A stochastic dominance based approach to consumer-oriented Kansei evaluation with multiple priorities

Abstract

Nowadays, with the increasing of aesthetic products, it becomes more and more important and quite difficult for consumers to choose their preferred products, especially the ones whose artistic and aesthetic aspects play a crucial role in consumer purchase decisions. Taking Kansei as one quality aspect of products, consumer-oriented Kansei evaluation focuses on evaluation of existing commercial products based on consumers’ Kansei preferences. This paper proposes a stochastic dominance based approach to consumer-oriented Kansei evaluation with multiple priorities. Particularly, given a consumer’s preferences, the concept of stochastic dominance is used to build an evaluation function for each Kansei attribute. Then, the importance weights captured by a priority hierarchy of Kansei attributes, together with the fuzzy majority, are incorporated into the aggregation of individual stochastic dominance degrees into an overall one. An application to the hand-painted Kutani cups in Ishikawa, Japan, is conducted to illustrate the effectiveness and efficiency of the proposed approach. It is seen that the proposed approach outperforms the existing research in terms of easy of use and better decision-support to the consumers.

Appendix: Deriving stochastic dominance degrees from uncertain profiles

In this section, we will introduce an approach to derive stochastic dominance degrees from uncertain profiles, which is based on our previous work (Yan et al. 2013b).

Let $Z_{1}$ and $Z_{2}$ be two independent discrete random variables with respective probability distributions $p_{1}$ and $p_{2}$ defined over a finite set of qualitative scale ${\mathbb {V}}=\left\{ V_{1},V_{2},\ldots ,V_{G}\right\}$ with $V_{1}<V_{2}<\cdots <V_{G}$, where

$$\begin{aligned} \sum _{z\in {\mathbb {V}}}p_{1}\left( z\right) =1 \quad \text{ and }\quad \sum _{z\in {\mathbb {V}}}p_{2}\left( z\right) =1. \end{aligned}$$

(29)

The probability distribution over the qualitative scale is referred to as uncertain profile in this paper.

Let $z_{1}$ and $z_{2}$ be possible outcomes of $Z_{1}$ and $Z_{2}$, respectively. Let $\Pr (z_{1}\ge z_{2})$, $\Pr (z_{1}=z_{2})$, and $\Pr (z_{1}\le z_{2})$ denote the probabilities of $z_{1}\ge z_{2}$, $z_{1}=z_{2}$, and $z_{1}\le z_{2}$, respectively. Since the two random variables $Z_{1}$ and $Z_{2}$ are stochastically independent, we have

$$\begin{aligned} \begin{aligned} \Pr \left( z_{1}\ge z_{2}\right)&= \sum _{z_{1}=V_{1}}^{V_{G}}\sum _{z_{2}=V_{1}}^{z_{1}}p_{1}\left( z_{1}\right) \cdot p_{2}\left( z_{2}\right) \\ \Pr \left( z_{1}= z_{2}\right)&= \sum _{z_{1}=V_{1}}^{V_{G}}p_{1}(z_{1})\cdot p_{2}\left( z_{1}\right) \\ \Pr \left( z_{1}\le z_{2}\right)&= \sum _{z_{1}=V_{1}}^{V_{G}}\sum _{z_{2}=z_{1}}^{V_{G}}p_{1}\left( z_{1}\right) \cdot p_{2}(z_{2}) \end{aligned} \end{aligned}$$

(30)

Accordingly, we have

$$\begin{aligned} \begin{aligned} {\Pr }(z_{1}>z_{2})&=\Pr \left( z_{1}\ge z_{2}\right) -\Pr \left( z_{1}= z_{2}\right) \\ {\Pr }(z_{1}<z_{2})&=\Pr \left( z_{1}\le z_{2}\right) -\Pr \left( z_{1}= z_{2}\right) \end{aligned} \end{aligned}$$

(31)

Due to the above analysis, we are able to give the definition of stochastic dominance degree of two random variables with discrete probability distributions defined over a qualitative scale as follows.

Definition 6

Let $Z_{1}$ and $Z_{2}$ be two independent discrete random variables with (discrete) probability distributions $p_{1}$ and $p_{2}$ over a qualitative scale ${\mathbb {V}}=\left\{ V_{1},V_{2},\ldots ,V_{G}\right\}$ with $V_{1}<V_{2}<\cdots <V_{G}$, where $\sum _{z\in {\mathbb {V}}}p_{1}(z)=1$ and $\sum _{z\in {\mathbb {V}}}p_{2}(z)=1$. Then the stochastic dominance degree of $p_{1}$ over $p_{2}$ (noted as $R_{p_{1}\succ p_{2}}$) is defined as

$$\begin{aligned} \begin{aligned} R_{12}&=R_{p_{1}\succ p_{2}}\\&={\Pr }(z_{1}\ge z_{2})-0.5{\Pr }(z_{1}=z_{2}) \end{aligned} \end{aligned}$$

(32)

where ${\Pr }(z_{1}\ge z_{2})$ is the probability of $z_{1}\ge z_{2}$ and $0.5{\Pr }(z_{1}=z_{2})$ in Eq. (32) may be regarded as the probability of $z_{1}>z_{2}$ when event $z_{1}=z_{2}$ occurs (Fan et al. 2010).

