這篇文章探討產品性質需求模型(pure characteristics demand model)的最優化問題，該模型是一種隨機係數需求模型並在消費者的效用函數中沒有特殊的邏輯誤差項。邏輯基礎班德氏分解法(logic-based Benders decomposition)是個出色解決方案不但能解決大規模問題，還能處理混合整數規劃，隨機規劃等。我們提出了一種透過邏輯基礎班德氏分解法將混整數規劃和約束規劃相結合的混合演算法，而模型的目標式是最大程度減少誤差項。原問題分解為由市場份額分配組成的主問題和許多複雜的子問題。切割生成規劃(cut generation program)專門處理混合整數子問題並基於對偶子問題信息獲得最小鬆弛有效班德氏切割(Benders cut)。本篇研究目的為快速分析2001 年至2015 年英國汽車市場的消費者的選擇行為。本篇研究結論是班德氏模型明顯優於整數規劃模型。我們也證明了所提出的班德氏切割是有效的，並且提出演算法收斂於最優性。

This thesis addresses an optimization problem to estimate pure characteristics demand models, a class of random coefficients demand models without the idiosyncratic logit error term in consumer's utility function. A logic-based Benders decomposition is an outstanding solution method for tackling large scale problems,
mixed-integer programming, stochastic programming, etc. The hybrid algorithm is proposed that combines mixed-integer programming and constraint programming via logic-based Benders decomposition with cut generation program to minimize the pure error. The original estimation problem is firstly decomposed into a market-share allocation master problem and a number of complex subproblems. The cut generation method is a strategy to deal with mixed-integer subproblems and obtain minimally relaxed valid Benders cut based on dual subproblem information. The purpose of this research is to quickly analyze consumers' choice behavior faced by UK vehicle market from 2001 to 2015. It is concluded that the Benders models substantially outperform the integer programming model. We prove that the proposed Benders cut is valid and the algorithm converge to optimality.

摘要………………………………………………………………………………………………………………………………………………………………i
Abstract……………………………………………………………………………………………………………………………………………………ii
List of Tables……………………………………………………………………………………………………………………………………v
Chapter 1 Introduction……………………………………………………………………………………………………………1
Chapter 2 Literature Review………………………………………………………………………………………………4
2.1 Pure Characteristics Demand Model…………………………………………………………………4
2.2 Classical Benders Decomposition………………………………………………………………………6
2.3 Logic-based Benders Decomposition…………………………………………………………………7
2.4 Benders Decomposition with Mixed-Integer Subproblem…………………9
Chapter 3 Pure Characteristic Demand Model for Estimating Pure Error……………………………………………11
3.1 Motivating Source Problem………………………………………………………………………………………11
3.2 Symbols Explanation………………………………………………………………………………………………………11
3.3 Consumer’s Utility Function…………………………………………………………………………………13
3.4 Market Shares Calculation………………………………………………………………………………………15
3.5 Maximization of Utility……………………………………………………………………………………………16
3.6 Mixed-Integer Programming Formulation………………………………………………………20
Chapter 4 Decomposition Algorithm………………………………………………………………………………22
4.1 Assignment Master Problem…………………………………………………………………………………………22
4.2 Sequencing Subproblem…………………………………………………………………………………………………23
4.3 Cut Generation Program………………………………………………………………………………………………24
4.3.1 Optimality Condition……………………………………………………………………………………………25
4.3.2 Generic Benders Cut………………………………………………………………………………………………30
4.3.3 Cut Generation Method…………………………………………………………………………………………36
4.3.4 Relaxed Cut Generation Method……………………………………………………………………38
4.3.5 Algorithm…………………………………………………………………………………………………………………………41
Chapter 5 Conclusions………………………………………………………………………………………………………………47
References………………………………………………………………………………………………………………………………………………49

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