Application of Choquet Integral-Fuzzy Measures for Aggregating Customers’ Satisfaction
Application of Choquet Integral-Fuzzy Measures for Aggregating Customers’ Satisfaction
Abdullah, Lazim;Awang, Noor Azzah;Othman, Mahmod
2021-09-22 00:00:00
Hindawi Advances in Fuzzy Systems Volume 2021, Article ID 2319004, 8 pages https://doi.org/10.1155/2021/2319004 Research Article Application of Choquet Integral-Fuzzy Measures for Aggregating Customers’ Satisfaction 1 2 3 Lazim Abdullah , Noor Azzah Awang, and Mahmod Othman Management Science Research Group, Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, Kuala Nerus 21030, Malaysia Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA, Shah Alam 40450, Selangor, Malaysia Fundamental and Applied Sciences Department, Universiti Teknologi PETRONAS, Seri Iskandar 32610, Perak, Malaysia Correspondence should be addressed to Lazim Abdullah; lazim_m@umt.edu.my Received 15 June 2021; Revised 27 August 2021; Accepted 6 September 2021; Published 22 September 2021 Academic Editor: Katsuhiro Honda Copyright © 2021 Lazim Abdullah et al. *is 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. Choquet integral is a type of aggregation operator that is commonly used to aggregate the interrelated information. Nowadays, this operator has been successfully embedded with fuzzy measures in solving various evaluation problems. Inspired from this new development, this paper aims to introduce a combined Choquet integral-fuzzy measures (CI-FM) operator that uses the Shapley value standard and interaction index to deal with the interactions between elements of information. *e proposed operator takes into account not only the importance of elements or their ordered positions but also the interaction among criteria during the evaluation process. A case of customers’ satisfaction over two fast restaurants in Malaysia is presented to illustrate the application of the proposed aggregation operator. Based on three customers’ satisfaction criteria, restaurant 1 and restaurant 2 received CI- FM scores of 0.711011 and 0.704945, respectively. Interestingly, the criterion “services” constantly received the highest rating across both restaurants. In addition, the proposed aggregation operator successfully identified which restaurant is superior in the eyes of customers. Finally, this study offers some research ideas for the future. uncertainty regarding the criteria. Considering this diffi- 1. Introduction culty, the notion of fuzzy measure is used in Choquet in- Current developments in information processing have in- tegral where interaction phenomena among the criteria can creased the need for an efficient information aggregation be modelled [3]. *e Choquet integral employs the notion of operator. *e interaction between criteria exists in a real fuzzy measure to indicate the weights or the importance of multicriteria decision-making analysis. When each deci- multiple interdependent criteria in decision-making [4]. In sionmaker provides their individual set of criteria, two or other words, the Choquet integral uses the concept of fuzzy more criteria might be redundant and complementary [1]. measure to quantify the importance of multiple interde- *e Choquet integral is one of the aggregation operators that pendent criteria. *e amalgamation of Choquet integrals is used to deal with the interactions between criteria. *e and fuzzy measure is further emphasized by Vu et al. [5]. intensity of this aggregation operator is reflected by the final *ey reaffirmed that Choquet integral is an aggregating Choquet integrated values after performing a series of function defined in terms of the fuzzy measure. According to computations. It was introduced by Choquet [2] with the them, a fuzzy measure is a set function that acts on the purpose to solve the interrelationship among criteria of domain of all conceivable amalgamations of a set of criteria. decision problems of which the final ordering of criteria is *e Choquet integral aggregation function with fuzzy established. However, ordering of criteria is not a measure is written as the Choquet integral-fuzzy measure straightforward process as there exists some extent of (CI-FM) and will be used throughout this paper. *e 2 Advances in Fuzzy Systems and convenience were significantly influenced their choice compatibility of this combination may be seen in the way that fuzzy measure permits the Choquet integral to allocate of fast-food outlets. *e survey also identified factors like the