Journal of Control Science and Engineering

Volume 2015, Article ID 943795, 12 pages

http://dx.doi.org/10.1155/2015/943795

## Material Selection in Engineering Design Using Choquet Integral-Based Linguistic Operators under Hybrid Environment

Engineering Training Center, Huaihai Institute of Technology, Lianyungang, Jiangsu 222005, China

Received 27 March 2015; Revised 1 July 2015; Accepted 14 July 2015

Academic Editor: Kalyana C. Veluvolu

Copyright © 2015 Anhua Peng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1], the variation in material and design resulted in significant difference in service performance: The pipe using heat-resistant casting steel failed after only 22,000 h of service while that using wrought INCOLOY alloy 800H remained in operation for 83,000 h. Therefore, material selection plays an important role in product cost and performance throughout its life cycle. An ever increasing variety of materials is available today, with each having its own characteristics, applications, advantages, and limitations. There is no material which satisfies all the relevant properties. For example, some materials are good enough to satisfy cost-related criteria but not so good in terms of some mechanical criteria, while some are good to satisfy a set of thermal criteria but not suitable in terms of cost, and so on. The large number of materials available to designers, coupled with complex interrelationship between the different selection parameters, often makes the material selection process a difficult task. The traditional material selection methods, such as those based only on designers’ experiences, try-and-error methods, or analogy methods, are often made in the following way: one chooses between a few materials which have been used for similar situations before. This often leads to a conservative choice and one also misses newly developed materials which may be suitable for the new modified situation.

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 [2]. However, drawing the Ashby chart requires a broad engineering knowledge, which sometimes makes it difficult for a practitioner to employ the method, and the material selection procedure is performed based on two performance indices per chart. Consequently, if more than two performance indices are required to be considered, then it should be done using a sequential process. In addition, Ashby’s charts normally offer a range or a list of materials to the designers to choose from, so they can only be used in material screening, not in material ranking. Fuzzy inference systems (FIS) and genetic algorithm (GA) are two typical knowledge-based methods and can be found widely used in material selection as in [3, 4]. However, a limitation of the FIS is that the inclusion of a new criterion increases exponentially the number of decision rules of an inference system. The main drawback of GA is that it requires users to have a level of specialized knowledge that is likely to be well beyond that possessed by most managers and organizational decision makers. Also a severe drawback of GA is that some feasible solutions cannot be generated by crossover operation [5]. As stated in [6], this research provides evidence that the MADM approaches have potential to greatly improve the material selection methodology, which motivates this paper to use MADM to address the phase change material selection problem.

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 [7], three kinds of ratings for attributes are considered: exact values, intervals, and linguistic terms, but employing the method of computing the interval distance to normalize, although simple in calculation, loses a lot of useful information. As stated in [8], some of these attributes can be expressed as numbers, like density or thermal conductivity; some are Boolean, such as the ability to be recycled; some, like resistance to corrosion, can be expressed only as a ranking (e.g., poor, adequate, and good); and some can only be captured in text and images. Moreover, in the material selection process especially in the initial screening stage, the growing complexity and uncertainty of decision situations make it less and less possible for a decision maker to consider all relevant aspects of a problem and necessitate the participation of multiple experts in decision making to consider every aspect completely, draw on collective wisdom, absorb all useful ideas, and finally improve decision making results. Due to the decision maker’s knowledge field, attitudes, motivations, and personality and the nature of evaluated attributes, the decision makers may provide the assessments with different formats. Such a type of MADM problems is called the fuzzy heterogeneous MADM problems with which seldom literature deals [9]. Consequently, it is very necessary and important to develop a normalization method which deals with the fuzzy heterogeneous information. In this paper, a method is proposed to transform the heterogeneous information to linguistic terms in the basic linguistic term set (BLTS).

