The performance of phase change materials directly influences the performance and cost of thermal energy storage, and it is the first important task to select the suitable phase change materials for use in a particular kind of applications. Due to the decision maker’s knowledge field and the nature of evaluated attributes, assessments are always with different formats, which were first unified into the linguistic terms in the basic linguistic term set. Two-additive fuzzy measures were used to model criteria interactions by pairs, and the special expressions of Marichal entropy and Choquet integral were derived, more convenient to use in practice. Fuzzy measures were identified based on the maximum of Marichal entropy, and, based on the Choquet integral, the linguistic hybrid weighted geometric averaging with interaction was developed for integrating the individual attributes’ ratings. The detailed decision making procedure was illustrated, with the material 33.2Cu as the optimal solution, which by comparison is reasonable and trustworthy.
Engineering design draws on tens of thousands of materials and on many hundreds of processes to shape, join, and finish them. One aspect of optimized design of a product or system is that of selecting, from this vast menu, the materials and processes that best meet the needs of the design, maximizing its performance and minimizing its cost. The selection of the most appropriate materials not only affects the capability of manufacturing systems and satisfaction of customers but also impacts environmental issues. Furthermore, material selection is the prerequisite for a chain of different engineering selection problems, for instance, process selection and machine selection. As pointed out by Tawancy et al. in [
To ease out the material selection procedure and make the right decision, a systematic and efficient approach is required. According to literature retrieval, these methods can roughly be classified as material selection charts, knowledge-based methods, and multiattribute decision making (MADM). Ashby has suggested material selection chart, also known as Ashby chart, for selecting materials in a given application, and it is widely used in the literature as [
Much literature using MADM deals with the material selection. However, in most of the literature on the material selection, only one kind of ratings for attributes was considered. In the literature [
Many ranking methods have been developed to aggregate each attribute’s rating for all alternatives, which can be classified as two different approaches: compensatory and noncompensatory models. Whether compensatory methods or noncompensatory methods, most of the ranking methods regard attribute’s relationships as independent. To all intents and purposes, the relationships among many attributes exhibit interdependences with various degrees, such as the relationship between hardness and elastic modulus, increased hardness usually leading to decreased elastic modulus, and that between strength and elongation at break, increased strength usually leading to decreased elongation at break. This has also given rise to the attention of many experts. As argued by Jahan et al. in [
In 1974, Sugeno introduced the concept of fuzzy measures, substituting the additive rigid constraints in classical theory of probability with monotony with weaker constraints, and in the process of MADM employing the integration operators based on fuzzy measures and integral not only takes into account the relative weights but also flexibly represents and treats any interactions among attributes. To the best knowledge of the authors, to date, no paper on material selection has used them to deal with the interdependences among attributes. Some literature as in [
When an attribute is related to qualitative aspects, it may be difficult to qualify it using some values, and it is very convenient to express with linguistic terms (e.g., when evaluating chemical stability of a material, terms like “very good,” “ good,” “average,” “ bad,” or “very bad” can be used). Suppose
With literature retrieval, four ways can be found to treat the linguistic variables: (i) based on the extension principle, (ii) based on the symbolic model, (iii) based on virtual linguistic terms, and (iv) based on 2-tuple fuzzy linguistic representation
For group decision making problems, experts may express linguistic preferences over attributes or alternatives with different cardinalities, so in the process of information integration we should first uniform the linguistic terms with different cardinalities into the ones in the BLTS. Let
The transformation function enjoys good properties of the one-to-one characteristic and simple calculation process and can do the inverse operation.
Suppose
If
Since the physical dimensions and measurements of the
For an interval fuzzy number
For a real number
For convenience,
Note that
Illustration of transforming a triangular number to a linguistic term in BLTS.
Illustration of transforming an interval to a linguistic term in the BLTS.
Illustration of transforming a real number to a linguistic term in the BLTS.
Let boundedness: monotonicity: if
From the perspective of MADM,
Although fuzzy measures constitute a flexible tool for modeling the importance of coalitions, they are not easy to handle in a practical problem, since we generally need to find
Let
Set function
Inversely, for a given Möbius representation, the corresponding fuzzy measure can be calculated as follows:
For any coalition
For a given Möbius representation, the corresponding interaction index can be calculated as
The relationship of
For coalition
According to (
Compared with (
Since the sum of all attributes’ Shapley values satisfies
According to the literature [
Determine
Determine
Up to now, many aggregation operators have been developed to aggregate linguistic ratings, as linguistic ordered weighted geometric averaging operator [
Linguistic weighted geometric averaging with interaction (LWGAI) operator of dimension If all the elements in According to (
where
In (
Linguistic hybrid weighted geometric averaging with interaction (LHWGAI) operator of dimension If According to normal distribution, the further a value is apart from the mean value, the smaller the value of its probability density function is, while the closer a value is to the mean value, the greater the value of its probability density function is, which coincides with notion mentioned above. Therefore the following formula is employed to determine the weights to weight the ordered position of the rating of attribute
where
Taking full advantage of solar power is one of the most important means to mitigate energy shortages, resource depletion, environmental pollution, and so forth, brought about by traditionally thermal power generation, but due to day alternating with night, climate change, and solar energy radiation intensity fluctuating with time within a day, solar energy is an intermittent, not a stable, energy source. Consequently, integration of the solar power with thermal energy storage (TES) is necessary for its effective utilization, as it can store solar power and release it whenever necessary, resulting in capacity buffer, stable power output, and increased annual utilization rate. There are three types of TES: sensible heat storage, phase change heat storage (latent heat storage), and thermochemical energy storage [
Procedure for phase change material selection.
