Constructing the lineup for a game is one of the traditions of baseball that is, if not arcane, then at least subject to much theory and contemplation. With the advent of sabermetrics, new conjecture has arisen with all sorts of mathematical ideas that, while potentially sound, will likely never see the light of day because of baseball’s resistance to change, no matter the ideas’ merits (or lack thereof).
Traditionally
Most baseball fans are familiar with the basic concepts of lineup construction — that you want fast runners who get on base a lot in the first two positions, then your best overall hitters to drive them in — and of course, the catcher and pitcher are usually 8th and 9th respectively (at least in the National League sans DH). Also, as a rule, hitters are better at hitting pitchers of the opposite handedness (so righty hitters like to hit lefty pitchers) — as well as the converse, that pitchers are better at pitching to batters of the same handedness. So, you want to take your opponent’s starting pitcher into account and not put, for example, a bunch of lefty hitters in a row to face a lefty pitcher.
The Wikipedia article on batting order has a pretty good overview of the more-or-less traditional approach, as do this outline at WebBall, Teamwork Batting Order, and the batting order entry at Baseball Reference.
Special OPS
Another approach, simplified here, is to arrange your hitters in order of descending OPS. On Base Percentage (OBP) plus Slugging Percentage (SLG), as Wikipedia says, “combines both the ability of a player to get on base and to hit for power. It is very popular in measuring the offensive worth of a player, because it is easy to calculate and has a strong positive correlation, at the team level, with runs per game.” Those accustomed to the traditional order will undoubtedly find the resulting lineup mighty peculiar, perhaps downright counterintuitive, but the basic idea is that you want your best hitters hitting more often. (The higher in a lineup you bat, the more plate appearances you get per game, on average.)
What follows are some reading resources for anyone interested in exploring a little deeper.
Reading
Does Batting Order Matter? - “Optimal Baseball Batting Order Construction: a case study using the 2000 Toronto Blue Jays.”
Evaluating Traditional Lineups - “I will be analyzing two lineups: the composite NL and AL lineups used from 1993 to 2004. I will use a form of the Markov process model to evaluate lineup strength, and will attempt to evaluate, not several, or a couple, or 100, or a few thousand, but all possible lineup combinations - all 362880 of them.”
Managing to Fill Out the Lineup - “So it appears that the advantage of scoring first is not as important as scoring often, and that runs scored in bunches are more likely to lead to a win than playing for one run.”
Baseball Musings Lineup Analysis tool - “Based on work by Cyril Morong and Ken Arneson. Fill in nine players, OBA and Slugging Percentage, then press submit.”
…and the related stories — also at Baseball Musings, Analyzing the Analysis “Don Scotto at Beyond the Boxscore spent a lot of time with the Lineup Analysis tool and he’s figured out how to build the lineup without the computer’s help.” …and at Ducksnorts (a Padres blog), Because Someone Has to Bat Second “Last week in the discussion we were trying to figure out who would and/or should bat second for the Padres in 2006. Reader Steve pointed us to a projected batting order at Padres.com, which shows Mike Cameron in the #2 hole.”