﻿ 面向客户偏好异质性的车身产品离散选择模型
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 同济大学学报(自然科学版)  2018, Vol. 46 Issue (5): 667-672.  DOI: 10.11908/j.issn.0253-374x.2018.05.014 0

### 引用本文

XU Kaixiang, LIU Haijiang, PAN Zhenhua. Discrete Choice Model of Auto-body Product for Customer Preference Heterogeneity[J]. Journal of Tongji University (Natural Science), 2018, 46(5): 667-672. DOI: 10.11908/j.issn.0253-374x.2018.05.014

### 文章历史

Discrete Choice Model of Auto-body Product for Customer Preference Heterogeneity
XU Kaixiang , LIU Haijiang , PAN Zhenhua
School of Mechanical Engineering, Tongji University, Shanghai 201804, China
Abstract: Based on the random parameter mixed logit model, a discrete choice model for customer preference heterogeneity was established for the hierarchy of auto-body product.According to the sampled data obtained from SP(stated preference) survey and parameter prior distribution setting, the Markov chain Monte Carlo simulation method was used to make the Bayesian estimation of parameters.Finally, the McFadden's likelihood ratio test proves that the random parameter mixed logit model is of optimal goodness-of-fit, and better than others to elucidate where the customer preference heterogeneity rooted in.This modeling approach helps to capture personalized customer needs, and helps manufacturers to anticipate mutiple preferences of potential customers and assists in the design and the development of auto-body products.
Key words: preference heterogeneity    mixed logit model    Bayesian estimaton

1 随机系数mixed logit模型的建立

 ${\mathit{U}_{\mathit{in}}}{\text{ = }}{\mathit{V}_{\mathit{in}}}{\text{ + }}{\varepsilon _{\mathit{in}}}$

 $\begin{gathered} {U_{in}} = {\mathit{\boldsymbol{X}}_{in}}{\mathit{\boldsymbol{\beta }}_n} + {\varepsilon _{in}} = \sum\limits_{k = 1}^K {{x_{ink}}{\beta _{nk}}} + {\varepsilon _{in}}, \hfill \\ {\mathit{\boldsymbol{\beta }}_n} = \mathit{\boldsymbol{\beta }} + \mathit{\boldsymbol{\theta Z}} + \mathit{\boldsymbol{ \boldsymbol{\varGamma} }}{\mathit{\boldsymbol{u}}_n} \hfill \\ \end{gathered}$ (1)

 ${\beta _{nk}} = {\beta _k} + {\theta _k}{z_n} + {\sigma _k}{u_{nk}} = {\beta _k} + {\theta _k}{z_n} + {\sigma _k}{\rm{exp}}({\omega _k}{s_n}){\zeta _{nk}}$ (2)

 ${V_{in}}\left( {ML} \right) = \mathit{\boldsymbol{\beta }}{\mathit{\boldsymbol{X}}_{in}} + \mathit{\boldsymbol{Z\theta }}{\mathit{\boldsymbol{X}}_{in}} + {\mathit{\boldsymbol{\beta }}_{Xn}}{\mathit{\boldsymbol{X}}_{in}}$ (3)

 ${L_{in}}({\mathit{\boldsymbol{\beta }}_n}|{\mathit{\boldsymbol{X}}_{in}}, {\mathit{\boldsymbol{\beta }}_{Xn}}) = {\rm{exp}}({V_{in}}({\mathit{\boldsymbol{\beta }}_n}))/\sum\limits_i {{\rm{exp}}} ({V_{in}}({\mathit{\boldsymbol{\beta }}_n}))$

 ${P_n}\left( i \right) = {\smallint _{{\mathit{\boldsymbol{\beta }}_n}}}{L_{in}}({\mathit{\boldsymbol{\beta }}_n}|{\mathit{\boldsymbol{X}}_{in}}, {\mathit{\boldsymbol{\beta }}_{Xn}})f({\mathit{\boldsymbol{\beta }}_{Xn}}){\rm{d}}{\mathit{\boldsymbol{\beta }}_{Xn}}$ (4)

