Operations Research Management Science Today - April 2003

<BODY bgcolor="#ffffff" link="cc0000" vlink="cc0000" text="#000000"> <img src="../cleardot.gif" width=1 height=1 vspace=4 border=0><br>   <table border=0 cellspacing=0 cellpadding=0 width=480><tr valign=top> <td width=10><img src="../cleardot.gif" width=1 height=1 hspace=4 border=0></td> <td width=470> <FONT face="verdana, arial, helvetica" SIZE=2><B><I><A HREF="../../ORMS.shtml" TARGET="_top">OR/MS Today</A> - April 2003</I></B></font><BR> <hr size="1"> <img src="cleardot.gif" width=1 height=1 vspace=4 border=0><br> <IMG SRC="art/Int-OR.jpg" WIDTH="468" HEIGHT="183" HSPACE="0" VSPACE="0" BORDER="0" ALT=" "> <font face="times new roman, times"> <img src="../cleardot.gif" width=1 height=1 vspace=4 border=0><br> <FONT face="verdana, arial, helvetica" SIZE=2><B>International OR</B></FONT><br> <IMG SRC="../barbluehook.gif" WIDTH="468" HEIGHT="4"><br> <img src="../cleardot.gif" width=1 height=1 vspace=8 border=0><br> <FONT SIZE="+2"><B>Batty over Cricket</B></FONT><BR> <img src="../cleardot.gif" width=1 height=1 vspace=2 border=0><br> <FONT SIZE="+1"><I>OR knows the score when it comes to the "ins" and "outs" of complicated sport</I></FONT><BR> <img src="../cleardot.gif" width=1 height=1 vspace=6 border=0><br> <FONT face="verdana, arial, helvetica" SIZE=2><B>By Tony Lewis, Michael Wright and Graham Rand</B></FONT><br> <hr size="1"> <img src="../cleardot.gif" width=1 height=1 vspace=12 border=0><br>    Many people's perception of cricket may be of English villages and delightful cricket grounds, as depicted in the accompanying photo of Shireshead Cricket Club, near Lancaster. Cricket, however, is an international sport that can involve big money. (England's failure to play a World Cup game in Zimbabwe in February may result in a $1.5 million fine). Cricket is also a business on which operations research has made a considerable impact.<BR> <img src="../cleardot.gif" width=1 height=1 vspace=6 border=0><br> Ten countries (Australia, Bangladesh, England, India, New Zealand, Pakistan, South Africa, Sri Lanka, West Indies and Zimbabwe) are full members of the International Cricket Council (ICC) and are qualified to play Test Matches (international matches played over five days). These top teams, plus four associate members, played in the 2003 Cricket World Cup in Africa earlier this year. Twenty-seven countries are associate members of the ICC, and another 49 countries are affiliated with the ICC. Two associate members, the United States and Canada, played the first international match in 1844.<BR> <img src="../cleardot.gif" width=1 height=1 vspace=6 border=0><br> Despite cricket's international nature, many readers are no doubt less than fully conversant with the game. The following text, culled from a tea towel, is a well-known introduction to cricket for foreigners:<BR> <img src="../cleardot.gif" width=1 height=1 vspace=12 border=0><br> <FONT face="verdana, arial, helvetica" SIZE=3><B>CRICKET</B></FONT><BR> <IMG SRC="../barbluehook.gif" WIDTH="468" HEIGHT="4"><br> <img src="../cleardot.gif" width=1 height=1 vspace=2 border=0><br> (As explained to a foreign visitor) <UL> <LI> You have two sides out in the field and one in.<BR> <img src="../cleardot.gif" width=1 height=1 vspace=2 border=0><br> <LI> Each man that's in the side that's in goes out and when he's out he comes in and the next man goes in until he's out.<BR> <img src="../cleardot.gif" width=1 height=1 vspace=2 border=0><br> <LI> When they are all out the side that's out comes in and the side that's been in goes out and tries to get those coming in out.<BR> <img src="../cleardot.gif" width=1 height=1 vspace=2 border=0><br> <LI> Sometimes you get men still in and not out.<BR> <img src="../cleardot.gif" width=1 height=1 vspace=2 border=0><br> <LI> When both sides have been in and out including the not outs ... </UL> That's the end of the game!