FUZZY PERFORMANCE MEASUREMENT SYSTEM FOR THE MAINTENANCE FUNCTION

Maintenance aims to sustain the manufacturing process by ensuring that the production tools are well managed. However, modern production means are more automated and therefore more complex, so they require a performing maintenance for which planning and implementation are increasingly difficult. This paper, therefore, emphasizes the elaboration of a maintenance performance measurement system. Therefore, for an appropriate and reliable performance evaluation, we adopt a multi-level multi-criteria approach where every criterion influencing the maintenance performance either indirectly or directly are specified. Thereafter, indicators for the specified criteria would be measured using fuzzy logic to overcome limitations due to lack of available resources and data. For a global maintenance measurement, the elementary measurements will be aggregated into a total measurement through a multiple-criteria decision-making method such as MACBETH.


Introduction
To remain in a competitive position, the manufacturing systems must be more productive, effective and economical [1].However, the complexity of these manufacturing systems, the large diversity of manufactured products and the uncertainty of the production environment are behind critical dysfunctions.These malfunctions and failures negatively affect production capacity, production quality and cost, as well as potential health and safety risks and accidents to personnel, plants and the environment [2].This is the reason behind maintenance, which is no more restricted to repair facilities, but rather involves preventing their dysfunction [3][4].Consequently, maintenance is now a vital element in the global business success for any manufacturing company [5].Therefore, a performing maintenance became a necessity and not a possibility.
Achieving the required performance begins with its measurement as without it there are practically any indications of improvement [6].A performance measurement system is a management tool that allows a company to evaluate how its activities are effective and efficient [7].This tool is structured in a multi-criteria approach, including a combination of performance indicators selected according to the company's objectives [8].The main challenge when measuring maintenance performance is defining and choosing indicators that reflect the corporate strategy and provide quantitative feedback on the maintenance strategies performance [9].Indeed, a suitable indicator would track and control performance, assist identifying performance gaps, and continuous improvement [5].Therefore, to make this system more useful, there is an obligation in defining the indicators to be measured, the way in which they should be evaluated and who is responsible for their implementation [10].
However, in an environment characterized by cost, time, resources and skill limitations, measuring the elementary performance is a big issue.For this reason, we propose that the fuzzy logic is used to quantify this elementary performance.Subsequently, by using the multi-criteria decision-making method MACBETH, the elementary performance measurements are aggregated into an overall measure that provides the total maintenance performance.Consequently, the suggested approach allows measuring the global performance and the local performances of the maintenance, as well as defining the necessary actions for a continuous maintenance performance enhancement.

The identification of the elementary maintenance performance measurement indicators
In order, to measure maintenance performance correctly, the strategic aspects are specified and aligned with the company's strategy.These strategic aspects are the financial aspect, the technical aspect, the internal process aspect, the support aspect, the human resources aspect and the health, safety and environment (HSE) aspect.Afterwards, they will be disaggregated until reaching elementary indivisible sub-criteria.This paper will focus

The elementary Performance Measurement using the Fuzzy Logic:
The fuzzy logic has been inserted by Zadeh (1965) [11], in order to overcome the complexity and fuzziness of the real systems, by representing them through linguistic values that will follow a mathematical treatment transformed into numerical ones [12].

Linguistic terms and triangular fuzzy number identification
First, to measure elementary performance, it is necessary to construct the appropriate rating scale.This performance is from 0 to 1.The 0 is a complete dissatisfaction of the target, and the 1 is a complete satisfaction of the target.Then for each criterion Ci represented in Figure 1, decision-makers will linguistically express their satisfaction degrees regarding to the objective Oi using a specific term.The result is reported in Table 1.After that, every linguistic term is assigned to the appropriate fuzzy number.In this study, the triangular fuzzy numbers are chosen for their simplicity [13-14-15].
p K ̃ is the triangular fuzzy number of a specific criterion Ci. p K ̃= (L K , M K , R K ) with K = VP, P, M, G, VG and E where L and R are two points on the left and right of the triangular fuzzy number representing the interval [LK, RK].Mk is the most possible value.The result is in the table 2.Then, each linguistic parameter is assigned with the corresponding triangular fuzzy number.The result is shown in Tables 3 and 4.

Aggregation of the fuzzy numbers
In order to get a consensus among the various evaluators, the satisfaction degrees .,n expressed by each evaluator on the evaluation criterion Ci should be aggregated into a unique TFN P ̃i = (L i , M i , R i ) using equation (1) [16]: By using equation ( 1), the various fuzzy numbers expressed by the decision-makers will be aggregated into a single fuzzy number reflecting the satisfaction degree associated to each elementary sub criterion Cij.The result is shown in Tables 5 and 6.

Defuzzification
In order to convert the TFN   ̅ related to each evaluation criterion Ci into crisp value.The   ̅ will be deffuzzify by the CFCS method (Converting the Fuzzy data into Crisp Scores) proposed by Opricovic [17].
In the presented study, the fuzzy logic is utilized for quantifying the elementary maintenance performance related to the criterion Ci.So   ̌= (  ,   ,   ) is the membership function of the elementary performance for the criterion Ci.
Then: The normalization of the TFN: The Left and Right normalized value   and   : The total normalized crisp value: Finally, the crisp values Pi, which refer to the elementary performance of the criterion Ci: By using this defuzzification method, the aggregated fuzzy numbers will be converted into a performance expression.Therefore, we will get the elementary performance Pi for each Ci criterion.The result is displayed in figure 2.
However, the elementary performance measurements are not suitable to provide a holistic maintenance performance assessment.Because of this, they must be aggregated to form a unified measure.

The overall maintenance performance measurement
The evaluation criteria and aspects have a different effect on the performance.As a result, to identify the relative importance of each evaluation criterion and aspect, a multi-criteria decision-making MCDA method such as MACBETH is needed.Indeed, this method allows quantifying the relative weight of each criterion by using the comparative attractiveness of each criterion as expressed linguistically by the decision-makers [18].
To do this, the decision-makers will express the attractiveness relating to the evaluation criteria of the same hierarchical level.By using the MACBETH method and the M-MACBETH software, each criterion and subcriterion will be weighted.The weights Wi of F1, F2, F3, and F4 are shown in Figure 3.After that, the overall maintenance performance is calculated through the aggregation of the elementary performance using the weighted arithmetic mean as in equation (3).The result is displayed in figure 4.

Figure 4 Aggregation of the elementary performances
As a result, the overall maintenance financial performance is 0.62.In other words, the company has achieved 62% of its maintenance financial target.In addition, the weighting of the criteria by MACBETH allows to precisely identify the most critical improvement areas, in this study the performance of the direct cost and that of the indirect cost are the most critical.Therefore, these two criteria must be prioritized for further improvement.

Conclusion
The performance measurement system of the maintenance function introduced in this paper is elaborated through three sections.Since the first part defines and prioritizes the performance measurement indicators that are the most appropriate.The second section, by using the fuzzy logic the elementary performances are quantified trough a qualitative evaluation.Subsequently, in the third section, these elementary measures are aggregated to an overall maintenance performance measure.All of this will give the company a synthesized and further detailed visibility on both local and global performance for an optimal maintenance improvement.

Figure 2
Figure 2 Financial elementary performances

Figure 3
Figure 3 Financial criteria weighing using M-MACBETH.

Table 2
Triangular fuzzy number TFN

Table 5
Aggregation of TFNs

Table 6
Aggregation of TFNs