A fuzzy-based decision support model for ... - Semantic Scholar

Expert Systems with Applications 38 (2011) 13342–13350

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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa

A fuzzy-based decision support model for engineering asset condition monitoring – A case study of examination of water pipelines H.C.W. Lau a,⇑, R.A. Dwight b a b

School of Management, University of Western Sydney, Australia School of Mechanical, Materials and Mechatronic Engineering, University of Wollongong, Australia

a r t i c l e

i n f o

Keywords: Engineering asset management Failure symptoms Fuzzy-based decision support Condition monitoring Water pipelines

a b s t r a c t Engineering asset management (EAM) is a multi-disciplinary activity that aims to tackle the issues of asset capability, life, safety, maintenance and reliability, taking into account economical and managerial factors. Condition monitoring is an important aspect of EAM as it is able to identify potential failure symptoms and suggest remedial actions prior to any operational interruptions. In general, conditions of assets can be investigated through various tests and then decision has to be made if the asset should be repaired or replaced or further in-depth test is needed. In the current practice, the decision to be made is normally based on human judgement and field experience which are subject to personal view and bias. As such, a more scientific and reliable decision support model is needed to help companies make the right decision which may be vital to ensure that daily operations will not be disrupted. In this paper, a decision support model characterized by its inclusion of fuzzy logic technology to achieve rule inference is proposed. This fuzzy-based decision support (FDS) model adopts the fuzzy reasoning approach to suggest the optimal action that needs to be taken to deal with the problem of asset conditions. This approach provides a relatively independent result based on available numerical (crisp) data. One of the main benefits of using FDS model is that its knowledge base accumulates experience through a learning process and becomes ‘‘smarter’’ over time. To demonstrate the feasibility of this approach, a case study related to the condition monitoring of water pipelines in an electro-plating plant based in China has been conducted. Results indicate that this model provides expertise advice and pre-warning signals, if any, of engineering asset conditions based on input crisp data of examined asset status, thereby enhancing the effectiveness of EAM. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Condition monitoring is important for steel pipelines which may leak water or even burst due to unattended damage caused by corrosion. In general, galvanic cells are created in the presence of oxygen and moisture and the subsequent effect is that corrosion will occur in the pipelines. This happens in a number of ways where steel pipelines are in contact with soil externally and water internally. This type of corrosion damage normally causes pipelines being dismantled, resulting in plant disruption and huge repair cost and eventually it becomes necessary to replace the entire pipeline at a great cost. Condition monitoring of pipeline integrity is essential to detect potential corrosion problem areas, so that the required corrosion protection techniques can be applied before the damage necessitates replacement of the pipeline. This paper focuses on the theory and implementation of a proposed ⇑ Corresponding author. Tel.: +61 (612) 96859488. E-mail addresses: [email protected], [email protected] (H.C.W. Lau), [email protected] (R.A. Dwight). 0957-4174/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.04.158

model which is able to provide a scientific analysis of the pipeline conditions in an independent and unbiased way. The proposed model can be used to check the integrity of engineering assets other than water pipelines such as gas pipelines and even rail lines. 2. Related studies There are a number of publications related to condition monitoring of engineering assets. A condition monitoring technique on wear assessment of engineering assets has been proposed with recommendable results evidenced by a case study conducted in an industrial environment (GarcìaMàrquez, Schmid, & Collado, 2003a). This group of researchers also suggested another condition monitoring approach to assess railway safety based on a reliability centered approach with a case study to validate the technique suggested (GarcìaMàrquez, Schmid, & Collado, 2003b). Jardine and his team also proposed a decision support model based on optimization of condition-based maintenance for engineering assets subject to vibration (Jardine, Joseph, & Banjevic, 1999). Sherwin and Al-Najjar (1999) have also initiated a model for condition

