## 1. Introduction

To reduce the failure risk of production equipments, preventive maintenance or replacement activities should be performed in appropriate schedules. The search of these appropriate schedules has led to the development and implementation of maintenance optimization policies for stochastic degrading production equipments. Indeed, the literature on this matter is already extensive, growing rapidly and also very heterogeneous. Accordingly, this chapter focuses only to some relevant and fundamental works on the maintenance theory. Early in [1, 2], several models appeared on the optimization of replacement or maintenance policies on infinite time horizon. In these works, the authors mainly discussed about the optimality conditions of theses maintenance models. Subsequently to these works, many extensions of the previous models were proposed on finite time span [3, 4] and also on infinite time horizon in the literature. For survey, the reader may refer, for example, to [5–8] and the references therein. We note that in most cited works, the authors assumed that the equipment lifetime distribution is parametrically characterized and well known. However, Coolen and his coauthors [9, 10] showed that this assumption impacts clearly the optimal replacement age and its cost per unit of time when the equipments undergo an age replacement policy (ARP). Recently in [11], de Jonge et al. pointed out also the weakness of the assumption on knowing of the equipment lifetime distribution and proposed a parametric modeling of ARP for new equipment with an uncertainty on the parameters of the equipment lifetime distribution. In this work, de Jonge and his coauthors used Bayesian approach to model the uncertainty on the parameters and figured out that this uncertainty has effects on the optimal policy (age and cost) under ARP.

Another way, most existing models merely rely only on a classical assumption which states that the operating environment is steady and has not any effect on the equipment characteristic and its lifetime distribution. Roughly speaking, they assume that the degradation process is the same during the equipment’s life cycle. This is a restrictive assumption in many industrial areas where production equipment may have experiences under different operating environments with their own severity degree that impacts the equipment performance. For example, the degradation process of the mining machinery is impacted by the severity level of the environment where the machinery is being exploited. Another example may be the engines used for oil extraction. The degradation process of such equipment depends on whether they are operated onshore or offshore. In some other industries, production equipments are first operated in a given environment and then moved to another location where this latter might be more or less severe than the first. In the same way, many companies operate their equipments at home for several years before shipping them to their subsidiaries in other countries where they would be subjected to more severe operating conditions. Therefore, suitable maintenance strategies, integrating the heterogeneous operating conditions, should be developed to assess the degradation of such equipments.

In this chapter, a preventive maintenance is investigated for such equipment subject to random failures. The equipments are assumed to have an experience under two operating environments. In fact, each operating environment is characterized by its own degree of severity, which impacts the equipment lifetime distribution. Therefore, the equipment lifetime distributions follow then a different distribution depending on the operating environments. To reduce the failure occurrence risk during operating under both operating environments, the equipment undergoes to an periodic preventive maintenance (PM). However, the equipment is subjected to minimal repair at failure. The objective consists then on evaluating the optimal age to perform periodic preventive repair in order to minimize the expected maintenance cost per unit of time. This expected cost is induced by the costs of minimal and preventive repairs. This policy was already discussed by Nakagawa in [12], in which Nakawaga considered that the equipment lifetime remains the same during the operation. Nakagawa analyzed mathematically the periodic and sequential maintenance policies. Therefore, our chapter can be considered as an extension of Nakawaga work.

The remainder parts of the chapter are organized as follows. The analyzed problem is briefly introduced in Section 2. This section proposes a mathematical formulation of the total maintenance cost. Section 3 focuses on the maintenance cost analysis in order to derive the optimal conditions which ensure minimal total cost per unit of time. In this same section, an heuristic is proposed to find the optimal number and period between preventive actions on both environments. Numerical experiments are conducted to illustrate the proposed approach on the one hand, and on the other hand, the accuracy and robustness of model are demonstrated through the simulation in Section 4. At the end, a conclusion and future works are drawn in the last section.

## 2. Mathematical formulation of the maintenance cost

In this section, modeling of the maintenance policy is going to be proposed. This modeling takes into account different hypotheses of our analysis. In fact, our equipment has to be used under two operating environments with different severities denoted by
_{.} The equipment operates successively on both environments in order to perform its missions. During this operation, the equipment undergoes by two types of maintenance actions. Roughly speaking, the equipment is going to be repaired minimally at failure and preventively after some
**Figure 1**.