2 模型参数的贝叶斯估计方法

 ${U_{jn,t}} = {\text{ }}{\mathit{\boldsymbol{X}}_{jn,t}}{\mathit{\boldsymbol{\beta }}_n} + {\varepsilon _{jn,t}}$

 $L({\mathit{\boldsymbol{y}}_n}|\mathit{\boldsymbol{b}}, \mathit{\boldsymbol{ \boldsymbol{\varSigma} }}) = \smallint L({\mathit{\boldsymbol{y}}_n}|{\mathit{\boldsymbol{\beta }}_n})\phi ({\mathit{\boldsymbol{\beta }}_n}|\mathit{\boldsymbol{b}}, \mathit{\boldsymbol{ \boldsymbol{\varSigma} }}){\rm{d}}{\mathit{\boldsymbol{\beta }}_n}$ (5)

 $H\left( {\mathit{\boldsymbol{b}},\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}|\mathit{\boldsymbol{Y}}} \right) \propto \prod L({\text{ }}{\mathit{\boldsymbol{y}}_n}|\mathit{\boldsymbol{b}},\mathit{\boldsymbol{ \boldsymbol{\varSigma} }})h\left( {\mathit{\boldsymbol{b}},\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}} \right)$

 $\begin{array}{l} H\left( {\mathit{\boldsymbol{b}},\mathit{\boldsymbol{ \boldsymbol{\varSigma} }},{\mathit{\boldsymbol{\beta }}_n}|\mathit{\boldsymbol{Y}}} \right) \propto \\ \prod\limits_n L ({\mathit{\boldsymbol{y}}_n}|{\mathit{\boldsymbol{\beta }}_n})\phi ({\mathit{\boldsymbol{\beta }}_n}|\mathit{\boldsymbol{b}},\mathit{\boldsymbol{ \boldsymbol{\varSigma} }})h\left( {\mathit{\boldsymbol{b}},\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}} \right) \end{array}$

Gibbs抽样是Metropolis-Hastings算法的一个特例, 它遍历所有参数区块, 在所有区组参数的最新抽样值条件下, 对每一候选参数进行抽样.Gibbs抽样可以从该后验分布进行抽样, 对于βn的抽样均是在其余2个参数的条件上进行的, 如下所示:

 $H({\mathit{\boldsymbol{\beta }}_n}|\mathit{\boldsymbol{b}}, \mathit{\boldsymbol{ \boldsymbol{\varSigma} }}, {\mathit{\boldsymbol{y}}_n}) \propto L({\mathit{\boldsymbol{y}}_n}|{\mathit{\boldsymbol{\beta }}_n})\phi ({\mathit{\boldsymbol{\beta }}_n}|\mathit{\boldsymbol{b}}, \mathit{\boldsymbol{ \boldsymbol{\varSigma} }})$ (6)

 $F = \frac{{L({\mathit{\boldsymbol{y}}_n}|{{\mathit{\boldsymbol{\tilde \beta }}}_{n, 1}})\phi ({{\mathit{\boldsymbol{\tilde \beta }}}_{n, 1}}|\mathit{\boldsymbol{b}}, \mathit{\boldsymbol{ \boldsymbol{\varSigma} }})}}{{L({\mathit{\boldsymbol{y}}_n}|{\mathit{\boldsymbol{\beta }}_{n, 0}})\phi ({\mathit{\boldsymbol{\beta }}_{n, 0}}|\mathit{\boldsymbol{b}}, \mathit{\boldsymbol{ \boldsymbol{\varSigma} }})}}$

3 车身产品离散选择模型的实证

 图 1 集成贝叶斯的层次选择模型 Fig.1 Integrated Bayesian hierarchy selection model

(1) 模型1.采用固定系数β进行贝叶斯参数估计.

(2) 模型2.采用随机系数βn进行贝叶斯参数估计.

 图 2 M1层次选择模型随机系数分布 Fig.2 Distribution of random coefficient of M1 level selection model
4 结语

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