<BR> <img src="../cleardot.gif" width=1 height=1 vspace=6 border=0><br> Cricket connoisseurs like to joke that the game is so complicated that it takes up to five days to play a match and nearly as long to explain it. Here is an explanation sufficient for understanding this article.<BR> <img src="../cleardot.gif" width=1 height=1 vspace=12 border=0><br> <FONT face="verdana, arial, helvetica" SIZE=3><B>Playing the Game</B></FONT><BR> <IMG SRC="../barbluehook.gif" WIDTH="468" HEIGHT="4"><br> <img src="../cleardot.gif" width=1 height=1 vspace=2 border=0><br> One-day cricket, a variant on the longer version of Test Match cricket or first-class cricket (three or four days), is a game between two teams each of 11 players. Upon the toss of a coin one team will "bat" and the other "field." The 11 players bat in pairs sequentially and continuously until a player is "out" (loses his wicket) and until either 10 of the batsmen are out or they run out of balls to be received which are grouped together into six-ball overs. Typical limitations are 50 overs (300 balls). All of this constitutes what is called one innings. The batting side tries to score as many runs as possible within its two limitations of overs and wickets. A total of 250 or more runs is quite common, but the average in international matches has been around 225.<BR> <img src="../cleardot.gif" width=1 height=1 vspace=6 border=0><br> At the end of the first innings the roles are reversed, and the objective of the team batting second, in their innings, is to exceed the first team's score within the same limitations of 50 overs and 10 wickets. One of the distinctions between baseball and cricket is that in baseball teams alternate from batting to fielding throughout the game, whereas in cricket all of a team's batting is done either before or after the opposition. It is this aspect of the game that has caused consternation when bad weather interrupts play, and the game has to be shortened after it has started.<BR> <img src="../cleardot.gif" width=1 height=1 vspace=6 border=0><br> Operations research has been applied to this situation so successfully that a new playing condition known as the Duckworth/Lewis rules (or D/L method) has been implemented worldwide. As a result, Tony Lewis, along with statistician and co-developer Frank Duckworth, is probably mentioned more frequently in the world media than any other operations researcher.<BR> <img src="../cleardot.gif" width=1 height=1 vspace=6 border=0><br> Whenever rain curtails an international match, the result is announced that A beat B by x runs, according to the Duckworth/Lewis method. The method allows the target score in one-day cricket to be reset when there are interruptions to play that result in a shortening of the innings of one or both sides. It's usually rain that does this, and so it's called the "rain-rule;" though a day's play has been shortened for other meteorological reasons including snow, hail, wind and sandstorms, as well as the occasional crowd riot. The rain rule is invoked for all such situations.<BR> <img src="../cleardot.gif" width=1 height=1 vspace=6 border=0><br> To illustrate the problem, suppose the team batting first, Team A, scored 250 runs in their 50 overs  that's an average of five runs per over. Then it rained between the innings, resulting in a delayed restart. To finish the game within the time allowed, Team B is only permitted 25 overs. What should be the revised target? The old answer, and one that survived for many years because of its simplicity, was to set the target in accordance with the Average Run Rate (i.e. five per over for 25 overs is 125) so to win the target is 126. This is very advantageous to Team B since they still have all 10 wickets with which to score the runs  the fewer overs there are to bat, the more risks that can be taken and a generally higher run rate can be sustained.