taste of food, nutritional value, and variety of menu that were priority to all feasible groupings of criteria in decision problems, allowing for much more aggregation flexibility. not significant to the choice made by rural consumers. On *e intricacy of the CI-FM integral lies on the expo- the other hand, Yan et al., [19] conducted a Kansei evalu- nential of 2n subsets, where n is the number of criteria in a ation on commercial products, which is an individual decision problem [6]. *e combination of fuzzy measures subjective judgement of a product’s aesthetic appeal. In this and Choquet integrals would result in a thorough aggre- evaluation, a three-phase group with no additive multi- gation method. *e unit interval [0, 1] is used to define both attribute Kansei evaluation model was proposed. After generating Kansei profiles, a target-oriented Kansei assess- the inputs and outputs of CI-FM. However, other intervals are also possible depending on the researchers’ preferences. ment function was presented to induce the Kansei satis- faction utility based on the consumer’s personal Kansei *e CI-FM encapsulates the nonadditive capacity property and correlates to a vast class of aggregation functions, preference. In the third phase, an evaluation function was proposed based on the Kansei assessment function and the allowing greater flexibility in the decision-making process [7, 8]. Subadditive or superadditive operators are used to Choquet integral. On a subset of Kansei characteristics, this integrate functions with respect to the fuzzy measures where entropy-based technique was used to estimate the fuzzy many extensions and generalizations of fuzzy sets could be measure. inserted into fuzzy measures. Groes et al., [9] stated that In another customers’ satisfaction research, Pelaez, ´ et al. Choquet integral plays a key role for recent advances in the [20] proposed a new purchase decision prediction model in decision theory that encompasses nonadditive measures. investigating the rank of consumer purchasing factors in digital ecosystem. Evaluations of alternatives and criteria From the above literature, it is clearly seen that the main application of CI-FM can be explained in the decision theory were made by considering the opinions expressed by con- sumers in digital ecosystems. *e opinions expressed are the and decision problems. According to Heilpern, [10] the CI- FM can be applied to many real decision-making cases in manifestations of customers’ satisfactions over the products, and this is the only input information that is available to the economics and business. It is also reported that CI-FM has been used in specific areas such as insurance [11], green evaluation model. *e suggested method discovers the building [12], airline services [13], business education implicit synergies within criteria and alternatives by evaluation [14], loan market matching [15], and financial extracting their weights. Finally, the model uses the CI-FM risk [16]. In customer service-oriented research, Pasrija and to suggest aggregation values which eventually determines a Srivastava [17] investigated the software quality using the purchase ranking. *e CI-FM is a tool for describing the CI-FM. *ey hypothesized that the CI-FM would be effective correlations between consumers’ satisfaction and criteria of multiple products. *e advantage of the CI-FM is mostly due in comparing the software solutions that would be executed. In their study, companies and clients need to make ap- to the use of integral and fuzzy measure in its calculations. *is computation allows to comprehensively reflect the propriate selections on which option to operate with a high- quality viewpoint of the ranking. *ese quality criteria, interaction between all possible combination of criteria in which are based on multiple points of view, are interde- consumers’ satisfaction decision problem where information pendent and ambiguous in nature. To evaluate user satis- about multiple products and their multiple interdependent faction, the theory of fuzzy set with a quality model was criteria are aggregated. Motivated by this advantage, this employed to estimate the software quality. In another paper extends the application of CI-FM to another real case customer services research, Vu et al., [5] examined customer study of consumers’ satisfaction decision problem. Specifi- preference using the CI-FM. *e aims of customers’ pref- cally, this paper aims to propose the Choquet integrated