H.C.W. Lau, R.A. Dwight / Expert Systems with Applications 38 (2011) 13342–13350

monitoring based on proposed inspection intervals. This approach proves to be quite practical based on testing results. In general, the research in the area of condition monitoring of engineering assets is more focused on providing a model or technique to assess the asset status (Lee & Lau, 1999). Christer (1999) and Christer and Wang (1995) focused on use of data derived from expert opinion to estimate the time to failure given condition information, for the purposes of determining appropriate condition monitoring inspection intervals. However, the study related to the development of a reliable and practical model which can suggest numerical results reflecting the actual conditions of the assets is yet to be considered as adequate. Normally, little or no factual information about the remaining time to failure is available in the practical situation. In this study, a fuzzy-based model supporting decision making with information provided related to the condition of assets is proposed and has been under test in an electro-plating plant. This proposed model capitalizes on the unique features of fuzzy logic principles. In fact, the first publication on fuzzy logic was made in the US by Lotfi Zadeh (Kosko, 1993). In short, fuzzy logic is one of the fastest growing technologies in the world since the beginning of the computer era (Berkan & Trubatch, 1996; Lau, Tso, & Ip, 1999a) and is generally regarded as the study of the methods and principles of reasoning in all its possible forms. In addition, its reasoning mechanism is based on fuzzy sets or sets of fuzzy rules (Ciliz, 2005; Kosko, 1993). Since 1965, different kinds of ways to use fuzzy logic are developed extensively. The application areas are generally related to process industry, biotechnology, manufacturing, electro-mechanical systems, traffic control, avionics and biomedical systems (Lau, Cheng, Lee, & Ho, 2008; Martins, Costa, Pires, & Dente, 2001; Verbruggen & Babuska, 1999). One of the areas in which fuzzy logic principles have been applied most extensively is in modeling for managerial decision-making (Duch, Adamczak, & Grabczewski, 2001; Lin, Hsu, & Sheen, 2007; Zimmermann, 1996). Also, the application of fuzzy logic principles in home appliances to save energy and optimize operation processes is not uncommon nowadays. Basically, washing machines incorporated with fuzzy logic features are known for the benefits of saving energy as well as the optimization of wear and tear on clothes. It is also believed that the Sendai subway system might run smoother and more economically with a fuzzy controller than with an automatic controller (Guo, Tsai, Lee, & Chang, 1998; Kosko, 1997; Lau, Wong, & Pun, 1999b). Other applications for fuzzy systems include areas such

Situation Analysis

3. Development of fuzzy-based decision support (FDS) model for EAM FDS model is developed for monitoring of conditions of engineering assets. The proposed model helps engineers and executives make decisions regarding refurbishment or replacement of water pipelines or even further investigation is needed. In short, FDS model consists of six stages including situation analysis, knowledge acquisition, data collection, fuzzification, fuzzy inference engine and defuzzification. Fig. 1 shows the conceptual model of FDS and indicates the inputs and outputs of the processes involved. The procedures comprising fuzzification to defuzzification can be grouped to form the fuzzy system which embraces the fuzzy logic principle. This is the core part of the whole system with details shown in Fig. 2. 3.1. Situation analysis The first stage is the situation analysis which is concerned with the investigation of the potential problem to be dealt with. Aims and missions are to be determined, such as problem analysis, data required, and knowledge domain. Problem analysis is related to the description of the seriousness of the involved problem. The cause of the problem is the focus of the investigation. Normally,

Input Data Set

Fuzzification

Data Collection

Required Knowledge Domain

as breakdown prediction for nuclear reactors in Europe and earthquake forecasting in China (Terano, Asai, & Sugeno, 1992). It can be seen that although the fuzzy logic principle has been adopted in various industrial areas, its applications in the engineering asset management aspects still need to be explored due to inadequate literature in this area. Recent study on the relevant topics indicates that whilst a number of approaches have been designed and implemented to improve the checking of engineering assets using various techniques and methodologies, it falls short of the approaches for the formulation of a model, with certain degree of intelligence, to cope with the uncertain factors related to actual asset conditions. The issue is addressed in this paper with the introduction of a fuzzybased decision support (FDS) model which is able to provide a crisp value to reflect the current status of the engineering assets, enhancing the efficiency of subsequent actions to be taken to rectify the conditions.

Required System Structure

Required Data Classification

Input Fuzzy Set

Fuzzy Inference Engine

Knowledge Acquisition Required Knowledge

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Output Fuzzy Set

Optimization System

Defuzzification

Fuzzy System

Crisp Value / Linguistic Value

Fig. 1. The schematic diagram of FDS model.