<BR> <img src="../cleardot.gif" width=1 height=1 vspace=6 border=0><br> The Duckworth/Lewis solution recognizes that Team B, for half its overs, still has all 10 wickets and so has more than half its resources. Based on OR modeling principles, the method compares resources available to the teams to revise the target. Duckworth and Lewis devised a method based on modeling the relationship between average runs that are scored from any position of overs remaining and wickets already lost. Historical data and cricket common sense are used to estimate the parameters of the model.<BR> <img src="../cleardot.gif" width=1 height=1 vspace=6 border=0><br> The model of the average runs for wickets lost and overs left looks like Figure 1. Look first at the heavy, top line. This represents the notion of the diminishing returns nature of extra overs. With few overs to bat, average runs per over are high, reflecting the greater risks that can be taken. When there are more overs available, then the average scoring rate per over declines because more overs are not accompanied by more wickets. For example, the average score for 40 overs has been around 200 (five per over), but for 50 overs the average has been around 225 or 4.5 per over. The lower curves represent more limited average scoring capability in the remaining overs when some wickets have been lost. The curves are not strictly proportional because of the lesser batting skill of later batsmen in a team. The curves flatten out as the number of overs increases. Extra overs are of no use, on average. From some point onwards, everyone is out and the point at which this occurs is lower the more wickets you have already lost.<BR> <img src="../cleardot.gif" width=1 height=1 vspace=6 border=0><br> <IMG SRC="art/Cricket_Px_Fig-1.gif" WIDTH="450" HEIGHT="300" HSPACE="0" VSPACE="0" BORDER="0" ALT=" "><BR> <img src="../cleardot.gif" width=1 height=1 vspace=2 border=0><br> <FONT face="verdana, arial, helvetica" SIZE=1><B>Figure 1: Model of average runs obtained, for overs left and wickets lost</B></font><BR> <img src="../cleardot.gif" width=1 height=1 vspace=6 border=0><br> The next step in the modeling is to convert the figures of these graphs into percentages  percentages of the average score in a 50-over innings. Fifty is chosen because this is the most common one-day innings length and the standard for One Day Internationals. From match records, which the D/L model reflects, the average is 225 runs. An extract of the resulting percentages can be seen in Table 1. A full version is available to match officials and in the published booklet. These percentages are used as a representation of a team's resources they still have available. The revised target is set using the comparative resources of the two teams (not the overs as in the old method).<BR> <img src="../cleardot.gif" width=1 height=1 vspace=6 border=0><br> <TABLE BORDER="1" CELLSPACING="0" CELLPADDING="6" BGCOLOR="#FFFFFF" width=436> <TR ALIGN="Left" VALIGN="bottom"> <TD><font size="2" face="verdana, arial, helvetica"><B>overs</B></TD> <TD><font size="2" face="verdana, arial, helvetica"><B>Wickets<BR> Lost</B></TD> <TD><font size="2" face="verdana, arial, helvetica"> </TD> <TD><font size="2" face="verdana, arial, helvetica"> </TD> <TD><font size="2" face="verdana, arial, helvetica"> </TD> <TD><font size="2" face="verdana, arial, helvetica"> </TD> </TR> <TR ALIGN="Left" VALIGN="Top"> <TD><font size="2" face="verdana, arial, helvetica">left</TD> <TD><font size="2" face="verdana, arial, helvetica">0</TD> <TD><font size="2" face="verdana, arial, helvetica">2</TD> <TD><font size="2" face="verdana, arial, helvetica">5</TD> <TD><font size="2" face="verdana, arial, helvetica">7</TD> <TD><font size="2" face="verdana, arial, helvetica">9</TD> </TR> <TR ALIGN="Left" VALIGN="Top"> <TD><font size="2" face="verdana, arial, helvetica">50</TD> <TD><font size="2" face="verdana, arial, helvetica">100.0</TD> <TD><font size="2" face="verdana, arial, helvetica">83.8</TD> <TD><font size="2" face="verdana, arial, helvetica">49.5</TD> <TD><font size="2" face="verdana, arial, helvetica">26.5</TD> <TD><font