{2 }+ … + ξ_{n} and nξ are identically distributed uncertain variables. In other words, if Φ is the common uncertainty distribution of ξ_{1}, ξ_{2}, …, ξ_{n}, then has also the uncertainty distribution Φ (but not independent of ξ_{1}, ξ_{2}, …, ξ_{n}).

Expected value is the average of an uncertain variable in the sense of uncertain measure. It is an important index to rank uncertain variables.

**Definition 5** (Liu, 2007). Let ξ be an uncertain variable. Then the expected value of ξ is defined by

provided that at least one of the two integrals is finite.

An uncertain process (Liu, 2008) is essentially a sequence of uncertain variables indexed by time. As an important uncertain process, a renewal process is an uncertain process in which events occur continuously and independently of one another in uncertain times.

**Definition 6** (Liu, 2008). Let ξ_{1}, ξ_{2}, … be iid positive uncertain variables. Define S_{0} = 0 and S_{n} = ξ_{1 }+ ξ_{2 }+ … + ξ_{n} for n ≥ 1. Then the uncertain process

is called an uncertain renewal process.

For the uncertain renewal process, Liu (2010) proved that N_{t}/t converges in distribution to 1/ξ_{1} which is an uncertain variable. Based on this, Liu (2010) proved the elementary renewal theorem, i.e.,

under the assumption that E[1/ξ_{1}] exists.

### 3. NEW THEOREMS ON UNCERTAIN AGE REPLACEMENT POLICY

Age replacement policy in uncertain environment has been proposed by Yao and Ralescu (2013). Age replacement means that a part is always replaced at failure or at time T. Assume that the lifetimes of the parts are iid uncertain variables ξ_{1}, ξ_{2}, … with a common uncertainty distribution Φ. Then the actual lifetimes of the parts are iid uncertain variables

which may generate an uncertain renewal process

Let B denote the “failure replacement” cost of replacing an element when it fails earlier than T , and A the “planned replacement” cost of replacing an element at the age T. Define

Then f(ξ_{i}∧T) is just the cost of replacing the ith element, and the average replacement cost before the time t is

Yao and Ralescu (2013) proved that

in the sense of convergence in distribution.

**Theorem 2** (Yao and Ralescu, 2013). Let ξ_{1}, ξ_{2}, … be a sequence of iid positive uncertain variables with a common uncertainty distribution Φ, and N_{t} be an uncertain renewal process with uncertain interarrivals ξ_{1}∧T, ξ_{2}∧T, … Assume that

with 0 < A < B . Then

In order to find the optimal replacement time, we just need to find T^{*} which solves the problem

Next, we discuss the optimal age replacement policy when the lifetime of unit follows different uncertainty distribution.

**Theorem 3.** Let ξ_{1}, ξ_{2}, … be a sequence of iid positive uncertain variables with a common linear uncertainty distribution L(a, b)(a > 0), and N_{t} be an uncertain renewal process with uncertain interarrivals ξ_{1}∧T, ξ_{2}∧T, …. Assume that

with 0 < A < B. Then the optimal age replacement time is

And the minimum average age replacement cost is

**Proof.** The linear uncertainty distribution L(a, b) (Liu, 2013) is

Then

We can find the optimal age replacement time and the minimum cost in three intervals

It is easy to verify that the expected cost of the second interval is the minimum cost. Therefore, the optimal replacement time is

The theorem is verified.

**Theorem 4.** Let ξ_{1}, ξ_{2}, … be a sequence of iid positive uncertain variables with a common zigzag uncertainty distribution Z (a, b, c)(a > 0), and N_{t} be an uncertain renewal process with uncertain interarrivals ξ_{1}∧T,ξ_{2}∧T, … Assume that

with 0 < A < B. Then the optimal age replacement times is

And the minimum average age replacement cost is

**Proof.** The uncertainty distribution of zigzag uncertain variable Z (a, b, c) (Liu, 2013) is

Then

where the expected cost of the fourth interval is a constant. We can find the optimal age replacement time and the minimum cost in each interval

It is easy to verify that the expected cost of the second or the third interval is the minimum cost. Therefore, the optimal replacement time is

The theorem is verified.

**Theorem 5.** Let ξ_{1}, ξ_{2}, … be a sequence of iid positive uncertain variables that have a common lognormal uncertainty distribution N(e,σ ) with σ <π /, and N_{t }be an uncertain renewal process with uncertain interarrivals ξ_{1}∧T, ξ_{2}∧T, …. Assume that

with 0 < A < B. Then the optimal replacement time is

under the condition described in Theorem 2.

**Proof.** The lognormal uncertainty distribution (Liu, 2013) is

Therefore,

Let C' (T) = 0, we can get the unique root

Obviously, T is the optimal replacement time T^{*}. The theorem is verified.

### 4. CONCLUSION

In this paper, we discussed an optimal age replacement policy when the age of the unit follows different uncertain distributions, and the optimal time was given to minimize the long-run average replacing cost. In addition, three theorems were developed on the analytically optimal replacement policy under the assumption that the uncertain lifetimes follow linear distribution, zigzag distribution and lognormal distribution.

### ACKNOWLEDGMENTS

This work is supported by the Fundamental Research Funds for the Central Universities under grant no. ZXH2012K005. The research is also supported by Uncertainty Theory Laboratory of Department of Mathematical Sciences, Tsinghua University, Beijing, China.

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