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Fuzzification Input Fuzzy Set

Fuzzy Inference Engine IF A IS T AND B IS P THEN C IS K IF D IS R OR E IS Q THEN C IS W IF G IS X AND H IS L THEN C IS M

Defuzzification Output Fuzzy Set

Rule Set

Input 1

Rule Block

Crisp Value

COA Input 2

Linguistic Value

Input 3

MOM Output 1

Fig. 2. The structure of fuzzy model.

problems are generated during the period of business transformation to meet market changes. Better analysis of the market situation and development process is the key to success at this stage which provides essential information to support the design of the structure of the fuzzy system as well as the selection of linguistic values. Generally speaking, the structure of the fuzzy system is closely related to the nature of the existing problem and the linguistic values are most likely the cause factors of the problem. Although different kinds of data are generated and used in a company, it is necessary to sort out the required data. Not all data is necessary for situation analysis as only the data related to the problem is selected and processed. The selected data will be classified into different types for the following data collection stage.

which form part of the fuzzy system. Knowledge acquisition can be conducted in different ways such as interviews of domain experts and data mining of historical data. For the interview of domain experts, implicit knowledge is normally obtained but requires a special process to turn it into explicit knowledge. The knowledge from the domain experts is the result of years of field experience, providing the basis of supporting the decisions of actions to be taken. The proposed FDS provides decision support which is built upon the knowledge and practical experience of practitioners in related areas. In general, the captured knowledge is represented in the form of rules as well as facts, all of which are derived based on the information obtained from the interview process with relevant domain experts. The block of rules created is considered as an important part of the FDS.

3.2. Data collection 3.4. Fuzzification Data is collected from different data classifications as mentioned above. Classified raw data needs to be mapped with the relevant category with the right format prior to being delivered to the fuzzy system as the input data set. The input data received from the external world may exist in dissimilar formats such as crisp numeric data, bivalent data, linguistic data, statistical data and a set of crisp value. Some data can be put into the fuzzy system directly while others may not be format-compatible. For example, some sets of data represented in the form of diagrams are obviously not supported for direct data integration with the system. An intermediate process is needed to translate the data into the acceptable format for direct data assimilation. Apart from the significance of the right format of the data, the size of data is also a factor for consideration. The data size can affect the accuracy of the input data set. Too few data cannot represent the whole set whilst a large data size makes the system slow in data processing. In general, decisions have to be made based on past experience and previous results.

Referring to Fig. 2, fuzzification is the first stage of operation in the fuzzy system. It is mainly concerned with the conversion of the input data set (from the data collection process) into fuzzy sets. In order to carry out this fuzzification process, it is necessary to specify two decisive factors including universe of discourse and membership function, both of which are needed to determine the overall features of the fuzzy sets. In short, the universe of discourse is the numerical range of the inputs, normally referring to the range of x-axis in the graph of the fuzzy set. It limits the input values which are to be constrained within the specified range. Basically, the input ranges are determined in accordance with the analysis results obtained from the previous stage. The membership function decides the characteristics of the fuzzy subset A:

A¼

n X

lA ðxi Þ=xi

i1

3.3. Knowledge acquisition Required knowledge needs to be extracted from the related domains followed by the procedure to turn it into IF-THEN rules

The above equation is the general mathematical expression of fuzzy subset A of X where X is the whole data set and x is the element of subset A, lA(xi) is the membership function of element xi in the universe of discourse when the support set is a finite set,

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µT(t) µT(t) U

BA

1

1 0 0

4

t

12

t

8

µT(t) U BA JA G VG

1

µT(t)

0 µT(t)

12

24

t

JA

1

G

1 0 0

12

18 t

µT(t)

8

16

t

VG

1

0

18

24

t

Fig. 3. The composition of fuzzy set.