size="2" face="verdana, arial, helvetica">7.6</TD> </TR> <TR ALIGN="Left" VALIGN="Top"> <TD><font size="2" face="verdana, arial, helvetica">40</TD> <TD><font size="2" face="verdana, arial, helvetica">90.3</TD> <TD><font size="2" face="verdana, arial, helvetica">77.6</TD> <TD><font size="2" face="verdana, arial, helvetica">48.3</TD> <TD><font size="2" face="verdana, arial, helvetica">26.4</TD> <TD><font size="2" face="verdana, arial, helvetica">7.6</TD> </TR> <TR ALIGN="Left" VALIGN="Top"> <TD><font size="2" face="verdana, arial, helvetica">30</TD> <TD><font size="2" face="verdana, arial, helvetica">77.1</TD> <TD><font size="2" face="verdana, arial, helvetica">68.2</TD> <TD><font size="2" face="verdana, arial, helvetica">45.7</TD> <TD><font size="2" face="verdana, arial, helvetica">26.2</TD> <TD><font size="2" face="verdana, arial, helvetica">7.6</TD> </TR> <TR ALIGN="Left" VALIGN="Top"> <TD><font size="2" face="verdana, arial, helvetica">25</TD> <TD><font size="2" face="verdana, arial, helvetica">68.7</TD> <TD><font size="2" face="verdana, arial, helvetica">61.8</TD> <TD><font size="2" face="verdana, arial, helvetica">43.4</TD> <TD><font size="2" face="verdana, arial, helvetica">25.9</TD> <TD><font size="2" face="verdana, arial, helvetica">7.6</TD> </TR> <TR ALIGN="Left" VALIGN="Top"> <TD><font size="2" face="verdana, arial, helvetica">20</TD> <TD><font size="2" face="verdana, arial, helvetica">58.9</TD> <TD><font size="2" face="verdana, arial, helvetica">54.0</TD> <TD><font size="2" face="verdana, arial, helvetica">40.0</TD> <TD><font size="2" face="verdana, arial, helvetica">25.2</TD> <TD><font size="2" face="verdana, arial, helvetica">7.6</TD> </TR> <TR ALIGN="Left" VALIGN="Top"> <TD><font size="2" face="verdana, arial, helvetica">10</TD> <TD><font size="2" face="verdana, arial, helvetica">34.1</TD> <TD><font size="2" face="verdana, arial, helvetica">32.5</TD> <TD><font size="2" face="verdana, arial, helvetica">27.5</TD> <TD><font size="2" face="verdana, arial, helvetica">20.6</TD> <TD><font size="2" face="verdana, arial, helvetica">7.5</TD> </TR> </TABLE> <img src="../cleardot.gif" width=1 height=1 vspace=2 border=0><br> <FONT face="verdana, arial, helvetica" SIZE=1><B>Table 1: Extract of resource percentages remaining.</B></font><BR> <img src="../cleardot.gif" width=1 height=1 vspace=6 border=0><br> A 50-over innings represents 100 percent, but starting a 25-over innings with all 10 wickets, the tables indicate that a team has 68.7 percent of the resources compared to 50 overs. If an innings is interrupted at any stage, the table is used to calculate resources lost due to a stoppage.<BR> <img src="../cleardot.gif" width=1 height=1 vspace=6 border=0><br> Consider a real example from one of the first actual uses of the D/L method  the final of the International Cricket Council Trophy in Kuala Lumpur in April 1997. Kenya scored 241 in 50 overs, then rain between the innings left Bangladesh with 25 overs. The score to beat was 68.7 percent of 241, which is 165.6, providing the target of 166 for Bangladesh. Generally this was thought to be fair, except, of course, by Bangladesh who would have much preferred to have been chasing the old average-run-rate target of 121. And the result? Bangladesh won the match and the trophy  but off the very last ball of the final over. The D/L method had helped produce a very exciting finish.<BR> <img src="../cleardot.gif" width=1 height=1 vspace=6 border=0><br> The method is flexible enough to handle stoppages at any stage of either innings, including within an innings when they come off the field and the same innings resumes but with fewer overs left to be received on the resumption. It handles multiple stoppages to either innings, and also some technical situations when one team is penalized overs.<BR> <img src="../cleardot.gif" width=1 height=1 vspace=6 border=0><br> The methodology is summarized in a booklet authored by Duckworth and Lewis. Several Web pages give a general summary (see references).