Ranking of asset maintenance process KPIs using Fermatean fuzzy Delphi and Fermatean fuzzy ELECTRE method

2.1 Process decomposition approach

The complexity of organizations requires a good knowledge of the process, as it involves various inputs, outputs and stakeholders. In order to measure the performance of the process, it is necessary to realize the decomposition of the process into sub-processes. One of the models for process decomposition is Hierarchy - Input - Output - Processing (HIPO) [11]. The HIPO method was chosen because it corresponds to hierarchically structured processes, follows the flow of information through the process, identifies the procedural flow from the input to the output of the process, and monitors the relationships between sub-processes and within the sub-processes themselves.

2.2 Approaches to evaluating KPIs

Evaluation of KPIs can be objective and subjective. The objective approach gives values that are measured and expressed in some measurement unit, while the subjective approach of evaluation gives values expressed in the form of linguistic expressions, based on the experience and intuition of the evaluator. In the development of the proposed model, a subjective approach for evaluating KPIs was used. Modeling values that are uncertain by nature require the application of a fuzzy set. The Fermatean fuzzy set was chosen for this model, for the reason that it has justified its application in previous research, as shown in the papers analyzed in Section 2.3.

The complexity of the organization requires the engagement of experts, who will contribute to verifying the obtained results of the model practically on the example of a complex organization. The formation of a group of experts who will evaluate KPIs belongs to the domain of group decision-making, which causes the use of models with the achievement of agreement on the obtained values of KPIs. The Delphi method was used, which, with its concept, can meet the stated requirements. The suitability of using the Delphi method in current practice has been confirmed in the papers analyzed in Section 2.5.

To obtain values for each KPI, ranking and determining priority KPIs for the process, it is most appropriate to use multi-criteria decision-making methods. The ELECTRE method extended with FFS is proposed in the model. Previous research and application of the ELECTRE method are analyzed in detail in Section 2.4.

2.3 Analysis of Fermatean fuzzy set

Higher-order fuzzy sets arose from the need to, in addition to the existing fuzzy sets, model uncertainties more qualitatively, taking into account the certainty and uncertainty of the decision-maker when evaluating or assessing the situation. The basis of FFS is intuitive fuzzy sets developed by Atanassov [7], from the need to include certainty or uncertainty in the decision made when making a decision in uncertainty modeling. The upgrade of intuitive fuzzy sets consists of Pythagorean fuzzy sets [54] as fuzzy sets of the second order, and then Fermatean fuzzy sets as fuzzy sets of the third order, all with the aim of quantifying uncertainty modeling as precisely as possible. In the literature review, FFS has recently played a major role in scientific research, where uncertainty modeling is used in various fields, such as: analysis of the advantages of the MABAC method that uses the r-rung orthopair fuzzy environment [50], research on the evolution of pattern recognition techniques [15], a statistical concept for medical decision-making applications [25], application of the Hernian mean operator on the example of electronic surveillance [40], analysis of hospital preparedness in the event of a disaster [24], medical waste disposal planning for health centers [1], pattern recognition based on similarity measures [9], extended capital budgeting technique [12], selection of antivirus mask [46], assessment of challenges for Industry 4.0 adoption for sustainable digital transformation [42], occupational risk assessment of flight school [16], warehouse location selection for the automotive industry [41], evaluation of green suppliers in a complex uncertain environment [56], multi-criteria decision making dealing with the lack of preference order of similarity to the ideal solution [5], using Fermatte type-2 phase sets in group decision making [2], analysis of trading performance of the European Union and Serbia [27], supplier selection [35]. From the previously analyzed literature, there is the author’s conviction that uncertainty modeling using FFS in practice becomes justified.

Based on [43–45] in the world scientific literature, Senapati & Yagwer introduced FFS in 2019, where they defined the following algebra:

Definition 1. Let X be a fixed set, a FFS in X is defined as, F ={  〈 x _j , μ _F  (x _j ) , ν _F  (x _j ) 〉 | x _j  ∈ X } where μ _F  (x _j )  (0 ⩽ μ _F  (x _j ) ⩽ 1) represent the membership and ν _F  (x _j )  (0 ⩽ ν _F  (x _j ) ⩽ 1) nonmembersship degrees of x _j  ∈ X to the set F, respectively and they satisfy the following condition: 0 ⩽ (μ _F  (x _j )) ³ + (ν _F  (x _j )) ³ ⩽ 1. For all x _j  ∈ X, if π F ( x j ) = 1 - ( μ F ( x j ) ) 3 - ( ν F ( x j ) ) 3 3 , then π _F  (x _j ) is the indeterminacy degree of x _j  ∈ X to the set F.

