*2.2.1 Establishment of risk assessment model for mechanical equipment*

After completing the scoring according to each evaluation index, it is necessary to comprehensively evaluate the risk level of the object. Therefore, it is necessary to comprehensively consider the rating vector *V* of its evaluation index and its corresponding weight value vector *W* [12]. The object's risk level evaluation index can be expressed as

$$Index(V, W) = F[v\_1 w\_1, v\_2 w\_2, \dots, v\_n w\_n].\tag{1}$$

Among them, *F*½ �� is a comprehensive evaluation function that reflects the degree of influence of each evaluation index on the object risk level and can be a function of various forms. Using a simpler linear weighted model, the formula for calculating the object risk level evaluation index, namely, *Index*, is as follows:

$$Index = \sum\_{i=1}^{n} v\_i w\_i \tag{2}$$

where *n* is the number of evaluation indicators; *vi* is the rating value of the *i*th evaluation index of the evaluated object; and *wi* is the weight value of the *i*th evaluation index of the evaluated object.

Therefore, the magnitude of evaluation value *Index* indicates the risk level, so that the objects can be sorted and screened based on their different risk levels.

From Eq. (2), we can know that the weight *wi* of the influencing factors will have a great influence on the final value of the risk level evaluation inde*x Index*. Therefore, the AHP method is used for this calculation above. The specific calculation steps are as follows:

**Step 1:** Constructing a judgment matrix *D* through pairwise comparisons among the evaluation indexes,

$$D = \begin{bmatrix} u\_{11} & u\_{12} & \dots & u\_{1n} \\ u\_{21} & u\_{22} & \dots & \dots & u\_{2n} \\ \dots & \dots & \dots & \dots & \dots \\ u\_{n1} & u\_{n2} & \dots & \dots & u\_{nn} \end{bmatrix} \tag{3}$$

*CI* <sup>¼</sup> ð Þ *<sup>λ</sup>*max � *<sup>n</sup> <sup>=</sup>*ð Þ <sup>n</sup>‐<sup>1</sup> (7)

where CR is the random consistency ratio of the judgment matrix; CI is the general consistency index of the judgment matrix; and RI is the mean random

For 2 to 9th-order judgment matrix, the value of RI is shown in **Table 13**.

consistency meets the above requirement. At this time, the maximum eigenvector of the judgment matrix *D* corresponds to the weight value of each factor. The priority of mechanical equipment can be determined according to the weight of

Because the scoring process of the influencing factors of the risk level of mechanical equipment has subjective factors and differences among individual experts, based on the AHP analysis method to determine the ranking of the influencing factors of each level of risk, the Monte Carlo simulation method is used for calculation [13]. In the calculation process of the Monte Carlo method, the weight of each evaluation factor can be changed by generating random numbers, so that the robustness of the risk level ranking of mechanical equipment is enhanced, and the ranking results are less affected by subjective factors. The logic block diagram of the Monte Carlo simulation is shown in

As shown in **Figure 1**, a certain random numbers in [0, 1] are generated in the

calculation process. The random numbers are regarded as the weight value of certain evaluation indexes and assigned with the priority order obtained in the previous calculation process [14]. In other words, for any group of random numbers, the largest random number will be assigned to the top priority, the smallest one will be assigned to the lowest priority, and the rest of random numbers will be assigned to the other evaluation indexes in order of priority from large to small. Then, in an MCS computation, the total score of all evaluation indexes can be calculated using Eq. (1), and the risk level of the mechanical equipment will be obtained and ranked according to the calculated *Index* . Through *N* times simulation calculations in the MCS, a number of ranking values are obtained based on different risk levels of the same mechanical equipment. Then, the risk level of a single equipment can be displayed from their sequence of cumulative frequency reaching 1, namely, the faster cumulative frequency of one mechanical equipment reaches

*n* **2 345** RI 0.00 0.58 0.9 1.12 *n* 6 789 RI 1.24 1.32 1.41 1.45

If CR<0*:*01, the consistency of the judgment matrix *D* is satisfactory, which means that the weight apportionment of each evaluation index is reasonable; if not, the judgment matrix *D* should be adjusted until the

**Step 5:** Performing consistency adjustment and weight ordering.

*2.2.2 Analysis of eliminating the subjective factors based on the MCS*

consistency index of the judgment matrix.

*Maintenance Decision Method Based on Risk Level DOI: http://dx.doi.org/10.5772/intechopen.91913*

each factor.

**Figure 1** [12].

**Table 13.**

**339**

"1," so that it will be a higher risk level.

*RI values of the 2 to 9th-order judgment matrix.*

where *uij* is a relative risk level value that of the *i*th evaluation index compared with the*j*th evaluation index and *ujiuji* is the relative risk level value that of the *j*th evaluation index compared with the *i*th evaluation index.

Thus, the value of *uji* is the reciprocal value of *uij*, namely *uji* � *uij* ¼ 1 . The definition and fundamental scale of the relative risk level are shown in **Table 12**.