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 [10], it can be highlighted that the correlation between criteria is realistic in material selection; thus ranking of materials without attention to the dependency of material properties causes doubtable final solution. As proposed by Karande et al. in [11], future research may aim at improving these methods so that the possible correlation between the considered criteria can be taken into account for arriving at the best material selection decision. Liu et al. in [12] proposed that considering the interrelationship of the material indices is one of the subjects that should receive some more attention in the process of material selection. It is indeed true, for a decision making model considering interdependences among attributes is more scientific, accurate than that not considering interdependences, which is only a special case in decision making problems. Jahan et al. in [10] proposed the correlation effects weighting to mitigate the effect of interdependences where the attribute with the greater intercriteria correlation with the other attributes was assigned a smaller correlation effects weighting. Peng and Xiao in [7] proposed the analytic network process (ANP), a relatively new MADM method based on analytic hierarchy process (AHP), to consider the feedback and interactions within and between sets of design criteria and alternatives. However, the ANP can only identify whether or not a criterion is affected by the corresponding control criterion but cannot identify whether the interactions between any two criteria are positive (superadditive) or negative (subadditive), and with ANP decision makers must construct so many comparison matrices, which incurs great burdens on the decision makers.

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 [13] applied Choquet integrals to supplier selection but under the presupposition that the fuzzy measures are already known or are only subjectively identified by experts, yet actually whether or not the fuzzy measures are accurate directly determines the accuracy of fuzzy integrals, and therefore how to determine the fuzzy measures is the key step. Literature [14], and so forth, employed fuzzy measures to identify fuzzy measures for each attribute or attribute coalition, but although it can greatly reduce the difficulty in identifying fuzzy measures, it can only express one kind of interactions, either all with positive interactions or with negative interactions, abating the power of interaction expressions and violating the actual situations. In this paper, to better capture the interactions among attributes, two-additive fuzzy measures were used to model criteria interactions by pairs and to derive the special expressions of Marichal entropy and Choquet integral, more convenient to use in practice. Fuzzy measures were identified based on the maximum of Marichal entropy. Two Choquet integral-based operators were proposed to obtain the overall ratings of each alternative, which were then used to sort all alternatives.

#### 2. Transforming Hybrid Information into Linguistic Terms in BLTS

##### 2.1. Linguistic Terms

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 is a finite and total discrete term set, where the middle term represents “average,” that is, a probability of approximately 0.5, and the remaining terms are ordered symmetrically around it. As for the properties of a linguistic term, refer to [15].

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 (where is a linguistic label from a predefined linguistic term set ; , , denotes the value of symbolic translation, particularly in a predefined linguistic term set). Since the first two methods take an approximation process, this inevitably produces the consequent information loss and hence the lack of precision. In comparison 2-tuple fuzzy linguistic representation involves no approximation process, does not give rise to information loss, is explicit enough in physical meanings, and therefore is used in this paper. Let , be the result of an aggregation of the indices of a set of labels assessed in a linguistic term set , that is, the result of a symbolic aggregation operation, with as the cardinality of set . Then can be represented as 2-tuple using the function [16]:where is the usual round operation, has the closest index label to , and denotes the difference between and in . Conversely let be a 2-tuple linguistic term; then can be represented as equivalent numerical value, , using the inverse function [16]:

##### 2.2. Making the Linguistic Terms Uniformed

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 be BLTS with cardinality of , and source linguistic term in set can be equivalently transformed into in the BLTS using the following function [7]:

The transformation function enjoys good properties of the one-to-one characteristic and simple calculation process and can do the inverse operation.

##### 2.3. Transforming the Other Information

###### 2.3.1. Normalization

Suppose is the rating of alternative in respect of criterion . Generally, criteria can be classified into two types: benefit () and cost () criteria. The larger the value of an alternative on the benefit criterion, the better the alternative, while the smaller the value of an alternative on the cost criterion, the better the alternative. If is a triangular fuzzy number then it is denoted by , or for short, whose membership function is given as follows:

If is an interval number then it is denoted by , or for short, whose membership function is given as follows:

Since the physical dimensions and measurements of the attributes are different, the raw data need to be normalized. For a triangular fuzzy number , it can be normalized as follows [9]:where .

For an interval fuzzy number , it can be normalized as follows [9]:where .

For a real number , it can be normalized as follows [9]:where .

###### 2.3.2. Transformation of Normalization Data Size into BLTS

For convenience, is hereafter employed to express the membership function of the triangular, interval, and real numbers. Letting be the set of fuzzy sets defined in , the BLTS, is transformed into using the function [17]:

Note that is not related to at all but represents the extent to which belongs to fuzzy linguistic term . Supposing the BLTS is with membership function of triangular fuzzy numbers defined as = , the transformations of a triangular fuzzy number, an interval number, and a real number into the linguistic terms are illustrated, respectively, in Figures 1–3.