Classical,fuzzy,hesitant fuzzy and intuitionistic fuzzy analytic hierarchy processes applied to industrial maintenance management

3.1 Analytic hierarchy process

In AHP, priorities for alternatives and criteria are obtained with the right eigenvector w of a comparisons matrix A , as in Equation 1, where λ_max is the maximum eigenvalue. Aw = λ max w(1)

For fully consistent matrices, a _ik  = a _ij a _jk and λ_max = n [20]. Otherwise, λ_max > n. The closer λ_max is to n, the more A can be considered consistent [21]. The Consistency Index CI (Equation 2) is a better measure for the consistency, since CI compares the deviation between λ_max and n with the degree of freedom n - 1. CI = λ max - n n - 1(2)

The Consistency Ratio RC = CI/  - RI, where RI is the Random Index [3], is a more complete parameter for the consistency of a comparison matrix. CR’s upper limit is 0.1 [10].

When more than one expert is available, they can provide individual comparisons matrices. If they are from the same company or department, that is, if they share common interest, their comparisons must be aggregated before w is computed. Saaty and Peniwati [22] suggested the aggregation of individual judgments by their geometrical mean.

A fuzzy set X ˆ is characterized by a membership function μ (x) which associates each element x to a real number in the interval [0, 1] [13]. Triangular fuzzy sets (TFS) are common membership functions used in engineering, because they are easy to manage [23, 24]. A TFS is often represented by a vector [l, m, u], being m the modal value, that is μ (m) =1, and l and u being the lower and upper limits, or μ (l) = μ (u) =0, as in Equation 3. μ ( x ) = { 0 x < l x - l m - l u - x u - m 0 l ≤ x < m m ≤ x < u x ≥ u(3)

In FAHP, all comparisons and all priorities are TFS: a ˆ ij = [ l ij , m ij , u ij ] and w ˆ i = [ l i , m i , u i ] . Then, for every a _ij experts now need to provide three values l _ij , m _ij and u _ij . Priorities can be estimated as in Equation 4, proposed by [25].

w ˆ i = [ ∑ j = 1 n l ij ∑ i = 1 n ∑ j = 1 n u ij , ∑ j = 1 n m ij ∑ i = 1 n ∑ j = 1 n m ij , ∑ j = 1 n u ij ∑ i = 1 n ∑ j = 1 n l ij ](4)

There are several formulas to defuzzify a TFS. Equation 5 is the simplest one adopted in FAHP. w i = l i + m i + u i 3(5)

Intuitionistic fuzzy sets incorporate a degree of hesitation which is defined as 1 minus the sum of membership μ (x) and non-membership v (x) values [8]. An intuitionistic triangular fuzzy set (ITFS) with μ (x) as in Equation 3 and v (x) as in Equation 6 is denoted as [l, m, u ; l′, m, u′]. v ( x ) = { 1 x < l ′ m - x m - l ′ x - m u ′ - m ′ 1 l ′ ≤ x < m m ≤ x < u ′ x ≥ u ′(6)

A TFS is a particular case of an ITFS, with l = l′ and u = u′. In IFAHP, all comparisons and all priorities are ITFS: a ˆ ij = [ l ij , m ij , u ij ; l ij ′ , m ij , u ij ′ ] and w ˆ i = [ l i , m i , u i ; l i ′ , m i , u i ′ ] . Priorities can be estimated as in Equation 7: w ˆ i = [ ∏ j = 1 n l ij , ∏ j = 1 n l ij ′ , ∏ j = 1 n m ij , ∏ j = 1 n m ij , ∏ j = 1 n u ; ∏ j = 1 n u ij ′ ](7)

Defuzzification of an ITFS is done by applying Equation 8, based on [26], where α is the highest value of μ (x). Usually, α = 1. w i = ( u i ′ - l i ) + ( u i ′ - l i ) 3 + l i ′ + α [ ( u i - l i ) + ( m i - l i ) 3 + l i ] 2(8)

Table 2 presents a triangular fuzzy intuitionistic scale. This scale was proposed by [26], to facilitate IFAHP application.

Hesitant fuzzy sets (HFS) are extensions of fuzzy multisets (FMS). FMS are based in the concept of bags, that is, a set with repeated elements [27]. A FMS has more than one membership value for x. For TFS [l, m, u], a triangular FMS could be [(0, 0.2)/  - l, (0.75, 0.8, 1)/  - m, (0, 0.1)/  - u].

HFS are similar to FMS, since a set of values of memberships are possible for a single element x. A representation of an HFS is {〈 x, h (x) |x ∈ X 〉 }, where h (x) denotes a set of membership values.

With more than two experts involved, an envelopment approach [8] improves data collection. It is particularly useful when experts hesitate to express their assessments [18]. Another advantage is that experts need to provide only one value for every a _ij .