Generally, analyzing and translating the design requirements (expressed as constraints and objectives) into required material’s properties (attributes) is the first step, and then we divide the required material properties into “rigid” and “soft” requirements. Any material with one property that cannot satisfy the “rigid” requirements can first be eliminated. High-temperature molten salt and aluminum-base alloy are two kinds of the most potential phase change materials, but the high-temperature molten salt suffers from lower thermal conductivity and solid-liquid delamination, and it is eliminated from the candidates not satisfying the design requirements and constraints. The five kinds of aluminum-base alloy are listed as the candidate materials, that is, 35Mg/6Zn (
Criteria’s types, expressions, and requirements.
Criteria | Types | Expressions | Requirements |
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Cost | Linguistic terms | The less the ratings, the better the attributes. The ratings of material cost, subject to various factors, are hardly exactly identified and so expressed in linguistic terms according to the knowledge of experts. |
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Beneficial | Linguistic terms | Materials with good chemical stability, though subject to repeated heat absorption and heat release, do not experience the problems of segregation, side reaction, and chemolysis and hence smaller attenuation in the heat storage capacity. |
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Beneficial | Intervals | Under the same phase change temperature, the greater the phase change latent heat is, the more the energy can be stored. |
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Beneficial | Exact values | For the materials with approximately equal phase change latent heat, the greater the density is, the greater the heat per volume can be stored, which lowers the cost of the heat storage equipment. |
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Beneficial | Triangular values | Greater thermal conductivity implies quicker speed in the process of heat storage and extraction and better performance in conductivity, and that absorbing or releasing the same heat requires less temperature gradient. |
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Cost | Linguistic terms | Materials with smaller high-temperature corrosivity are compatible with many other materials, which implies a wide range of material selection for the heat storage pieces of equipment, lowering their cost. |
Raw ratings on each attribute with respect to each alternative and each expert.
PCMs |
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Expert 1 | ||||||
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2380 |
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3424 |
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2700 |
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2730 |
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2300 |
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Expert 2 | ||||||
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2380 |
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3424 |
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2700 |
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2730 |
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2300 |
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Expert 3 | ||||||
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2380 |
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3424 |
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2700 |
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2730 |
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2300 |
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Expert 4 | ||||||
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2380 |
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3424 |
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2700 |
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2730 |
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2300 |
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In this paper, suppose the BLTS is
Individual and corrective normalized ratings.
PCMs |
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Expert 2 | ||||||
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Expert 3 | ||||||
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Expert 4 | ||||||
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Collective normalized ratings | ||||||
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Since
Since the expert preferences are related to expert’s social status, prestige, knowledge structures, expectations, and so forth, consequently in group decision making, the preferences among experts maybe exhibit interactions. If these respects of experts are similar, the relationship of experts exhibits a negative synergetic interaction, resulting in overestimation if neglected, while if they are greatly different, it exhibits a positive synergetic interaction, resulting in underestimation if neglected. Determine the range of the interaction indexes
Interaction coefficient ranges between any two experts.
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[0.0000, 0.0000] | [−0.0816, −0.0272] | [0.0296, 0.0888] | [−0.0296, 0.0296] |
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[−0.0816, −0.0272] | [0.0000, 0.0000] | [−0.0272, 0.0272] | [0.0272, 0.0816] |
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[0.0296, 0.0888] | [−0.0272, 0.0272] | [0.0000, 0.0000] | [0.0381, 0.1143] |
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[−0.0296, 0.0296] | [0.0272, 0.0816] | [0.0381, 0.1143] | [0.0000, 0.0000] |
For the attribute
With the same method used in identifying each expert’s Shapley values, we can obtain the attribute’s Shapley values as
This study has contributed to the material selection literature by (i) considering hybrid information, including real values, interval values, triangular fuzzy numbers, and linguistic variables with different cardinalities, and proposing a method to transform the heterogeneous information to linguistic terms in the BLTS; (ii) providing a feasible and effective method to determine 2-additive fuzzy measures based on the principle of maximum entropy and further simplifying the expressions of Marichal entropy and Choquet integral which, after simplification, is more convenient to use in practice; and (iii) proposing the LHWGAI operator considering not only the interactions between attributes but the ordered positions of the attribute ratings. Compared with [ The expressions of attribute ratings in this paper only cover linguistic terms, real values, interval values, and triangular fuzzy numbers. There exist the other expressions such as interval-valued fuzzy numbers [ The proposed decision model for multiple attribute material selection considering attribute interactions under hybrid environment is a general method and can be easily extended to deal with other management decision making problems such as strategic management, human resource management, supply chain management, and investment management.
The 2-tuple linguistic representation in a linguistic term set with cardinality being
The 2-tuple linguistic representation in the basic linguistic term set
Alternative
Criterion
Expert
The raw rating of alternative
The normalized rating of alternative
The collective normalized rating of alternative
The overall rating of alternative
Membership function
Fuzzy measure for an expert
Fuzzy measure for an attribute
Shapley value for an expert
Shapley value for an attribute
Möbius representation for an expert
Interaction index between any two experts
Interaction index between any two attributes
Multiattribute decision making
The basic linguistic term set
Linguistic weighted geometric averaging with interaction
Linguistic hybrid weighted geometric averaging with interaction
Analytic network process
Analytic hierarchy process.
On behalf of the coauthors, the authors declare that there is no conflict of interests regarding the publication of this paper.