Introduction
Problem statement
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RQ1. Which criteria limit the entire model, i.e. even a small decrease in their value will significantly affect the entire result? Similarly, it is important to determine which criteria should be invested in in the first place, so that the index value will increase to the greatest extent.
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RQ2. How to fine-tune the model for the needs of a given enterprise, based on detailed data obtained from group of experts?
The framework of maintenance sustainability performance assessment
Maintenance sustainability issues
Composite Maintenance Sustainability Index
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Step 1. Determination the importance of decision perspective/criteria with linguistic variablesThe first step in developing a maintenance sustainability index focuses on weighting individual elements (perspectives and criteria). The determination of the importance of the evaluated perspectives and criteria is carried out by a team of experts. Since the step is carried out by a team of experts and because of their subjectivity and cognitive differences, linguistic variables are used and then aggregated by fuzzy arithmetic. The Fuzzy Number Ordered Weighted Average (FN-OWA) operator was used in the model for averaging expert evaluations, which is an OWA operator, designed especially for fuzzy numbers and linguistic data (Chen and Chen 2003). The main reason why FN-OWA was chosen is that it has the ability to aggregate not only quantitative data, but also can handle linguistic as well as crisp data. Moreover, it is an idempotent operator, which means that an operator retains the same linguistic state as if all input criteria had equal values (Sadiq and Tesfamariam 2008). Aggregated fuzzy weights are then de-fuzzified in order to be applied in constructing fuzzy measure.
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Step 2. Construction of fuzzy measureOne important issue in sustainable assessment is the need to express not only the importance of individual features but also interactions between them. There are three kind of interactions: synergy, inhibitory and non-interaction. The classic probability theory only uses the third one, and thus is unrealistic in this case. Fuzzy measure can be applied to all three situations. It represents the importance of a criterion in the set of criteria and can be considered as generalization of probability measure. In order to lower the number of coefficients (which increases exponentially with number of perspectives/criteria) and satisfy the monotonicity and continuity, CMSI model uses λ-measure (see Jasiulewicz-Kaczmarek and Żywica (2018)). There are normally three kinds of interactions between two criteria A and B (µ denotes importance/weight): synergetic interaction, which can be represented by µ (A ∪ B) > µ(A) + µ(B); inhibitory interaction, which can be represented by µ(A ∪ B) < µ(A) + µ(B); non-interaction, which can be represented by µ(A ∪ B) = µ(A) + µ(B).One of the problems that can occur in the case of expert assessments is the situation in which individual criteria will be rated so high (close to 1) that pairs, triples etc. of criteria, due to the monotonicity of the fuzzy measure, will have very similar values (effectively equal to 1). While in the case of aggregation using a weighted average such a situation is not a problem, in the case of a fuzzy measure and the Choquet integral it can lead to undesirable results (shallowing/equalization of criteria weights and total omission of interactions among criteria). To significantly reduce the impact of these problems on the aggregation result, the q-measure proposed by Mohamed and Xiao (2003) was applied in CMSI model. q-measure is an extension of λ-measure that simplifies its computations and allows for choosing the λ parameter. The applied algorithm finds such a re-scaling of the data that models best the interactions between the criteria.
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Step 3. Calculate the aggregate value (CMSI)The purpose of aggregation functions is to combine multiple numerical inputs, into a single numerical value, which in some sense represents all the inputs. Many aggregation functions have a limitation mainly from their natural assumption that input criteria are independent of each other (Beliakov and Wu 2019). This is not always true in reality, where the independence of criteria cannot be assumed. Some interaction among different criteria does exist, including the independence, the complement, and the correlation (Grabisch and Roubens 2000). To take all of the interaction among the attributes into account, it has been proposed to use the fuzzy measure in the calculation of the overall aggregation result (Grabisch and Roubens 2000). The fuzzy integral is used in the sustainable maintenance assessment to combine assessments since it does not require criteria to be independent. Thanks to stability of Choquet integral under positive linear transformations, the exact numerical scale in relation to which the calculations are made is not relevant. This way, collecting data from experts is way simplified, and it allows assessment with use of linguistic scale.To apply the model, we need to present it to the decision-makers in order to collect data. The assessment team from the factory assess the issues describing individual criteria in a numerical scale (maturity matrix). The assessments are then aggregated using the model. CMSI measures the organization’s maintenance sustainability in range [0; 1]. The value of CMSI closer to ‘0’ indicates that the maintenance is un-sustainable; whereas a value closer to ‘1’ means that, the maintenance structure is sustainable and contributes to the sustainability of the manufacturing system. The calculated CMSI value can be then used to determinate the relative importance between perspectives and criteria. This way the CMSI can stand as help for decision-makers by directing their attention to the areas that need improving.
