# Bidding strategy using multivariate distribution and EM algorithm

In a number of industries, a significant percentage of business is acquired by means of aggressive sealed bidding. Construction contracts undoubtedly are a common instance of this. In Hong Kong, construction market is a significant business and there are more than 10,000 registered construction businesses. The ideal approach to allocate construction projects to construction companies is by employing competitive sealed bidding. Every bidding contractor establishes a bidding price using a cost-plus method. The moment the contractor’s cost is estimated, a mark-up as a % of the cost is added, and the sum is the tender price. The problem experiencing a competing contractor is in establishing the tender price at an acceptable level. A contractor may lose to the rivals if his tender is high. In contrast, the project defintely won’t be profitable if the tender price is way too low. For that reason, the development of an effective bidding strategy is a major factor in the survival of construction businesses.

Contents

Chapter 1: Introduction
1.1 Background to Competitive Bidding
1.2 Classical Competitive Bidding Model
1.3 General Bidding Model
1.4 Overview of the Thesis
Chapter 2: Literature Review
2.1 Introduction
2.2 Competitive Bidding Models
2.2.1 Friedman’s Model
2.2.2 Gates’s Model
2.2.3 Carr’s Model
2.2.4 Limitation of the Classical Bidding Model
2.3 The EM Algorithm
2.3.1 Application of the EM Algorithm
2.3.1.1 Missing Data
2.3.2 Convergence Properties of the EM Algorithm
Chapter 3: Competitive Bidding Models
3.1 Two Main Criticisms of Previous Models
3.2 Contractor Groups
3.3 Measure of Competitiveness
3.3.1 Index of competitiveness – Bid to Cost Ratio
3.3.2 Index of competitiveness – Bid to Lowest Bid Ratio
3.4 Non-Serious Bids
Chapter 4: EM Algorithm
4.1 Introduction to the EM Algorithm
4.2 The EM Algorithm Applied to Competitive Bidding Models
4.3 Three Sets of Data Due to Missing Values
4.4 The EM Algorithm Applied to the Bid to Cost Ratio
4.5 The EM Algorithm Applied to Bid to Lowest Bid Ratio
4.5.1 Relation between Truncated Bivariate Normal Distribution and Truncated Normal Distribution
4.5.2 The Expectation Step of the EM Algorithm
4.5.2.1 Obtaining Good Initial Values using the Least Square Method
4.6. Maximization Step of the EM Algorithm
4.7 Verification of the EM algorithm for Missing Values
Chapter 5: Application to Competitive Bidding Models
5.1 Data Used and Contractor Groups
5.2 Non-Serious Bids
5.3 Competitive Bidding Models
5.3.1 The Cost Model
5.3.1.1 Index of Competitiveness in the Cost Model
5.3.1.2 Log-Normal Distributions and their Parameters
5.3.2 The Lowest Bid Model
5.3.2.1 Index of Competitiveness in the Lowest Bid Model
5.3.2.2 Truncated Log-Normal Distribution
5.3.2.3 Moments of Univariate Truncated Normal Distribution
5.4 The Building of a Multivariate Distribution
5.5 Application of the EM algorithm
5.5.1 EM Algorithm for Finding the Correlation Coefficient in the Cost Model
5.5.2 EM Algorithm for Finding the Correlation Coefficient in the Lowest Bid Model
5.6 Probability of Winning and Optimal Mark-up
5.6.1 Probability of winning & optimal mark-up for the Cost Model
5.6.2 Probability of winning and optimal mark-up for the Lowest Bid Model
Chapter 6: Conclusions……….

Source: City University of Hong Kong