Definition 5. Let F _i  = (μ _i , ν _i ) i = 1, 2, . . , n be a number of FFNs and ω = (ω ₁, ω ₂, . . . , ω _n )  ^T be weight vector of F _i with ∑ i = 1 n ω i = 1 , then a Fermatean fuzzy weighted power average (FFWPA) operator is a function , where

FFWPA ( F 1 , F 2 , . . . , F n ) = ( ( ∑ i = 1 n ω i μ i 3 ) 1 3 , ( ∑ i = 1 n ω i υ i 3 ) 1 3 )(1)

Definition 6. Let F _i  = (μ _i , ν _i ) i = 1, 2, . . , n be a number of FFNs and ω = (ω ₁, ω ₂, . . . , ω _n )  ^T be weight vector of F _i with ∑ i = 1 n ω i = 1 then a Fermatean fuzzy weighted power geometric (FFWGA) operator is a function , where FFWPG ( F 1 , F 2 , . . . , F n ) = ( ( 1 - ∏ i = 1 n ( 1 - μ i 3 ) ω i ) 1 3 , ( 1 - ∏ i = 1 n ( 1 - υ i 3 ) ω i ) 1 3 )(2)

Definition 7. Let F ₁ = (μ ₁, ν ₁) and F ₂ = (μ ₂, ν ₂) two FFS. Then the Euclidean distance between F ₁ and F ₂ is defined as: d e ( F 1 , F 2 ) = 1 2 ( [ μ 1 3 - μ 2 3 ] 2 + [ υ 1 3 - υ 2 3 ] 2 + [ π 1 3 - π 2 3 ] 2 )(3)

Definition 8. Calculating the standard the deviation based on the expression [42]: σ 2 = 1 m ∑ i = 1 , . . . , e ( ω j - ω j e ) 2 ,(4) where: ω j - ω j e = d e , and ω _j is represents the aggregated value of subprocess relative importance (PPi) obtained by FFWPG, while ω j e represents the relative importance (PPi) assigned by the ith expert. The variance σ is a measure of the consensus for the obtained relative importances.

There are a number of methods and approaches for alternative ranking. In this, the multi-criteria decision-making method ELECTRE (Elimination and Choice translating reality) developed by Bernard Roy in 1986 with his collaborators was used. The ELECTRE method enables the decision-maker to choose an alternative with maximum advantage and minimum conflicts in function of various criteria, that is, it provides a clear picture of the value of each alternative. The method is based on pairwise comparisons of alternatives, which means that each alternative is compared with all other alternatives. The main advantage of this method is that it does not allow compensation between criteria and any normalization procedure, because normalization distorts the original data. The ELECTRE method is often used in research, especially in complex areas. The ELECTRE method has found application in the case of consumer decision-making [53], choice of alternative filling in bed production [49], evaluation of performance and team selection [23], performance analysis of food suppliers [28], selection of the most optimal supplier of drones [30], management of a project to remove construction obstacles for schools where there are children with developmental disabilities [38], selection of means of transport for the needs of the logistics system [48], comparative selection of a bicycle path for sustainable tourism in Franciacorta [8], application of the AHP-ELECTRE decision-making method for solving health and safety problems [37], assigning a machine maintenance strategy [26], improved ELECTRE II method with linguistic m-polar fuzzy sets [4], risk assessment with hesitation [3], evaluation the contractor selection [47], etc.

According to Wu [52] and Zhou [57], the fuzzy Electra extended FFS method algorithm is presented in seven steps.

