A Choquet Integral Toolbox and Its Application in Customer Preference Analysis

Vu, Huy Quan; Beliakov, Gleb; Li, Gang

doi:10.1016/b978-0-12-411511-8.00009-8

Cited by 14 publications

(12 citation statements)

References 15 publications

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“…; Vu et al . )

C_{v} (x) = \sum_{i = 1}^{N} x_{(i)} (v_{S (i)} - v_{S (i + 1)})

or the equivalent expression:

C_{v} (x) = \sum_{i = 1}^{N} (x_{(i)} - x_{(i - 1)}) v_{S (i)}

where: S (i) is a collection of subsets defined by S (i) = { (i),… (N) }; v is the vector of fuzzy measures stored following binary indexing system, with v 0 = 0 and v N = 1. For instance, if x = { x 1 , x 2 , x 3 } and x 2 ≤ x 1 ≤ x 3 , we have

\begin{matrix} left \\ v = {v_{0} ​, v_{1} ​, v_{2} ​, v_{12} ​, v_{3} ​, v_{13} ​, v_{23} ​, v_{123}} = {0, v_{1} ​, v_{2} ​, v_{12} ​, v_{3} ​, v_{13} ​, v_{23} ​, 1} \\ x_{()} = {x_{2} ​, \end{matrix}

…”

Section: Methodsmentioning

confidence: 99%

“…Considering all possible orders of the attributes ( x ( ) ), and the corresponding collection of subsets S (j) , an alternative expression of usual expression of Shapley value (Grabisch and Labreuche ; Vu et al . ) is, for a fuzzy measure ( v ),

ϕ_{i} (v) = \frac{1}{n!} \sum_{j = 1}^{n!} [Δ_{i j}],

(e.g., Castro et al . ) being Δ ij = v S ( j ) – v S ( j – 1 ) (marginal contribution of attribute i to subset S ( j ) ), and satisfying

\sum_{i = 1}^{N} ϕ_{i} (v) = 1

where: v S ( j ) is the fuzzy measure of the subset containing attribute j ; v S ( j – 1 ) is the fuzzy measure of the subset of predecessors of attribute j . …”

Section: Methodsmentioning

confidence: 99%

“…) and Rfmtool (Vu et al . ), are free toolboxes, distributed as standard packages for GNU R statistical system, which can facilitate the analyses of consumer preferences.…”

Section: Methodsmentioning

confidence: 99%

“…≤x(N). For a general fuzzy measure (v), Choquet integral is given by (Grabisch et al 2006;Vu et al 2013)…”

Section: Methodsmentioning

confidence: 99%

“…) and Rfmtool (Vu et al . ) toolboxes, considering 2‐additivity. In Tables and 3 are shown the analysis specifications for R package, using Kappalab – considering LSE estimation (Table ), and using Rfmtool – that performs least absolute deviance estimation (Table ).…”

Section: Examplementioning

confidence: 99%

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A Technical Note on the Use of Choquet Integral to Analyze Consumer Preferences: Application to Meat Consumption

Alfonso

2013

Journal of Sensory Studies

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This paper deals with the use of the Choquet integral to analyze meat consumer preferences. First, we present the method in comparison with multiple linear regression. Second, we illustrate the application in a data set of panelist evaluation of fresh pork loin chops for three attributes (color, lean content and overall acceptability) and purchase intention. Kappalab and Rfmtool packages for the GNU R statistical system were used for data analysis. The Choquet integral allowed to analyze the contribution of each attribute and their combinations, overcome the problems of multicollinearity in linear regression, and showed a similar or better predictive ability than linear model. In conclusion, the Choquet integral can be considered a complementary tool of usual statistical analysis of food consumer preferences. Practical Applications This paper suggests the use of the Choquet integral as a complementary statistical tool that can facilitate the analyses of complex sensory evaluations and the behavior of consumers. The Choquet integral allows estimating the importance of different sensorial attributes on the final consumer decision taking into account the dependencies existing between them. It allows a better understanding of the relationships existing between sensorial attributes especially in the presence of multicollinearity. Furthermore, the comparison of the behavior of different groups of consumers can be performed easily.

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“…; Vu et al . )

C_{v} (x) = \sum_{i = 1}^{N} x_{(i)} (v_{S (i)} - v_{S (i + 1)})

or the equivalent expression:

C_{v} (x) = \sum_{i = 1}^{N} (x_{(i)} - x_{(i - 1)}) v_{S (i)}

\begin{matrix} left \\ v = {v_{0} ​, v_{1} ​, v_{2} ​, v_{12} ​, v_{3} ​, v_{13} ​, v_{23} ​, v_{123}} = {0, v_{1} ​, v_{2} ​, v_{12} ​, v_{3} ​, v_{13} ​, v_{23} ​, 1} \\ x_{()} = {x_{2} ​, \end{matrix}

…”

Section: Methodsmentioning

confidence: 99%

ϕ_{i} (v) = \frac{1}{n!} \sum_{j = 1}^{n!} [Δ_{i j}],

(e.g., Castro et al . ) being Δ ij = v S ( j ) – v S ( j – 1 ) (marginal contribution of attribute i to subset S ( j ) ), and satisfying

\sum_{i = 1}^{N} ϕ_{i} (v) = 1

where: v S ( j ) is the fuzzy measure of the subset containing attribute j ; v S ( j – 1 ) is the fuzzy measure of the subset of predecessors of attribute j . …”

Section: Methodsmentioning

confidence: 99%

“…) and Rfmtool (Vu et al . ), are free toolboxes, distributed as standard packages for GNU R statistical system, which can facilitate the analyses of consumer preferences.…”

Section: Methodsmentioning

confidence: 99%

“…≤x(N). For a general fuzzy measure (v), Choquet integral is given by (Grabisch et al 2006;Vu et al 2013)…”

Section: Methodsmentioning

confidence: 99%

Section: Examplementioning

confidence: 99%

See 3 more Smart Citations