X ¼ fx1 ; x2 ; x3 ; x4 ; . . . ; xn g

Equation 2

y In FDS model, the universe of discourse is normally divided into several regions which belong to different predicates such as Unacceptable (U), Barely Acceptable (BA), Just Acceptable (JA), Good (G) and Very Good (VG). Fig. 3 shows how the predicate functions form the composition of the fuzzy set. These predicate functions have special shape, height and line style to represent their membership function. In general, triangles and trapezoids are the most commonly used shapes as they are simpler in terms of calculation. The input fuzzy set comprises several membership values from different fuzzy inputs. The expression of membership value is shown below.

Equation 1 y1

(x1, y1)

x Fig. 4. The intersection with a fuzzy set.

la ¼ lk ðxÞjx¼a ¼ b where a and b are real numbers, representing crisp input data and membership value respectively, lk(x) is the fuzzy set. These values are the intersection of the two equations represented by the crisp input data and the function of predicate. The intersection point (x1, y1) shown in Fig. 4 is the same point of the two equations in graph. The y-coordinate, y1, is the membership value relative to the crisp input data. In some cases, the crisp input equation intersects two different equations belonging to different predicate functions that can generate two membership values, such as p1 and p2 as shown in Fig. 5. In the case of condition monitoring of water pipelines, the three fundamental inputs are Input 1: Conditions based on Visual Examination (V). Where V is the fuzzy set, vi is the elements in the data set, lT(vi) is the membership function

Equation 2

p

P2 p1

(x1, p2) Equation 1 (x1, p1) Equation 3

x Fig. 5. The two intersection points of the fuzzy sets.

V¼

n X i1

lV ðv i Þ=v i

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µ H(h)

µV(v) U

BA JA G

VG

1

1

0

70

80

90

U BA JA G

0

V

Fig. 6. The fuzzy set lV(v).

60

70

VG

80

H

Fig. 7. The fuzzy set lH(h).

and

V ¼ fU; BA; JA; G; VGg where U is the Unacceptable, BA is the Barely Acceptable, JA is the Just Acceptable, G is the Good, VG is the Very Good. The numerical value of V is the percentage of coating and lining remaining in good condition Input 2: Strength based on Hardness Testing (H). Where H is the fuzzy set, hi is the elements in the data set, lH(hi) is the membership function

H¼

n X

lH ðhi Þ=hi

µT(t) 1

0

U BA JA G

8

10

VG

12

T (mm)

i1

Fig. 8. The fuzzy set lT(t).

and

H ¼ fU; BA; JA; G; VGg where, U is the Unacceptable, BA is the Barely Acceptable, JA is the Just Acceptable, G is the Good, VG is the Very Good. The hardness was done on a test machine to determine the bulk hardness (as shown in Fig. 7). Input 3: Minimum wall thickness using ultrasonic equipment (T). Where T is the fuzzy set, mi is the elements in the set, lM(mi) is the membership function

T¼

n X

lT ðti Þ=ti

i1

and

T ¼ fU; BA; A; RL; VGg where, U is the Unacceptable, BA is the Barely Acceptable, JA is the Just Acceptable, RL is the Good, VG is the Very Good (see Fig. 8). Output: Chance of Failure (CF).Where CF is the fuzzy set, cfi is the elements in the set, lCF(cfi) is the membership function and the expression is:

CF ¼

n X

lCF ðcfi Þ=cfi

i1

and

CF ¼ fEU; VU; U; NL; P; VP; EPg where, EU is the extremely unlikely, VU is the very unlikely, U is the unlikely, NL is the not that likely, L is the likely, VL is the very likely, EL is the extremely likely.

3.5. Fuzzy inference engine Fuzzy inference engine is the second stage of the fuzzy system. Its main operation is to convert the input fuzzy set into output fuzzy set through an inference process which includes rule block for-