Step 1. Formation of the decision-making matrix X _ij , where the decision-maker evaluates alternatives, i.e. KPIs according to all criteria, i.e. sub-processes PP _i . Grades are expressed using Fermatean phase numbers X _ij  = (μ _ij , v _ij ) X i j = K P I 1 K P I 2 ⋮ K P I i [ P P 1 P P 2 ⋯ P P j ( μ 11 , v 11 ) ( μ 12 , v 12 ) ⋯ ( μ 1 j , v 1 j ) ( μ 21 , v 21 ) ( μ 22 , v 22 ) ⋯ ( μ 2 j , v 2 j ) ( μ 31 , v 31 ) ( μ 32 , v 32 ) ⋯ ( μ 3 j , v 3 j ) ( μ i 1 , v m 1 ) ( μ i 2 , v m 2 ) ⋯ ( μ i j , v i j ) ](5)

Step 2. Different methods can be used to determine the weight or importance of criteria ω _j , usually the decision maker decides which method will be used to determine the importance of each criterion, while the condition must be met that ∑ i = 1 n ω i = 1 , where 0 ⩽ ω _j  ⩽ 1. In this paper, the Defley method extended with Fematean fuzzy set was used for the first time.

Step 3. Construct the Concordance and Discordance sets:

a) strong concordance set is portrayed as C _kl , medium concordance set is portrayed as , weak concordance set is portrayed as :

C k l = { j | μ k j ≥ μ i j , v k j < v i j i π k j < π i j } C k l ′ = { j | μ k j ≥ μ i j , v k j < v i j i π k j ≥ π i j } C k l ′ ′ = { j | μ k j ≥ μ i j , v k j ≥ v i j }(6)

b) strong discordance set is portrayed as D _kl , strong discordance set is portrayed as , strong discordance set is portrayed as :

D k l = { j | μ k j < μ i j , v k j ≥ v i j , π k j ≥ π i j } D k l ′ = { j | μ k j < μ i j , v k j ≥ v i j , π k j < π i j } D k l ′ ′ = { j | μ k j < μ i j , v k j < v i j }(7)

Step 4: Identify the weights of concordance and discordance sets: W ′ = { ω c , ω c ′ , ω c ′ ′ , ω D , ω D ′ , ω D ′ ′ } .

a) the weights of strong concordance set ω _c , the weights of medium concordance set ω _c′, the weights of weak concordance set ω _{c ^′′} is computed with as follows: ω c = ∑ f , g = 1 f ≠ g m ∑ j ∈ C f g ω j × d ¯ ( A f j , A g j ) × { ∑ f , g = 1 f ≠ g m ∑ j ∈ C f g ω j × d ¯ ( A f j , A g j ) × ∑ f , g = 1 f ≠ g m ∑ j ∈ C ' f g ω j × d ¯ ( A f j , A g j ) + ∑ f , g = 1 f ≠ g m ∑ j ∈ C ' ' f g ω j × d ¯ ( A f j , A g j ) } − 1 ω c ' = ∑ f , g = 1 f ≠ g m ∑ j ∈ C ' f g ω j × d ¯ ( A f j , A g j ) × { ∑ f , g = 1 f ≠ g m ∑ j ∈ C f g ω j × d ¯ ( A f j , A g j ) × ∑ f , g = 1 f ≠ g m ∑ j ∈ C ' f g ω j × d ¯ ( A f j , A g j ) + ∑ f , g = 1 f ≠ g m ∑ j ∈ C ' ' f g ω j × d ¯ ( A f j , A g j ) } − 1 ω c ' ' = ∑ f , g = 1 f ≠ g m ∑ j ∈ C ' ' f g ω j × d ¯ ( A f j , A g j ) × { ∑ f , g = 1 f ≠ g m ∑ j ∈ C f g ω j × d ¯ ( A f j , A g j ) × ∑ f , g = 1 f ≠ g m ∑ j ∈ C ' f g ω j × d ¯ ( A f j , A g j ) + ∑ f , g = 1 f ≠ g m ∑ j ∈ C ' ' f g ω j × d ¯ ( A f j , A g j ) } − 1(8)