Let a ij k be the pairwise comparison, between j and j, provided by expert k = 1, 2, 3 . . . K. A hesitant triangular fuzzy set a ˆ ij is obtained with Equations 9 to 11. l ij = min k a ij k(9) u ij = max k a ij k(10) m ij = ∑ k = 1 K a ij k - l ij - u ij K - 2(11)

4.1 Object of study

The typical industrial plant analyzed here is located in the southeastern Brazilian state of Sao Paulo. This plant belongs to a global manufacturer of components, primarily, for the automobile industry, civil construction and railroad equipment. The plant manufactures structural components for excavators, heavy machines, tractors, and trains. Its main customers and suppliers are also industrial plants belonging multinational groups, located in southeastern Brazil.

At the end of 2018, three managers were involved with maintenance management: A 42-year old electrical engineer with five years of experience in the plant, plus 15 years in another company; a 33-year old technologist in industrial maintenance with four years of experience in the plant; and a 30-year old mechanical engineer with three years of experience in the company.

Despite their different ages, experiences and job titles, their opinions were equally considered, since they all were involved with maintenance management.

Among other responsibilities, managers were in charge on the selection of maintenance strategies for plant’s equipment. One of the four following alternatives could be adopted, based only on the managers’ common sense:

Corrective Maintenance (A1) is when the intervention is at the moment, or after, a failure occurs. The failure makes the equipment unavailable, or performs worst. Corrective maintenance can be planned, or not. Often, when unplanned, the intervention is immediate and without service preparation. When planned, the intervention team can be prepared for the intervention.

Based on the literature on maintenance management and in company’s key performance indicators (KPI), five criteria were identified:

Safety (C1): The required safety levels have increased lately. For maintenance strategy, it represents conditions to avoid undesirable results, such as accidents, failure, mistakes, and so on. It also refers to the controllability of reducing known threats to an acceptable level, both in terms of personal safety, plant safety, and environmental safety.

4.2 Analytic hierarchy process application

The three managers were considered experts of equal importance, but they were individually interviewed for AHP application. Their individual pairwise comparisons matrices were aggregated by geometrical mean (Section 3.1). Table 3 presents an aggregated comparisons matrix for the criteria.

For the matrix in Table 2, λ_max = 5.55, CI ≈ 0.14 and CR ≈ 0.12. Comparisons matrices with CR > 0.1 represent an alert that comparisons may not be logically connected. After review in a group meeting, the experts decided to keep the priority vector. Then, Value Added (C4) and Quality (C3) are the highest priority criteria. This result is reasonable considering the high competition in the company’s market.

The experts provided individual pairwise comparisons matrices among alternatives regarding to each criterion. Table 4 presents a decision matrix, with overall priorities and the CR for every aggregated comparison matrix. C4 had CR > 0.1. Overall CR was also greater than the upper limit of 0.1. So, the comparisons needed to be reviewed. However, after the decision matrix was presented to the experts, they agreed with the overall priorities, indicating A1 as the best alternative.

Then, that was a situation: data collected from experts did not pass the consistency check. However, results from this inconsistency were welcomed by experts. They recognized some comparisons were hard to make and they were not sure about them. So, a fuzzy approach was applied.

The experts were invited to fuzzify their comparisons. Then, every previous comparison was assumed as a modal value for a TFS; lower and upper values were added with their review. Table 5 presents the aggregated comparisons matrix among the criteria.

Criteria C3 and C4 still have the highest priorities. Now, their priorities are closer.

Table 6 presents a new decision matrix, with defuzzified local priorities.

Corrective maintenance (A1) still has the highest priority, even with FAHP. The experts reported felt more confident of their data with FAHP than AHP.

The comparisons originally provided by the experts were based on AHP Fundamental Scale [8]. Table 7 presents an aggregated matrix of IFS obtained by converting original comparisons with Table 2.

For the matrix in Table 7, λ_max = 5.98, CI ≈ 0.25 and CR ≈ 0.22. Then, with, comparisons needed to be reviewed. However, ordinal priorities are the same as the FAHP application [1–5], and almost the same as the AHP application [1–5]. Cardinally, with IFAHP, priorities are more dispersed: C4 priority increases from 41% to 51%; C5 priority decreases from 7% to 4%.

Table 8 presents a new decision matrix with IFAHP. As with AHP and FAHP, A1 had the highest priority.

With IFAHP, the overall priority of A1 is over 50%. Therefore as for criteria, priorities of the alternatives are more disperse with IFAHP than AHP and FAHP.

Table 9 presents an aggregated fuzzy comparisons matrix and priorities of criteria, with HFAHP approach proposed by [18].

For the matrix in Table 9, λ_max = 7.56, CI ≈ 0.64and CR ≈ 0.57. Then, with CR > 0.1, comparisons needed to be reviewed. However, ordinal priorities are the same in FAHP and IFAHP applications [1–5]. Cardinally, with HFAHP, priorities are less dispersed than IFAP: the priority of C4 decreases from 51% to 38%; the priority of C5 increases from 4% to 7%.

Table 10 presents a new decision matrix with IFAHP. As with AHP, FAHP and IAHP, A1 had the highest priority.

With IFAHP, the overall priority of A1 is over 50%. Therefore as for criteria, priorities of the alternatives are more disperse with IFAHP than AHP and FAHP.