mation, rule composition, rule firing, implication and aggregation (as shown in Fig. 9). Rule block consists of a number of fuzzy rules which are interrelated and normally operate based on certain set criteria. The number of rules is determined in line with the complexity of the associated fuzzy system. A fuzzy rule is composed of two parts namely IF-part and THEN-part. Unlike conventional rule-based mechanism, fuzzy rules allow the use of imprecise, uncertain and ambiguous terms. In this research, a number of rules are established for suggesting actions to support the responsive replenishment system. The antecedents (statements before ‘‘THEN’’) and consequents (statements after ‘‘THEN’’) of the rules contain fuzzy data. Rules can be displayed in table format which can easily be searched. Fig. 6 shows the three-dimensional rule block including three input dimensions and the indicator (as shown in Fig. 6) is able to pinpoint the position of the associated three-dimensions. For instance, within the rule ‘‘IF Visual IS Just Acceptable AND Hardness IS Good AND Minimum Thickness IS Barely Acceptable THEN Chance of Failure IS Not That Likely, IF-part contains three inputs which connect two times with the operator ‘‘AND’’. The intersection cell of the column and the row is the output result, i.e. ‘‘NL’’. Any combination of the three inputs (such as Visual, Hardness and Minimum Thickness) can be found in the rule block based on the IF-THEN rule, thereby deriving the required result. The input fuzzy set, which is related to specific predicate function, has one membership value by adding the crisp input data. As there are three crisp inputs in each rule to cope with three different input fuzzy sets, three membership values are calculated. Composition is the process to calculate the three membership values into the finalized rule input. Different basic logic operators have different calculation methods. For example, using the rule above, three values are calculated from the input fuzzy sets. Rule firing is the process for screening out which fuzzy rule is to be activated. The activated rules are fired and selected for further analysis. There are many rules in the fuzzy system but only some rules are fired subject to the operating conditions. This activating process is called ‘‘firing’’.

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U (Visual) U

BA JA

G VG

U EL EL EL VL VL BA EL EL VL VL VL JA EL VL VL VL L G

VL VL VL

VG VL VL

L

L

L

BA (Visual)

L

L

U BA JA G VG

U VL VL VL BA VL VL L JA VL L L G

L

L

VG

L

L

L

L L L

L L NL

NL NL

NL NL NL U BA JA

Indicator: Visual (V)

G

L L L

L L NL NL L NL NL NL NL NL NL U

NL NL NL

VG NL NL

U

U

U

U

U

Hardness (H)

Min. Thickness (T)

JA (Visual) U BA JA G VG

G (Visual) U BA JA

G VG

U NL NL NL BA NL NL U JA NL U U

U U U U U VU

G

U

VG U

IF Visual IS Good AND Hardness IS Unacceptable AND Min. thickness IS Just Acceptable, THEN Chance of Failure IS Not that Likely

U

U VU VU

U VU VU VU U BA JA G VG

VG (Visual) U BA JA G VG U U U VU VU

U U VU VU VU

U VU VU VU EU

VU VU VU EU EU

VU NC EU EU EU

Fig. 9. The rule table of the fuzzy system.

The term implication, in the context of fuzzy logic, refers to the THEN-part calculation of the rules using the results of rule composition with different implication operators such as Mamdani Operator, Larsen Operator and Lukasiewicz Operator. Different operators generate different implication results involving various mathematics calculations and expressions, such as

lðx; yÞ ¼ /½lA ðxÞ; lB ðyÞ where implicator operator is denoted by /, input membership function by lA(x) and output membership function by lB(y). The three operators mentioned above are the most commonly used ones. Implication operators can be classified into two groups; one group is for generating the directly proportional result that the higher composition result can have a higher value of implication result, e.g. Mamdani Operator and Larsen Operator, whereas the other group can generate inversely proportional result such as Lukasiewicz Operator. They are used in different cases with different characteristics. New resultant fuzzy set is formed after the implication. Each fired rule has each implication result. Aggregation is the method for fusing all implication results into the final one. It is the last procedure in the fuzzy inference engine. All the implication results are processed to form the output fuzzy set by aggregation operator. Different implication operators match with different aggregation operator such as Union and Intersection. Whilst Mamdani Operator and Larsen Operator are used by Union operator, Lukasiewicz Operator is used by Intersection operator. Union aggregation maximizes all the fuzzy sets generated in implication and Intersection aggregation maximizes the same parts of all the fuzzy sets. 3.6. Defuzzification Defuzzification is the final stage in the fuzzy system. Crisp value or linguistic value can be obtained through the process of defuzz-