b) the weights of strong discordance set ω _D , the weights of medium discordance set ω _D′, the weights of weak discordance set ω _{D ^′′} is computed with as follows: ω D = ∑ f , g = 1 f ≠ g m ∑ j ∈ D f g ω j × d ¯ ( A f j , A g j ) × { ∑ f , g = 1 f ≠ g m ∑ j ∈ D f g ω j × d ¯ ( A f j , A g j ) × ∑ f , g = 1 f ≠ g m ∑ j ∈ D ' f g ω j × d ¯ ( A f j , A g j ) + ∑ f , g = 1 f ≠ g m ∑ j ∈ D ' ' f g ω j × d ¯ ( A f j , A g j ) } − 1 ω D ' = ∑ f , g = 1 f ≠ g m ∑ j ∈ D ' f g ω j × d ¯ ( A f j , A g j ) × { ∑ f , g = 1 f ≠ g m ∑ j ∈ D f g ω j × d ¯ ( A f j , A g j ) × ∑ f , g = 1 f ≠ g m ∑ j ∈ D ' f g ω j × d ¯ ( A f j , A g j ) + ∑ f , g = 1 f ≠ g m ∑ j ∈ D ' ' f g ω j × d ¯ ( A f j , A g j ) } − 1 ω D ' ' = ∑ f , g = 1 f ≠ g m ∑ j ∈ D ' ' f g ω j × d ¯ ( A f j , A g j ) × { ∑ f , g = 1 f ≠ g m ∑ j ∈ D f g ω j × d ¯ ( A f j , A g j ) × ∑ f , g = 1 f ≠ g m ∑ j ∈ D ' f g ω j × d ¯ ( A f j , A g j ) + ∑ f , g = 1 f ≠ g m ∑ j ∈ D ' ' f g ω j × d ¯ ( A f j , A g j ) } − 1(9)

Step 5. Formation of the consent matrix C pq is the result of approvals and their weights; it is realized in accordance with the following: G = [ - g 12 . . . g 1 m g 21 - . . . g 2 m g ( m - 1 ) 1 . . . . - g ( m - 1 ) m g m 1 g m 2 g m ( m - 1 ) - ]

where g k i = ω c × ∑ j ∈ C k i ω j + ω c ′ × ∑ j ∈ C ' k i ω j + ω c ' ′ × ∑ j ∈ C ' ' k i ω j(10)

so that the consent matrix can be written: K = [ - k 12 . . . k 1 m k 21 - . . . k 2 m k ( m - 1 ) 1 . . . . - k ( m - 1 ) m k m 1 k m 2 k m ( m - 1 ) - ](11)

where k _ki  = g ^* - g _ki , g ^* is the maximum consensus index.

Step 6. Formation of the matrix of discrepancies D pq is the result of discrepancies and their weights, realized in accordance with the following: H = [ - h 12 . . . h 1 m h 21 - . . . h 2 m h ( m - 1 ) 1 . . . . - h ( m - 1 ) m h m 1 h m 2 h m ( m - 1 ) - ](12)

where h ki = max j ∈ D ki ω D * × d ( X kj , X ij ) max j ∈ J × d ( X kj , X ij ) , ω D * = { ω D , ω D ′ , ω D ′ ′ }(13)

the discrepancy matrix is: L = [ - l 12 . . . l 1 m l 21 - . . . l 2 m l ( m - 1 ) 1 . . . . - l ( m - 1 ) m l m 1 l m 2 l m ( m - 1 ) - ] ,(14)

where l _ki  = h ^* - h _ki ,

h ^* is the maximum discrepancy index.

Step 7. Forming a dominance matrix Rwhich represents the aggregation of the safety matrix and the uncertainty matrix, and it has the following form: R = [ - r 12 . . . r 1 m r 21 - . . . r 2 m r ( m - 1 ) 1 . . . . - r ( m - 1 ) m r m 1 r m 2 r m ( m - 1 ) - ] ,(15)

where r ki = l ki k ki + l ki

Ranking of alternatives is realized in accordance with the following: T ki = 1 m - 1 ∑ i = 1 , l ≠ k m r ki , k = 1 , 2 , . . . , m(16) where the best alternative is the one with the maximum value A = max{ T _k  }.