**Step 2:** Calculating the maximum eigenvalue *λmax* of the judgment matrix *D* using the system of homogeneous linear equations as follows:

$$\begin{cases} (\boldsymbol{u}\_{11} - \boldsymbol{\lambda})\boldsymbol{o}\_{1} + \boldsymbol{u}\_{12}\boldsymbol{o}\_{2} + \cdots + \boldsymbol{u}\_{1n}\boldsymbol{o}\_{n} = \mathbf{0} \\ \boldsymbol{u}\_{21}\boldsymbol{o}\_{1} + (\boldsymbol{u}\_{22} - \boldsymbol{\lambda})\boldsymbol{o}\_{2} + \cdots + \boldsymbol{u}\_{2n}\boldsymbol{o}\_{n} = \mathbf{0} \\ \text{....} \\ \boldsymbol{u}\_{n1}\boldsymbol{o}\_{1} + \boldsymbol{u}\_{n2}\boldsymbol{o}\_{2} + \cdots + (\boldsymbol{u}\_{nn} - \boldsymbol{\lambda})\boldsymbol{o}\_{n} = \mathbf{0} \end{cases} \tag{4}$$

**Step 3:** Determining the eigenvector relative to the maximum eigenvalue *λmax*, which is given by

$$W = (a\_1, a\_2, \dots, a\_n) \tag{5}$$

**Step 4:** Checking the consistency of the judgment matrix *D*,

$$\text{CR} = \text{CI}/\text{RI} \tag{6}$$


**Table 12.** *The definition and fundamental scale of the relative risk level.* *Maintenance Decision Method Based on Risk Level DOI: http://dx.doi.org/10.5772/intechopen.91913*

where *n* is the number of evaluation indicators; *vi* is the rating value of the *i*th

Therefore, the magnitude of evaluation value *Index* indicates the risk level, so that the objects can be sorted and screened based on their different risk levels. From Eq. (2), we can know that the weight *wi* of the influencing factors will have a great influence on the final value of the risk level evaluation inde*x Index*. Therefore, the AHP method is used for this calculation above. The specific calcula-

**Step 1:** Constructing a judgment matrix *D* through pairwise comparisons among

*u*<sup>11</sup> *u*<sup>12</sup> …… *u*1*<sup>n</sup> u*<sup>21</sup> *u*<sup>22</sup> …… *u*2*<sup>n</sup>* …… …… …… …… *un*<sup>1</sup> *un*<sup>2</sup> …… *unn*

where *uij* is a relative risk level value that of the *i*th evaluation index compared with the*j*th evaluation index and *ujiuji* is the relative risk level value that of the *j*th

Thus, the value of *uji* is the reciprocal value of *uij*, namely *uji* � *uij* ¼ 1 . The definition and fundamental scale of the relative risk level are shown in **Table 12**. **Step 2:** Calculating the maximum eigenvalue *λmax* of the judgment matrix *D*

> ð Þ *u*<sup>11</sup> � *λ ω*<sup>1</sup> þ *u*12*ω*<sup>2</sup> þ ⋯ þ *u*1*<sup>n</sup>ω<sup>n</sup>* ¼ 0 *u*21*ω*<sup>1</sup> þ ð Þ *u*<sup>22</sup> � *λ ω*<sup>2</sup> þ ⋯ þ *u*2*<sup>n</sup>ω<sup>n</sup>* ¼ 0 ⋯⋯⋯⋯ *un*1*ω*<sup>1</sup> þ *un*2*ω*<sup>2</sup> þ ⋯ þ ð Þ *unn* � *λ ω<sup>n</sup>* ¼ 0

**Step 3:** Determining the eigenvector relative to the maximum eigenvalue *λmax*,

using the system of homogeneous linear equations as follows:

**Step 4:** Checking the consistency of the judgment matrix *D*,

**Important scale Definition** Equally important Moderately important Strongly more important Very strongly important Extremely more important 2, 4, 6, 8 Situation between the above levels

*W* ¼ ð Þ *ω*1,*ω*2, ⋯,*ω<sup>n</sup>* (5)

*CR* ¼ *CI=RI* (6)

(3)

(4)

evaluation index of the evaluated object; and *wi* is the weight value of the *i*th

evaluation index of the evaluated object.

*Operations Management - Emerging Trend in the Digital Era*

tion steps are as follows:

the evaluation indexes,

*D* ¼

evaluation index compared with the *i*th evaluation index.

8 >>><

>>>:

*The definition and fundamental scale of the relative risk level.*

which is given by

**Table 12.**

**338**

$$CI = (\lambda\_{\text{max}} - n) / (\text{n-1}) \tag{7}$$

where CR is the random consistency ratio of the judgment matrix; CI is the general consistency index of the judgment matrix; and RI is the mean random consistency index of the judgment matrix.