ification. The crisp value is commonly used in various industrial and control systems where only exact numerical values are needed during actual control operations, such as raise the temperature by 2 °C. There are many defuzzification methods including center of area, maximum possibility, mean of maximum possibilities, center of mass of highest intersected region, etc. Normally, the most suitable method is used subject to the conditions of operations. Center of area (COA) and mean of maximum possibilities (MOM) are the most commonly used techniques due to its simplicity and ease of use. 4. Case study To validate the feasibility of the FDS model, a case study has been conducted in an industrial setting. In this paper, the condition monitoring of water pipelines in an electro-plating plant in China has been chosen as the reference site for validating this approach. This is a huge plant requiring huge amount of water for the daily operations. It is important that the water pipelines are in good condition otherwise the plant operations will be disrupted severely. In the existing practice, an initial examination is conducted based on three simple processes including visual inspection of cut out pipe sections and couplings, hardness test of the samples using a hardness testing machine and the minimum thickness examination of samples using an ultrasonic machine. After the results of these three checking operations are obtained, experts are called in to judge if the water pipelines need to be refurbished, replaced or a more detailed examination is needed. However, the judgment is based on human experience and personal perception of the conditions of water-pipelines and therefore is not reliable and in most cases, conflicts arise when there are different views about the decision that needs to be taken. The proposed FDS model is featured by its capability to provide a numerical result which is essential to support decision making

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µT(t)

Table 1 The corresponding data for different inputs.

V (%) H (Hardness unit) T (mm)

1st Section

2nd Section

3rd Section

Average

71 80.5 10

83 70 12

62 58 9.5

72 69.5 10.5

1

1

VG

0.5

µV(v)

0

U BA JA G

U BA JA G

VG

8

10.5 12

T (mm)

Fig. 12. The membership value of lT(10.5).

0.6 Table 2 Rules generation

0.4 0

72

80

V (%)

90

(a) Rule 1 IF

Fig. 10. The membership value of lV(72). THEN (b) Rule 2 IF

µH(h) 1 0.9

U BA JA G

VG THEN (c) Rule 3 IF

0.1 0

THEN

60

69.5

80

H

Fig. 11. The membership value of lH(69.5).

and in this case, it is the probability of failure based on existing conditions. What the FDS model requires are the three inputs with crisp values including V (visual inspection), H (hardness) and M (minimum thickness). Based on these three inputs, a crisp result can be obtained. In this case study, 3 sections were extracted from pipeline samples and the average values of the data based on examinations of these sections were used as inputs of the FDS model (as shown in Table 1). According to the situation analysis, the universe of discourse for

(d) Rule 4 IF

THEN (e) Rule 5 IF

THEN (f) Rule 6 IF

THEN

(a) v in fuzzy set V is 0–100% condition. (b) h in fuzzy set H is 0–100. (c) t in fuzzy set T is 0–15 mm thickness. As shown in the following figure, the input crisp value for V is fuzzified. In this particular case, the visual inspection result (V) is 72% and it cuts the BA predicate at 0.4 and U predicate at 0.6 (as shown in Fig. 10). The Hardness Test (H) is 69.5 and it cuts the BA predicate at 0.1 and JA predicate at 0.9 (as shown in Fig. 11). The minimum thickness examination (T) result is 10.5 and it cuts the JA/G predicates at 0.5 (as shown in Fig. 12). As shown in Table 2, 8 Rules are generated based on the rules set in Fig. 9. The following Table 3 shows how the minimum membership function values are chosen for the associated rules. The eight results are then put into the implication process which is used to determine the output fuzzy set. Mamdani Opera-

(g) Rule 7 IF

THEN (h) Rule 8 IF

THEN

Visual Examination IS Unacceptable AND Hardness IS Barely Acceptable AND Minimum wall thickness IS Just Acceptable Chance of Failure IS Very Likely

Visual Examination IS Barely Acceptable AND Hardness IS Barely Acceptable AND Minimum wall thickness IS Just Acceptable Chance of Failure IS Likely

Visual Examination IS Unacceptable AND Hardness IS Just Acceptable AND Minimum wall thickness IS Just Acceptable Chance of Failure IS Very Likely

Visual Examination IS Barely Acceptable AND Hardness IS Just Acceptable AND Minimum wall thickness IS Just Acceptable Chance of Failure IS Likely