<BR> <img src="../cleardot.gif" width=1 height=1 vspace=12 border=0><br> <FONT face="verdana, arial, helvetica" SIZE=3><B>OR Invades Cricket</B></FONT><BR> <IMG SRC="../barbluehook.gif" WIDTH="468" HEIGHT="4"><br> <img src="../cleardot.gif" width=1 height=1 vspace=2 border=0><br> The first breakthrough in the implementation of the Duckworth/Lewis method occurred in 1995 when Frank and Tony got the ear of the chief executive of the ICC, as well as the ear of the cricket secretary of the England and Wales Cricket Board (ECB). The officials were fairly impressed but asked the researchers to do some more analysis with international data and to present their findings in July 1996 to the full council of the ICC, which they did.<BR> <img src="../cleardot.gif" width=1 height=1 vspace=6 border=0><br> <TABLE BORDER="0" CELLSPACING="0" CELLPADDING="5" align=right> <TR ALIGN="Left" VALIGN="Top"> <TD><IMG SRC="art/Cricket_Px_Newman-grfx.gif" WIDTH="159" HEIGHT="300"> <BR> <img src="../cleardot.gif" width=1 height=1 vspace=2 border=0><br> <FONT face="verdana, arial, helvetica" SIZE=1><B><I>Cartoon courtesy of Nick Newman</I></B></font><BR> </TD> </TR> </TABLE> After a few months of reflection and another presentation, Zimbabwe agreed to use the method for England's forthcoming tour, and England decided to trial the method domestically in 1997. Shortly after that, the ICC decided to use it for their Trophy competition in Kuala Lumpur in 1997. The next year, several other countries took it on board, especially since it was becoming increasingly likely that it would be used at the 1999 World Cup in England. Sri Lanka and Australia, the last of the major countries, came on board to try out the method ahead of the World Cup. It turned out, to everyone's surprise, that the weather in England was fine for the World Cup competition, although the method did get substantial use in the warm-up matches.<BR> <img src="../cleardot.gif" width=1 height=1 vspace=6 border=0><br> After the World Cup, the ICC decided to give the method a two-year full international trial in all countries under its jurisdiction. When the trial was successful, the Duckworth/Lewis method became the International Standard rain-rule subject to triennial review.<BR> <img src="../cleardot.gif" width=1 height=1 vspace=6 border=0><br> Since the method can be implemented using just the tables and a pocket calculator, it is beginning to be used not only at international and first-class levels but also at lower levels of cricket including age internationals all the way down to club levels. The method has been used more than 350 times to date, and very few of these have caused undue comment after the initial period of learning how it works. It has been perceived as complicated, but only until the principles are explained and understood. The perception of complexity may be the result of the sight of the full "over by over" table, reawakening horrible memories of school math in cricket journalists who are not usually numerate (see cartoon). At the matches the adjusted target score is posted in advance of the resumption of play so everyone knows what's going on. When rain threatens, players are sometimes spotted consulting the tables at the end of each over to determine targets for the short-term.<BR> <img src="../cleardot.gif" width=1 height=1 vspace=12 border=0><br> <FONT face="verdana, arial, helvetica" SIZE=3><B>OR Helps Scheduling</B></FONT><BR> <IMG SRC="../barbluehook.gif" WIDTH="468" HEIGHT="4"><br> <img src="../cleardot.gif" width=1 height=1 vspace=2 border=0><br> Scheduling is another area where OR has been applied to cricket, just as it's been applied to scheduling in other sports, such as baseball, as well as many other non-sporting situations. Scheduling problems are probably familiar to all of us in one guise or another  how to accommodate hard constraints (e.g. certain teams can't play on the same ground at the same time; all teams must play each other once or twice; etc.). Television and sponsors have requirements that need to be met. In addition, there numerous soft constraints that are desirable, if not necessary, such as limits on the number of days between matches and umpire pairings, not to mention the cost and time of travel.