The Delphi method is a group decision-making process based on the anonymous and iterative collection of opinions by experts in a specific field. The advantage of the method is the anonymity of the experts, which frees them from the pressure to agree or oppose the opinions of the majority of the group. This creates an environment where experts freely express their opinions. The next advantage of the method is that it enables multiple iterations of gathering the opinions of experts, because after each iteration, the experts are given feedback on the collective opinion of the group. This phase allows the experts to revise and adjust their answers. This act enables the group’s consensus to be reached. Different methods for reaching consensus can be found in the literature, such as the adaptive approach [31]. In this paper, the consensus check is based on testing the hypothesis about the difference of means combined with the algebra of Fermatean fuzzy sets. The Delphi method usually uses professionals from a certain field, experts, which results in high-quality and reliable results of the decision made. Another advantage of the Delphi method is the efficient collection of opinions using a survey, which enables quick data collection and analysis. Bearing in mind that it is necessary to make a decision based on the opinion of experts, where the assessment is formed on the basis of experience in the form of linguistic statements, which causes uncertainty, the existing Delphi method will be expanded with an intuitive fuzzy set, more precisely Fermatean fuzzy set (FFSDelfi), which until now, based on the reviewed works, it was not used. The traditional Delphi method with modifications was used to select an intermodal route for cargo transportation from Korea to Central Asia [51], assessment of key competencies for lifelong learning [6], development of an optimal water supply plan using integrated fuzzy Delphi and fuzzy ELECTRE III methods—Basin case study Gamasiab [33], the selection of performance factors in many service industries [22], the fuzzy evaluation method is particularly suitable for expressing opinions and preferences of experts characterized by uncertainty, ambiguity, unobservability and scarcity [55], evaluation of forest roads [10].

5.1 Discussion and results

Based on the obtained results, it can be considered that KPIs with a higher priority, that is, KPIs that have the greatest impact on the technological maintenance process, are KPIs whose values are in the interval from 0.3 to 0.7 and that they can be called special KPIs. While for KPIs whose values are in the interval from 0 to 0.2999, they will be KPIs that do not directly affect the maintenance process, but are not excluded from consideration because their importance is a necessary prerequisite for the initiation, implementation and completion of the technological maintenance process, and as such they can be called general KPIs. The model generated the following special

KPIs: availability of workshop capacities (KPI₁₂), availability of general and special tools and equipment (KPI₅), availability of experts (KPI₁₀), degree of conformity of procedures (KPI₉), level of required knowledge and skills (KPI₆), material provision of spare parts and consumables (KPI₃), user satisfaction (KPI₂) and availability of technical literature (KPI₁₁). The general KPIs that represent a prerequisite for the maintenance process are: the degree of application of occupational safety measures (KPI₄), the possibility of monitoring modern trends (KPI₈), the availability of protective equipment in the overhaul process (KPI₁) and the degree of coordination with other organizational units in the overhaul institute (KPI₇). General KPIs provide, through their processes, other units of the overhaul system that are not in the direct technological process. Based on the results, it can be concluded that for professionals, general KPIs do not represent a focus for the maintenance process, because protective equipment is provided up to compliance standards, all safety measures at work are prescribed to prevent injury to persons at work, coordination with other organizational units in terms of finances, procurement of required funds, as well as infrastructural support of workshop capacities functions without interruption. The possibility of monitoring modern repair trends as an indicator depends on the availability of literature for new means, the possibility of participating in seminars, visiting other repair systems that deal with the maintenance of more modern means, is specific and more difficult to achieve due to the reason of providing access to such activities and repair systems abroad. Ensuring the aforementioned conditions for monitoring modern overhaul trends is the responsibility of organizational units outside the technological process, so KPI₈ is also classified as general KPIs in the real system. Special KPIs require constant measurement, monitoring and improvement. The availability of workshop capacities (KPI₁₂) has the greatest impact on the maintenance process, and as such can represent a bottleneck in the maintenance process. It depends on infrastructural support, compliance with the line of the technological maintenance process, and most of all it depends on the allocation of capacity according to the requirements of the users of the funds. The problem arises when maintenance requirements exceed workshop capacities. Availability of general and special tools and equipment (KPI₅) indicator that presents the availability of tools and equipment in the maintenance process, its correctness and attestation. Disadvantages are manifested in the appearance of tool malfunctions, retention of tools and equipment in the certification process, unavailability of certain tools on the market. Availability of experts (KPI₁₀) indicator that represents the occupancy of experts. The lack of professionals is present in most companies, due to various trends (planned and natural outflows, migrations, leaving for better positions, retraining for other jobs, etc.), the impossibility of replacement in case of absence, as well as the possibility of young people entering the technological process. The degree of conformity of the procedures (KPI₉) presents the conformity of the procedures of the technological maintenance process with standards, rules, regulations, normatively regulated. The problems affecting this indicator are the up-to-dateness of procedures, the availability of rules and regulations for new assets coming for overhaul. Material provision with spare parts and consumables (KPI₃) represents the basis of the maintenance process. The problems affecting the indicator are mostly related to the area of procurement of spare parts and consumables, i.e. the reliability of purchased spare parts. Global problems in the form of the Covid pandemic have not bypassed the industrial sector for the production of spare parts. Deliveries of spare parts are delayed, smaller quantities are delivered compared to the required quantities. The reliability of the delivered spare parts is questionable because the vehicle often fails after overhaul. The level of required knowledge and skills (KPI₆) requires knowledge of tools, failure defects, assembly disassembly and assembly. What can reduce the value of this indicator is the ability to overhaul new assets. The availability of technical literature (KPI₁₁) is one of the tools of experts. The variety of assets puts experts in a disadvantageous situation when they do not have technical literature describing the maintenance of the asset, diagrams and images of assemblies and subassemblies, factory numbers of spare parts, etc. User satisfaction (KPI₂) presents an assessment of the work of the unit that deals with maintenance, indicates failures in the work on the asset itself, which can be useful to management.