Accordingly, the stochastic dominance degree of $p_{2}$ over $p_{1}$ (noted as $R_{p_{2}\succ p_{1}}$) is defined as

$$\begin{aligned} \begin{aligned} R_{21}&=R_{p_{2}\succ p_{1}}\\&={\Pr }(z_{2}\ge z_{1})-0.5{\Pr }(z_{1}=z_{2}) \end{aligned} \end{aligned}$$

(33)

where ${\Pr }(z_{2}\ge z_{1})$ is the probability of $z_{2}\ge z_{1}$ and $0.5{\Pr }(z_{1}=z_{2})$ in Eq. (33) may be viewed as the probability of $z_{2}>z_{1}$ when event $z_{1}=z_{2}$ occurs (Fan et al. 2010).

Extending two random variables to a vector of N random variables ${\mathbb {Z}}=(Z_{1},Z_{2},\ldots ,Z_{N})$, we are able to derive a matrix $\mathbf {R}$ of stochastic dominance degrees of the N discrete random variables as

$$\begin{aligned} \mathbf {R}=\begin{array}{lcccc} &{} Z_{1} &{} Z_{2} &{} \ldots &{} Z_{N} \\ Z_{1} &{} {R}_{11} &{} {R}_{12} &{} \ldots &{} {R}_{1N} \\ Z_{2} &{} {R}_{21} &{} {R}_{22} &{} \ldots &{} {R}_{2N} \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ Z_{N} &{} {R}_{N1} &{} {R}_{N2} &{} \ldots &{} {R}_{NN} \\ \end{array} \end{aligned}$$

(34)

Such a matrix of stochastic dominance degrees has the following interesting corollaries (Yan et al. 2013b).

Corollary 1

Let $\mathbf {R}=\left[ R_{nl}\right] _{N\times N}$ be a matrix of stochastic dominance degrees of the N discrete random variables, then we have ${R}_{nl}+{R}_{ln}=1,\forall n,l=1,2,\ldots ,N$.

Corollary 2

Let $\mathbf {R}=\left[ R_{nl}\right] _{N\times N}$ be a matrix of stochastic dominance degrees, the stochastic dominance degree of one discrete random variable over itself is

$$\begin{aligned} R_{nn}=0.5,\quad n=1,2,\ldots ,N. \end{aligned}$$

Corollary 3

Let $\mathbf {R}=\left[ R_{nl}\right] _{N\times N}$ be a matrix of stochastic dominance degrees, then the sum of all the elements of $\mathbf {R}$ is $N^{2}/2$, that is

$$\begin{aligned} \sum _{n=1}^{N}\sum _{l=1}^{N}R_{nl}=\frac{N^{2}}{2}. \end{aligned}$$

Corollary 4

Let $\mathbf {R}=\left[ R_{nl}\right] _{N\times N}$ be a matrix of stochastic dominance degrees, then we have $0\le {R}_{nl}\le 1,\forall n,l=1,\ldots ,N$.

Interestingly, the matrix $\mathbf {R}$ of stochastic dominance degrees with respect to a vector of N random variables ${\mathbb {Z}}=(Z_{1},Z_{2},\ldots ,Z_{N})$ satisfies the following properties of fuzzy preference relations.

Property 1

When ${R}_{nl}=1$, it indicates that $Z_{n}$ is absolutely preferred to $Z_{l}$, i.e., indicates the maximum degree of preference of $Z_{n}$ over $Z_{l}$.

Property 2

When $0.5<{R}_{nl}<1$, it indicates that $Z_{n}$ is slightly preferred to $Z_{l}$.

Property 3

When ${R}_{nl}=0.5$, there is no preference (i.e., indifference) between $Z_{n}$ and $Z_{l}$.

Property 4

When $0<{R}_{nl}<0.5$, it indicates that $Z_{l}$ is slightly preferred to $Z_{n}$.

Property 5

When ${R}_{nl}=0$, it indicates that $Z_{l}$ is absolutely preferred to $Z_{n}$.

Therefore, the matrix of stochastic dominance degrees of the random variable set ${\mathbb {Z}}$ is in fact a matrix of fuzzy preference relations formulated as $\mu _{\mathbf {R}}:(Z_{n},Z_{l})\in {\mathbb {Z}}\times {\mathbb {Z}}\longrightarrow R_{nl}\in [0,1]$, where $n,l=1,\ldots ,N,$ and $R_{nl}$ reflects the degree of fuzzy preference of $Z_{n}$ over $Z_{l}$. Moreover, it is obvious that the matrix of fuzzy preference relations satisfies the condition of fuzzy reciprocity such that $R_{nl}+R_{ln}=1,\forall n,l=1,\ldots ,N$.