Extension of CMSI model for decision support system
Goals
Design principles of an extended model
Model fine-tuning
Limiting by the least grown criteria
Proposed criteria evaluation method
Evaluation of the proposed model
Structure and development of decision support system
Numerical experiments and discussion
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The derivative calculated based on considered criterion—the higher the derivative, the higher the influence of this criterion on the CMSI.
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The percent—the difference (in percent), between the CMSI calculated with given criterion with current value and value 0.
Financial (FP) | Maintenance stakeholders (MS) | Maintenance process (MP) | Innovation and development (ID) | |||||||||||
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Criterion | F1 | F2 | S1 | S2 | S3 | S4 | P1 | P2 | P3 | P4 | P5 | Id1 | Id2 | Id3 |
Perspective | ||||||||||||||
Value | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 |
Shapley | 1.087 | 0.913 | 1.226 | 1.150 | 0.951 | 0.674 | 0.874 | 1.278 | 1.054 | 1.070 | 0.751 | 1.111 | 1.096 | 0.793 |
Derivative | 1.999 | 1.770 | 1.058 | 1.024 | 0.930 | 0.780 | 0.617 | 0.759 | 0.689 | 0.694 | 0.581 | 1.097 | 1.089 | 0.915 |
Percent | 20.64 | 19.36 | 12.04 | 11.73 | 10.96 | 9.91 | 8.3 | 9.62 | 8.93 | 8.98 | 8.02 | 13.1 | 13.03 | 11.66 |
CMSI value | 0.3 |
Financial (FP) | Maintenance stakeholders (MS) | Maintenance process (MP) | Innovation and development (ID) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Criterion | F1 | F2 | S1 | S2 | S3 | S4 | P1 | P2 | P3 | P4 | P5 | Id1 | Id2 | Id3 |
Perspective | ||||||||||||||
Value | 0.7 | 0.7 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.7 | 0.7 | 0.7 |
Shapley | 1.087 | 0.913 | 1.226 | 1.150 | 0.951 | 0.674 | 0.874 | 1.278 | 1.054 | 1.070 | 0.751 | 1.111 | 1.096 | 0.793 |
Derivative | 1.533 | 1.357 | 1.291 | 1.250 | 1.135 | 0.953 | 0.758 | 0.933 | 0.847 | 0.853 | 0.714 | 0.841 | 0.835 | 0.702 |
Percent | 25.44 | 23.46 | 8.02 | 7.81 | 7.3 | 6.6 | 5.53 | 6.4 | 5.94 | 5.98 | 5.34 | 14.99 | 14.91 | 13.34 |
CMSI value | 0.451 |
Perspective | Criterion | Symbol | Initial value of the parameter |
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Financial (FP) | Maintenance stakeholders costs | (f1) | 0.4 |
Maintenance costs | (f2) | 0.4 | |
Maintenance Stakeholders (MS) | Production and quality | (s1) | 0.5 |
Safety and health | (s2) | 0.4 | |
Environment | (s3) | 0.5 | |
Communication and cooperation with stakeholders | (s4) | 0.6 | |
Maintenance Process (MP) | Analysis and improvement | (p1) | 0.2 |
Execution and measurement | (p2) | 0.2 | |
Planning and scheduling maintenance processes | (p3) | 0.5 | |
Management of external services | (p4) | 0.8 | |
Management of spare parts and consumables | (p5) | 0.4 | |
Innovation and development (ID) | Competences of maintenance workers | (id1) | 0.6 |
Maintenance infrastructure | (id2) | 0.4 | |
Satisfaction of maintenance workers | (id3) | 0.4 |
Financial (FP) | Maintenance stakeholders (MS) | Maintenance process (MP) | Innovation and development (ID) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Criterion | F1 | F2 | S1 | S2 | S3 | S4 | P1 | P2 | P3 | P4 | P5 | Id1 | Id2 | Id3 |