<BR> <img src="../cleardot.gif" width=1 height=1 vspace=6 border=0><br> A Test and County Cricket Board employee reportedly used to shut himself away in virtual hermetic seclusion for three or four weeks in January and February each year in order to schedule the first-class umpires in England. When Mike Wright heard about this, he volunteered to tackle the problem using OR methodology. Wright's work is detailed in the paper, "Scheduling English cricket umpires" that appeared in the Journal of the Operational Research Society in 1991 [3]. Along with important technical points regarding the construction of objective functions for complex problems of this general type, the paper is full of fascinating details for the cricket aficionado.<BR> <img src="../cleardot.gif" width=1 height=1 vspace=6 border=0><br> Wright's first paper on scheduling was followed by a second the following year [4]. A new team, Durham, was to be added to the domestic cricket championship. It was perceived, and so it turned out, that they would be a weak team. Unfortunately, at that time there was not a balanced fixture list. Each team played some teams twice, and some teams once. Those that played Durham twice would probably have an advantage. The question facing Wright: How can fairness be established? The paper explains how the issue was resolved in conjunction with the client, resulting in a four-year schedule that was as fair as possible. The solution was never implemented, however, as the structure of the championship was changed so that every team played every other team exactly once.<BR> <img src="../cleardot.gif" width=1 height=1 vspace=6 border=0><br> A third paper in this stream describes the work that Wright did to schedule the fixtures in the English domestic cricket season, using a novel form of tabu search [5]. Wright developed an approach using "intensification" and "diversification"  two opposing strategies used in local search. Broadly speaking, intensification involves the gradual improvement of a solution, while diversification allows a major change to be made to avoid a local optimum. The main technical advance reported in this paper concerns the precise approach to diversification, influencing the search process to move into potentially fruitful new areas. He has since further developed this approach and used it for related problems, such as school timetabling [6].<BR> <img src="../cleardot.gif" width=1 height=1 vspace=6 border=0><br> The English cricket authorities have been so pleased with Wright's work that he has been providing schedules for them on an annual basis since 1990. At the end of each season in September, he receives detailed requirements and preferences from the English Counties for the following season. His model takes these into account as well as other ECB requirements. The allocation of umpires comes later, early in the year, again taking umpires' preferences into account as well as those of the ECB.<BR> <img src="../cleardot.gif" width=1 height=1 vspace=6 border=0><br> OR's contribution to cricket was so substantial that the U.K.'s Operational Research Society awarded medals to Duckworth, Lewis and Wright for their work [1, 5] in this arena.