For 2 to 9th-order judgment matrix, the value of RI is shown in **Table 13**. **Step 5:** Performing consistency adjustment and weight ordering.

If CR<0*:*01, the consistency of the judgment matrix *D* is satisfactory, which means that the weight apportionment of each evaluation index is reasonable; if not, the judgment matrix *D* should be adjusted until the consistency meets the above requirement. At this time, the maximum eigenvector of the judgment matrix *D* corresponds to the weight value of each factor. The priority of mechanical equipment can be determined according to the weight of each factor.

### *2.2.2 Analysis of eliminating the subjective factors based on the MCS*

Because the scoring process of the influencing factors of the risk level of mechanical equipment has subjective factors and differences among individual experts, based on the AHP analysis method to determine the ranking of the influencing factors of each level of risk, the Monte Carlo simulation method is used for calculation [13]. In the calculation process of the Monte Carlo method, the weight of each evaluation factor can be changed by generating random numbers, so that the robustness of the risk level ranking of mechanical equipment is enhanced, and the ranking results are less affected by subjective factors. The logic block diagram of the Monte Carlo simulation is shown in **Figure 1** [12].

As shown in **Figure 1**, a certain random numbers in [0, 1] are generated in the calculation process. The random numbers are regarded as the weight value of certain evaluation indexes and assigned with the priority order obtained in the previous calculation process [14]. In other words, for any group of random numbers, the largest random number will be assigned to the top priority, the smallest one will be assigned to the lowest priority, and the rest of random numbers will be assigned to the other evaluation indexes in order of priority from large to small. Then, in an MCS computation, the total score of all evaluation indexes can be calculated using Eq. (1), and the risk level of the mechanical equipment will be obtained and ranked according to the calculated *Index* . Through *N* times simulation calculations in the MCS, a number of ranking values are obtained based on different risk levels of the same mechanical equipment. Then, the risk level of a single equipment can be displayed from their sequence of cumulative frequency reaching 1, namely, the faster cumulative frequency of one mechanical equipment reaches "1," so that it will be a higher risk level.


**Table 13.** *RI values of the 2 to 9th-order judgment matrix.*

failure effect, and failure cause of the three categories of mechanical equipment are determined based on the FMEA method. Then, the MDMETs on the mechanical equipment of the petrochemical industry are established by referencing the logic decision diagram in the reliability centered maintenance (RCM) theory. The

1.The failure consequence of Class A of mechanical equipment has little or no influence on the function of the whole system or causes lower maintenance costs. Increasing the spare part inventory or decreasing the failure frequency for Class A of mechanical equipment cannot affect the production process. A

MDMET of Class A of mechanical equipment is shown in **Figure 2**.

2.When Class B of mechanical equipment has been failed, it might result in severe failure consequences, but it usually does not influence personnel safety and environment. The failure frequency of Class B of mechanical equipment could be reduced through reasonable maintenance strategies, so that failure consequences could be decreased as well. But these maintenance strategies might cause higher maintenance costs. A MDMET of Class B of mechanical equipment is shown in **Figure 3**, which includes corrective maintenance, time-

3.The failure of Class C of mechanical equipment might endanger the personnel

safety, pollute the environment, and cause the significant economic consequences. In order to ensure the operation reliability and maintenance economy of Class C of mechanical equipment, some special maintenance

MDMETs are shown in detail as follows [18–20]:

*Maintenance Decision Method Based on Risk Level DOI: http://dx.doi.org/10.5772/intechopen.91913*

based maintenance, and hidden failure detection.

**Figure 2.**

**Figure 3.**

**341**

*MDMET of Class A of mechanical equipment*

*MDMET of Class B of mechanical equipment.*

**Figure 1.** *Logic block diagram of the Monte Carlo simulation.*

### **2.3 Research on maintenance decision-making methods based on the risk level of mechanical equipment**

According to statistical data about the priority orders of the mechanical equipment from their evaluation risk levels in the previous step, their cumulative frequency can be plotted with a curve chart. Based on the principle of establishing a cumulative frequency curve chart, the percentage of the area on the right side of the curve from the total area can be taken as another representation evaluating the risk level of the mechanical equipment [15]. A larger area percentage is indicating that one mechanical equipment has a higher risk level. According to their different area percentages, namely, the different risk levels among them, the mechanical equipment can be divided into three categories, including Classes A, B, and C. Class A is an area percentage of 0–30% of mechanical equipment. Class B is an area percentage of 30–80% of mechanical equipment. Class C is an area percentage of 80–100% of mechanical equipment [16, 17].

According to different own characteristics and failure modes of the mechanical equipment in the petrochemical industry, the existing maintenance methods include Lubrication, Service, Corrective Maintenance, Time-based Maintenance, Hidden Failure Detection, and Condition-based Maintenance. In order to reasonably establish maintenance decisions of mechanical equipment tree (MDMET) and effectively implement the existing maintenance methods above, the failure mode,
