INTEGER OPTIMIZATION FOR THE SELECTION OF A TWENTY20 CRICKET TEAM by Mark Lourens Submitted in fulfillment of the requirements for the degree of Magister Scientiae in the Faculty of Science at the Nelson Mandela Metropolitan University January 2009 Supervisor: Mr G. D. Sharp Co-Supervisor: Prof. J. W. Gonsalves Abstract During the last few years, much effort has been devoted to measuring the ability of sport teams, as well as that of the individual players. Much research has been on the game of cricket, and the comparison, or ranking, of players according to their abilities. This study continues preceding research using an optimization approach, namely, a binary integer programme, to select an SA domestic Pro20 cricket team. Keywords Cricket Team selection Optimization Binary integer programming i Acknowledgements I would like to sincerely express my appreciation to the following: • My Lord Jesus, Thou art my God, and I will praise thee: thou art my God, I will exalt thee. O give thanks unto the LORD; for he is good: for his mercy endureth for ever. • My wife, Colleen, thank you for all your help, support and encouragement when it was really needed. • My supervisor, Mr. Gary Sharp, thank you for all your help, guidance and encouragement you gave through the period of this study. • My co-supervisor, Prof. J. W. Gonsalves, thank you for the input contributed during my study either directly, or indirectly. • The NMMU Research Office for the financial support provided in the form of the NMMU Postgraduate Research Scholarship. • The NMMU Director of Research Capacity Development, Dr Shaleen Els, for all your support and advice. • The NRF for Grant No. TTK 2005072900024. • UTi Material Handling for the financial support, as well as allowing me the much needed study leave. ii Declaration In accordance with Rule G4.6.3, I hereby declare that this dissertation is my own work and that it has not previously been submitted for assessment to another University or for another qualification. Full Name: Mark Lourens Student Number: 201337584 Qualification: Magister Scientiae in Mathematical Statistics Signature: Date: 18 th April 2009 iii Contents Abstract ............................................................................................................................... i Acknowledgements ............................................................................................................ ii Declaration ........................................................................................................................ iii Contents ............................................................................................................................. iv 1. Introduction ................................................................................................................... 1 1.1 Introduction to Cricket ......................................................................................... 1 1.2 Objective ............................................................................................................... 4 2. Literature Review .......................................................................................................... 7 3. Methodology ................................................................................................................ 16 3.1 Introduction ........................................................................................................ 16 3.2 The Data ............................................................................................................. 16 3.3 Definition of Statistics ........................................................................................ 17 3.4 Optimization – Theory and Models .................................................................... 22 3.4.1 The Linear Programming Model ................................................................. 22 3.4.2 The Binary Integer Programming Model .................................................... 23 4. The Theory behind the Models .................................................................................. 26 4.1 Model 1 – H.H. Lemmer (2002, 2004) ............................................................... 26 4.1.1 Bowling ....................................................................................................... 26 4.1.2 Batting ......................................................................................................... 28 4.2 Model 2 – H. Gerber & G.D. Sharp (2006) ........................................................ 31 4.3 Model 3 – M. Lourens ........................................................................................ 36 5. The Results of the Models ........................................................................................... 44 5.1 Model 1 – H.H. Lemmer (2002, 2004) ............................................................... 44 iv 5.1.1 Bowling ....................................................................................................... 45 5.1.2 Batting ......................................................................................................... 46 5.2 Model 2 – H. Gerber & G.D. Sharp (2006) ........................................................ 48 5.3 Model 3 – M. Lourens ........................................................................................ 52 5.4 Discussion of the Results .................................................................................... 55 6. Conclusion and Further Work ................................................................................... 63 Appendices ....................................................................................................................... 65 A. Player Statistics ........................................................................................................... 65 A.1 Batting Statistics ................................................................................................. 65 A.2 Bowling Statistics ............................................................................................... 67 A.3 Fielding Statistics ............................................................................................... 69 A.4 Wicket-keeping Statistics ................................................................................... 72 B. Player Ability Measures ............................................................................................. 73 B.1 H.H. Lemmer’s Ability Measures ...................................................................... 73 B.1.1 Bowling Ability Measure (CBR) ................................................................ 73 B.1.2 Batting Performance Measure (BP) ............................................................ 75 B.2 H. Gerber and G.D. Sharp’s Ability-Index Coefficients .................................... 77 B.3 M. Lourens’s Ability-Index Coefficients ........................................................... 81 References ........................................................................................................................ 85 v Chapter 1 Introduction 1.1 Introduction to Cricket The game of cricket has been in existence since at least the mid-16 th century; however the first official test match was not played until 1877, between England and Australia. This form of cricket remained the standard until 1963, when limited-overs cricket was introduced to counter declining interest in the longer version of the game. 1 Limited-overs international cricket, better known as one day international (ODI) cricket, is the most common version of the game seen in the media lately. An ODI match is completed in one day with both teams having 50 overs, unlike test cricket, which is played over five days. A more recent addition to the game of cricket is the Pro20, also known as Twenty20 cricket. Pro20 cricket was first played domestically in the United Kingdom in 2003 1 to draw more spectators to the game. It consists of each team having just one innings of 20 overs each, comparable to the ODI game of 50 overs. There are several differences making the Pro20 game more exciting and more spectator-friendly than the normal ODI game; these differences are summarized as follows2: • Each team shall bat for 20 overs, unless all out earlier. • A free hit for a “front foot” no-ball. • No-ball penalized by 2 runs. • 7-over fielding restrictions. • A minimum of 5 bowlers, bowling a maximum of 4 overs each. • A minimum of 7 overs per side constitutes a match. 1 MSN Encarta. 2 Retrieved from http://www.standardbank.co.za/site/pro20/rules.html 1 These differences bring about a different approach to batting and bowling strategies, which in turn affect measures of performance. The first strategy affected by these differences, is the batting strategy. There are two common measures of performance for batting, namely batting average and strike rate. Both these measures need to be as high as possible to identify a good batsman. This, however, may not always be the case as these measures in themselves have some shortcomings and may be very misleading. The batting average is calculated as the total number of runs scored by a batsman divided by the total number of innings in which the batsman is dismissed. An advantage of this measure is that it generally provides a good idea of how well a batsman has scored over the selected number of innings he has batted in. A limitation in using this measure as a measure of performance is that