### 2.1. Preventive maintenance cost

During operation, the equipment undergoes by preventive action after each
_{,} respectively, on the first and second environments. Therefore, the total preventive repair costs

during the length of operation

### 2.2. Minimal repair cost

The minimal repair is performed regardless of the preventive actions. The minimal repair is performed at failures in order that equipment reaches the same reliability just before failing. Each minimal repair costs
_{,} respectively, on the first and second environments. Therefore, the cost of minimal repair, on the

where

We also deduce the total minimal cost on the second environment as follows

In addition, the operation on the

After, the equipment moves on to the second environment to operate between

That involves a minimal cost on this period

To reduce the complexity during computing, we assume that the duration

Indeed, the hypothesis

Eq. (14) is equivalent to

where

### 2.3. Total maintenance cost

From previous Eqs. (13) and (14), we deduce a mathematical formulation of the total maintenance cost according to the set of parameters

Based on the equation, the next section is going to analyze the optimality according to the different parameters such as the number and the duration between the preventive repairs.

## 3. Optimality analysis

Herein, the maintenance cost is rewritten in order to integrated the impacts of preventive maintenance (PM) on the equipment lifetime distribution. We assume that a preventive action allows to reduce the age of equipment to zero and increase the hazard function. **Figures 1** and **2** point out the impact of PM on the equipment hazard and reliability functions. The hazard function is defined after PM as follows

where

with

In the second operating environment, the hazard function at

with

The total cost due to the minimal repair in the second environment becomes

By considering Eqs. (15), (18), and (19), the total cost per unit of time is rewritten as follows

(23) |

### 3.1. Optimality according to *n*_{1} and *n*_{2}

Let us assume that there is a pair

#### 3.1.1. Local optimality

The local optimality concerns the direct neighbors of the optimal pair such as

| () |

This system is equivalent to

with

In fact

and

The right-hand side of the previous equation shows that
_{,} then there exists a unique

| () |

This is equivalent to

with

This equation implies

and

Therefore,

#### 3.1.2. Global optimality

The global optimality compares the optimal pair to

| () |

This is equivalent to

with

With

and

Therefore,

### 3.2. Optimality according to *x*_{1} and *x*_{2}

For given number of preventive actions
_{,} the optimal durations

This implies

By dividing, we obtain

| () |

| () |

and from Eq. (32), we deduce

The uniqueness is tough to establish due to the number of parameters and the complexity of the proposed cost model here. To make the research of optimal solution easy, we propose a handy heuristic based on the optimal derived conditions in this chapter. The next section describes step by step the proposed heuristic which leads to a suitable solution for our optimization problem.

### 3.3. Numerical resolution of problem

Herein, an algorithm is drawn in order to find the optimal pairs for

Initialize the pair
Put
STEP (A) Research optimal
Compute
Research
Research
Research
Keep corresponding
Go to step (A)
End. | () |

## 4. Numerical application

We consider an equipment whose lifetime distribution function is Weibull with the same shape parameter
_{,} while
_{,} respectively, on the first and second. The preventive actions impact the lifetime distribution of equipment. The impact factors due to PM are equal to

## 5. Conclusion

This chapter shows how to solve Nakagawa maintenance policy problem for an equipment which operates simultaneously on two environments. Each environment impacts the lifetime distribution function of our equipment. Nakagawa’s maintenance problem is modeled under lifetime distribution changing in operation. The proposed model is deeply analyzed in order to derive the conditions under which optimal pairs exist and are reachable. To reach these pairs, algorithm was proposed to find the optimal solution for the periodic preventive maintenance on infinite horizon. The model is handy and suitable for production equipments which have to experience under different operating environments with their own severity degree that impacts the equipment performance such as onshore or offshore.

For future work, we plan to propose a statistical modeling by ignoring the hypothesis on the knowledge of the equipment lifetime distribution and perform an extension of the analysis by considering an finite-time horizon/span.