Visual Examination IS Unacceptable AND Hardness IS Barely Acceptable AND Minimum wall thickness IS Good Chance of Failure IS Very Likely

Visual Examination IS Barely Acceptable AND Hardness IS Barely Acceptable AND Minimum wall thickness IS Good Chance of Failure IS Likely

Visual Examination IS Unacceptable AND Hardness IS Just Acceptable AND Minimum wall thickness IS Good Chance of Failure IS Very Likely

Visual Examination IS Barely Acceptable AND Hardness IS Just Acceptable AND Minimum wall thickness IS Good Chance of Failure IS Likely

tor is selected as the implication operator in this case to implicate the result of the rules based on the following the mathematical expression,

lðx; yÞ ¼ /½lA ðxÞ; lB ðyÞ ¼ lA ðxÞ ^ lB ðyÞ

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H.C.W. Lau, R.A. Dwight / Expert Systems with Applications 38 (2011) 13342–13350 Table 3 The composition results of the rule of IF-part Rule Rule Rule Rule Rule Rule Rule Rule Rule

Composition result 1 2 3 4 5 6 7 8

(0.6 ^ 0.1 ^ 0.5) = 0.1 (0.4 ^ 0.1 ^ 0.5) = 0.1 (0.6 ^ 0.9 ^ 0.5) = 0.5 (0.4 ^ 0.9 ^ 0.5) = 0.4 (0.6 ^ 0.1 ^ 0.5) = 0.1 (0.4 ^ 0.1 ^ 0.5) = 0.1 (0.6 ^ 0.9 ^ 0.5) = 0.5 (0.4 ^ 0.9 ^ 0.5) = 0.4

µ CF (cf) EU

VU

U

NL

L

VL

EL

1

0.5 0.4 0 0

50

100

CF (%)

Fig. 14. The aggregation result.

where implicator operator is denoted by /, input membership function by lA(x) output membership function by lB(y) and intersection function by ^. The implication results are showed in Fig. 13. The eight results are fused for aggregation by using aggregation operator, Union (_), to generate the final fuzzy set and the result is shown in Fig. 14. To determine the crisp value, defuzzification process is required. The method of center of area is selected for this case due to its simplicity of use and the general equation is shown below:

PN

j¼1 wj C j Aj

Y ¼ PN

j¼1 wj Aj

where w, C, A denote the weight, center of gravity and area of each individual implication result, respectively. The calculated results of each polygon are shown in Table 4. The preliminary calculated results are

a

b

c

d

Fig. 13. Implications. (a) Rule 1 and 5. (b) Rule 2 and 6. (c) Rule 3 and 7. (d) Rule 4 and 8.

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Table 4 Data for defuzzification process

References

Polygon

Area (A)

Center of gravity (C) (%)

Weight (w)

ACw

Aw

Rule Rule Rule Rule Rule Rule Rule Rule

2.375 2.375 9.375 8 2.375 2.375 9.375 8

75.0 62.5 75.0 62.5 75.0 62.5 75.0 62.5

1 1 1 1 1 1 1 1

1.781 1.484 7.031 5 1.781 1.484 7.031 5

2.38 2.38 9.38 8 2.38 2.38 9.38 8

Rule8 X

1 2 3 4 5 6 7 8

wCA ¼ 30:59375

Rule1 RX ule8

wA ¼ 44:25

Rule1 PN j¼1 wj C j Aj PN j¼1 wj Aj

¼ 69:14%

After calculation, the center of area is 69.14% which represents the Chance of Failure of the asset under examination. This numerical data is vital for determining if the water pipelines are to be refurbished, replaced or a more stringent examination is needed.

5. Conclusion The suggestion of the probability of engineering asset failure proves to be helpful in supporting decision making based on the feedback from the engineers of the plant. In general, this case example indicates that the incorporation of the fuzzy logic approach in building a decision support model for EAM is able to upgrade the decision support functionality in the electro-plating plant. However, the results obtained, so far, is by no means perfect although it demonstrates that the suggested FDS is basically viable and therefore it is justifiable to have further investigation along this line of research.

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