erences have always been to support the strategic planning values for two fast-food restaurants in Terengganu Malaysia using the CI-FM. *ese values are used to suggest the and decision-making of business managers. In their study, the CI-FM has been employed as an aggregation function preference of consumers over the criteria and also the more preferred fast-food restaurant over the other. To the best of technique, and they also developed a new toolbox to facil- itate the computations. *e CI-FM and toolbox have been authors knowledge, this is the first identifiable work of CI- used to bring advantages to researchers and managers FM application to customers’ satisfaction. *e basic con- worldwide in performing more effective business and cepts of the fuzzy measures, Choquet integral, and Shapley knowledge management decision. value standard are recalled in the following section. From customers’ services research, it is now turning to consumer related research in which goods, products, or 2. Preliminary services are primarily employed to meet the needs and satisfactions of customers. Oni and Matiza [18], for example, In this section, mathematical definitions of the fuzzy established the value of CI-FM of the main factors that measures and fuzzy integrals are presented. *ese two influenced rural-consumer choice of fast-food outlets. In definitions are required to understand the whole compu- their study, a quantitative survey was conducted, and the tational procedures that are encompassed in this paper. number of sample data was collected from two hundred over *e fuzzy measure is an important tool of aggregating respondents. *e study further investigated that the tradi- information that is characterized by vague and uncertainty. tional main factors, such as the value of money, accessibility, *ere are two types of fuzzy measures which are additive and Advances in Fuzzy Systems 3 nonadditive. If the fuzzy measure is additive, this particular measure is also known as Sugeno measure in which the fuzzy measure is termed as the λ-fuzzy measure. *ese two following additional property is satisfied: types of λ-fuzzy measures are defined as follows. g(A∪ B) � g(A) + g(B) + λg(A)g(B). (1) Definition 1 (see [3, 21]). Fuzzy measures with λ � 0 where λ> − 1 for all A, B⊆g(X) and A∩ B � φ. In (1), λ � 0 Let a universal set X � x , x , x , . . . , x and g(x) be 1 2 3 n indicates that the λ-fuzzy measure g is an additive fuzzy the worth of all possible subsets. A fuzzy measure g on the measure and there is no interaction between A and B. set of criteria X is a set function g(x) ⟶ [0, 1], satisfying the boundary and monotonic properties: Definition 2 (see [3, 21]). Fuzzy measures with λ≠ 0 (i) g(φ) � 0, g(X) � 1 (boundary) If λ≠ 0, then it indicates that λ-fuzzy measure g is (ii) If A⊆ B⊆ X, then g(A)≤ g(B)≤ g(X) (monotonic) nonadditive and A and B are interacting with each other. If λ> 0, then g(A∪ B)> g(A) + g(B), which infers that the set A fuzzy measure is additive if g(A∩ B) � 0 (disjoints), {A, B} has a multiplicative impact. If λ< 0, then then the union of two sets can be written as g(A∪ B)< g(A) + g(B), which infers that the set {A, B} has g(A∪ B) � g(A) + g(B). *is particular fuzzy measure is a substitutive impact. If X is a finite set, then ∪ A � X. i�1 i termed as λ-fuzzy measure, a special kind of fuzzy measure *e λ-fuzzy measure g meets this requirement: defined on g(X) and satisfying the finite λ-rule. *is fuzzy ⎧ ⎨ ⎫ ⎬ ⎪ 1 1 + λg x − 1 , if λ≠ 0, ⎪ i ⎩ ⎭ i�1 g(X) � g∪ x � (2) i�1 ⎪ g x , if λ � 0, i�1 where x ∩ x � ∅ for all i, j � 1, 2, 3, . . . , n and i≠ j. It can (i) Boundary condition: μ(φ) � 0 and μ(X) � 1 i j be denoted that g(x ) for a subset with a single element x is i i (ii) Monotonicity: If A, B ∈ P(X) and A⊆B, then called fuzzy density and can be written as g � g(x ). i i μ(A)≤ μ(B) *e value λ may be determined in a unique way from For A, B ∈ P(X) with A∩ B ∈ φ, the fuzzy measure is g(X) � 1, which is equivalent to solve the following said to be as follows: equation: (i) An additive measure, if μ(A∪ B) � μ(A) + μ(B) λ + 1 � 1 + λg . (3) (ii) A superadditive measure, if μ(A∪ B)> μ(A) + μ(B) i�1 (iii) A subadditive measure if μ(A∪ B)< μ(A) + μ(B) Conditions of fuzzy measures are given in Definition 3. Also, Definition 3 (see [22]). Condition of fuzzy measure A fuzzy measure on X is a set function μ: P(X) ⟶ [0, 1], satisfying the following: μ(A∪ B) � μ(A) + μ(B) + λμ(A)μ(B), λ ∈ [− 1,∞), ∀A, B ∈ P(X) and A∩ B � φ. (4) λ + 1 � 1 + λμ