<BR> <img src="../cleardot.gif" width=1 height=1 vspace=6 border=0><br> <CENTER><TABLE BORDER="0" CELLSPACING="0" CELLPADDING="1" BGCOLOR="#990000" width=440> <TR ALIGN="Center" VALIGN="Middle"> <TD><IMG SRC="art/shim.gif" VSPACE=1 BORDER=0 HEIGHT=1 WIDTH=1> <TABLE BORDER="0" CELLSPACING="0" CELLPADDING="6" BGCOLOR="#FFFFFF" width=436> <TR ALIGN="Left" VALIGN="Top"> <TD> <CENTER><font color="#990000" size="3" face="verdana, arial, helvetica"><B>Useful Web Sites</B></font></CENTER><BR> <font size="2" face="verdana, arial, helvetica"> <A HREF="http://www.cricket.org/link_to_database/ABOUT_CRICKET/EXPLANATION/CRICKET_EXPLAINED_FOR_NOVICES.html" TARGET="_new">www.cricket.org/link_to_database/ABOUT_CRICKET/<BR> EXPLANATION/CRICKET_EXPLAINED_FOR_NOVICES.html</A><br> <img src="../cleardot.gif" width=1 height=1 vspace=2 border=0><br> <A HREF="http://www.cricket.org/link_to_database/ABOUT_CRICKET/RAIN_RULES/DUCKWORTH_LEWIS_2002.html" TARGET="_new">www.cricket.org/link_to_database/ABOUT_CRICKET/<BR> RAIN_RULES/DUCKWORTH_LEWIS_2002.html</A><br> <img src="../cleardot.gif" width=1 height=1 vspace=2 border=0><br> <A HREF="http://www.cricket.org/link_to_database/ABOUT_CRICKET/RAIN_RULES/DL_FAQ.html" TARGET="_new">www.cricket.org/link_to_database/ABOUT_CRICKET/<BR> RAIN_RULES/DL_FAQ.html</A><BR> </TD> </TR> </TABLE> <IMG SRC="art/shim.gif" VSPACE=1 BORDER=0 HEIGHT=1 WIDTH=1> </TD> </TR> </TABLE> </CENTER> <img src="../cleardot.gif" width=1 height=1 vspace=12 border=0><br> <FONT face="verdana, arial, helvetica" SIZE=3><B>References</B></FONT><BR> <IMG SRC="../barbluehook.gif" WIDTH="468" HEIGHT="4"><br> <img src="../cleardot.gif" width=1 height=1 vspace=2 border=0><br> <OL> <font size=1> <LI> Duckworth, F.C. and Lewis, A.J, 1998, "A fair method of resetting the target in interrupted one-day cricket matches," <I>Journal of the Operational Research Society,</I> Vol. 49, pgs. 220-227. <LI> Duckworth, Frank and Lewis, Tony, 1999, "Your comprehensive guide to the Duckworth/Lewis method of target resetting in one-day cricket," University of the West of England, Bristol. <LI> Wright, M.B., 1991, "Scheduling English cricket umpires," <I>Journal of the Operational Research Society,</I> Vol. 42, pgs. 447-452. <LI> Wright, M.B., 1992, "A fair allocation of county cricket opponents," <I>Journal of the Operational Research Society,</I> Vol. 43, pgs. 195-201. <LI> Wright, M.B., 1994, "Timetabling county cricket fixtures using a form of tabu search," <I>Journal of the Operational Research Society,</I> Vol. 45, pgs. 758-770. <LI> Wright, M.B., 1996, "School timetabling using heuristic search," <I>Journal of the Operational Research Society,</I> Vol. 47, pgs. 347-357.</font> </OL> <img src="../cleardot.gif" width=1 height=1 vspace=4 border=0><br> <IMG SRC="../barbluehook.gif" WIDTH="468" HEIGHT="4"><br> <img src="../cleardot.gif" width=1 height=1 vspace=2 border=0><br> <FONT face="verdana, arial, helvetica" SIZE=2><I><B>Tony Lewis</B> (<A HREF="mailto:ajlewis@brookes.ac.uk">ajlewis@brookes.ac.uk</A>) is a senior lecturer in the Business School at Oxford Brookes University. He no longer plays cricket but watches avidly. <B>Michael Wright</B> (<A HREF="mailto:m.wright@lancaster.ac.uk">m.wright@lancaster.ac.uk</A>) and <B>Graham Rand</B> (<A HREF="mailto:g.rand@lancaster.ac.uk">g.rand@lancaster.ac.uk</A>) are senior lecturers in Lancaster University's Management School, and still play cricket. </i><BR>  <img src="../cleardot.gif" width=1 height=1 vspace=5 border=0><br>  <img src="../barblue.gif" width=468 height=1 border=0 alt=" "><br> <img src="../cleardot.gif" width=1 height=1 vspace=8 border=0><br> <font face="verdana, arial, helvetica" size="1"> <CENTER><script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script></CENTER><br> <img src="../cleardot.gif" width=1 height=1 vspace=8 border=0><br> <LI> <A HREF="tocfr.html" Target="_top"><I>Table of Contents</A><br> <img src="../cleardot.gif" width=1 height=1 vspace=4 border=0><br> <LI> <A HREF="../../ORMS.shtml" TARGET="_top"><I>OR/MS Today</I> Home Page</A><br>  <img src="../cleardot.gif" width=1 height=1 vspace=8 border=0><br> <img src="../barblue.gif" width=468 height=1 border=0 alt=" "><br> <b><i>OR/MS Today</i></b> copyright � 2003 by <A HREF="http://www.informs.org/" target="_new">the Institute for Operations Research and the Management Sciences</a>. 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