In order to improve the indicators, the obtained results were analyzed, potential problems affecting individual KPIs were considered, and a proposal for measures to improve them was made. Workshop capacities need to be expanded and equipped with tools and equipment for maintenance in accordance with the introduction of new resources, worn-out tools should be replaced with new and more modern ones. For the layout of tools and equipment, use the Lean concept, such as “5 s principles” [19], send tools and equipment for certification at intervals so that the workshop does not remain without them for a long period of time. KPI₁₀ should be increased through the training of experts in the maintenance of related assets so that replaceability is possible, then the earlier entry of young experts into the maintenance process before older people end their working life. This results in the transfer of experience to young people as well as continuity in the maintenance process. By introducing new means of requiring and training maintenance personnel. Therefore, before new assets are put into use, it is necessary to carry out: training of maintenance professionals, which will increase the value of KPI₆, procurement of technical literature that follows the asset maintenance process that affects KPI₁₁, procurement of those spare parts whose operational reliability is rated as lower value, thereby creating a base that will influence KPI₃. Fulfillment of the aforementioned proposals will also improve user satisfaction with maintenance services (KPI₂).

A deeper analysis of KPIs can lead to the conclusion that even if KPIs are considered individually, there is still an interdependence between them, because the consequence of one KPI chain transfers the problem to other KPIs.

5.2 Managerial implications

Measuring the performance of business organizations is a complex and time-consuming process. Existing performance measurement models require modifications, which may lead to a violation of the basic concept of the performance measurement model. In order to depict the state of the process as realistically as possible, a hybrid model is proposed that can be widely applied in various areas where the subjective opinion of the decision maker plays a key role. Evaluating KPIs that are qualitative in size requires a subjective assessment, with the primary focus on harmonizing or reaching consensus. Reducing bias and influence in decision-making requires finding a mode, which in this model is presented as a solution with several teams of experts, where one of the teams focuses on the KPIs and PPi that will be measured, and the other team performs their evaluation. Uncertainty modeling is represented in the model by an intuitive fuzzy approach of the q^th order (where q = 3), in order to reduce fluctuations in subjective assessment.

In order to check whether and how uncertainty modeling by changing the q^th order affects the results of the proposed model, with the same input data, the model is tested for the values q = (4,5,6,7,8,9,10). Algebra for applying q-rung orthopair fuzzy set is given in the work of Pinar and Boran [35], and the obtained results are given in Table 5 and graphically shown in Fig. 2.

OPEN IN VIEWER

Fig. 2

Displaying the rank of KPIs for the values of q = (3,4,5,6,7,8,9,10).

Comparing the results obtained using the Fermatean fuzzy set (q = 3) with the results when q takes values from 4 to 10, it can be seen that the model gives the same or similar rank for KPI₄, KPI₅, KPI₇, KPI₈, KPI₁₀ and KPI₁₂, that is, less rank changes for KPI₁, KPI₂, KPI₃, KPI₆, KPI₉ and KPI₁₁.