Initial parameters for an analysed company | ||||||||||||||
Value | 0.4 | 0.4 | 0.5 | 0.4 | 0.5 | 0.6 | 0.2 | 0.2 | 0.5 | 0.8 | 0.4 | 0.6 | 0.4 | 0.4 |
Shapley | 1.087 | 0.913 | 1.226 | 1.150 | 0.951 | 0.674 | 0.847 | 1.278 | 1.054 | 1.070 | 0.751 | 1.111 | 1.096 | 0.793 |
Derivative | 1.615 | 1.430 | 0.748 | 1.077 | 0.660 | 0.386 | 1.165 | 1.377 | 0.879 | 0.608 | 0.968 | 0.796 | 1.230 | 1.062 |
Percent | 19.76 | 18.52 | 11.95 | 9.18 | 10.67 | 10.45 | 4.04 | 4.67 | 8.93 | 11.73 | 6.82 | 14.84 | 11.60 | 10.27 |
CMSI value | 0.411 | |||||||||||||
Sustainability parameters with improved criterion p1 and p2 (after first year) | ||||||||||||||
Value | 0.4 | 0.4 | 0.5 | 0.4 | 0.5 | 0.6 | 0.3 | 0.3 | 0.5 | 0.8 | 0.4 | 0.6 | 0.4 | 0.4 |
Derivative | 1.812 | 1.604 | 0.755 | 1.088 | 0.667 | 0.39 | 1.065 | 1.258 | 0.803 | 0.555 | 0.884 | 0.804 | 1.242 | 1.073 |
Percent | 19.29 | 18.10 | 11.90 | 9.24 | 10.67 | 10.46 | 5.74 | 6.66 | 9.02 | 11.72 | 6.95 | 14.67 | 11.55 | 10.27 |
CMSI value | 0.428 | |||||||||||||
Sustainability parameters with improved criterion p1 and p2 (after second year) | ||||||||||||||
Value | 0.4 | 0.4 | 0.5 | 0.4 | 0.5 | 0.6 | 0.4 | 0.4 | 0.5 | 0.8 | 0.4 | 0.6 | 0.4 | 0.4 |
Derivative | 1.854 | 1.641 | 0.733 | 1.113 | 0.682 | 0.399 | 0.811 | 0.955 | 0.706 | 0.488 | 0.777 | 0.98 | 1.514 | 1.308 |
Percent | 18.66 | 17.5 | 12.12 | 9.55 | 10.93 | 10.73 | 6.9 | 8.09 | 8.7 | 11.31 | 6.64 | 14.51 | 11.49 | 10.25 |
CMSI value | 0.442 |
Financial (FP) | Maintenance stakeholders (MS) | Maintenance process (MP) | Innovation and development (ID) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Criterion | F1 | F2 | S1 | S2 | S3 | S4 | P1 | P2 | P3 | P4 | P5 | Id1 | Id2 | Id3 |
Initial parameters for an analysed company | ||||||||||||||
Value | 0.4 | 0.4 | 0.5 | 0.4 | 0.5 | 0.6 | 0.2 | 0.2 | 0.5 | 0.8 | 0.4 | 0.6 | 0.4 | 0.4 |
Shapley | 1.087 | 0.913 | 1.226 | 1.150 | 0.951 | 0.674 | 0.847 | 1.278 | 1.054 | 1.070 | 0.751 | 1.111 | 1.096 | 0.793 |
Derivative | 1.615 | 1.430 | 0.748 | 1.077 | 0.660 | 0.386 | 1.165 | 1.377 | 0.879 | 0.608 | 0.968 | 0.796 | 1.230 | 1.062 |
Percent | 19.76 | 18.52 | 11.95 | 9.18 | 10.67 | 10.45 | 4.04 | 4.67 | 8.93 | 11.73 | 6.82 | 14.84 | 11.60 | 10.27 |
CMSI value | 0.411 | |||||||||||||
Sustainability parameters with improved criterion s2 and id2 | ||||||||||||||
Value | 0.5 | 0.5 | 0.5 | 0.4 | 0.5 | 0.6 | 0.2 | 0.2 | 0.5 | 0.8 | 0.4 | 0.6 | 0.4 | 0.4 |
Derivative | 0.960 | 0.850 | 0.986 | 1.421 | 0.871 | 0.509 | 1.133 | 1.338 | 0.854 | 0.591 | 0.941 | 0.914 | 1.412 | 1.220 |
Percent | 21.89 | 20.42 | 12.25 | 9.63 | 11.04 | 10.83 | 3.82 | 4.42 | 8.44 | 11.09 | 6.45 | 14.44 | 11.38 | 10.12 |
CMSI value | 0.435 | |||||||||||||
Sustainability parameters with improved criteria: f1, f2, s2 and id2 | ||||||||||||||
Value | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.6 | 0.2 | 0.2 | 0.5 | 0.8 | 0.4 | 0.6 | 0.5 | 0.4 |
Derivative | 1.367 | 1.210 | 0.737 | 0.714 | 0.651 | 0.380 | 1.149 | 1.357 | 0.866 | 0.599 | 0.954 | 0.927 | 1.432 | 1.654 |
Percent | 21.70 | 20.29 | 11.77 | 11.44 | 10.58 | 10.35 | 3.67 | 4.25 | 8.12 | 10.67 | 6.20 | 14.45 | 12.94 | 9.50 |
CMSI value | 0.453 |