"Wins above replacement," or WAR, is a complex statistical measure of a major league baseball player's

performance that sums up a player's total contribution to a team using various offensive and defensive

statistics. It reflects the number of additional wins a team has gained with a player relative to the

number of expected wins with a replacement$-$level player at a minimal cost. The highest WAR measure

for a player last season was $10\mathrm{.}0.$ The Sox's general manager is evaluating $10$ free agent players to

possibly sign to contracts for next season. Following are the WAR measures $($from last season$),$ contract

demands, and ages of these players.

Contract$ Contract

Player WAR $($millions$)$ Years Age

Mike Bass $4\mathrm{.}62\mathrm{.}125$

Josh Donald, p $3\mathrm{.}710\mathrm{.}028$

Miguel Gomez $5\mathrm{.}917\mathrm{.}4231$

Carlos Cabrera $9\mathrm{.}124\mathrm{.}0532$

Matt Carper, p $7\mathrm{.}218\mathrm{.}0334$

Evan Longier $4\mathrm{.}34\mathrm{.}327$

Paul Goldsmith $2\mathrm{.}73\mathrm{.}1136$

Joey Veto $5\mathrm{.}216\mathrm{.}5329$

Robinson Canoe $8\mathrm{.}124\mathrm{.}0628$

Matt Fists, p $7\mathrm{.}59\mathrm{.}3231$

For those players bargaining for multiyear contracts, the dollar values are spread equally over the years

of the contract. The general manager has some specific guidelines for signing new players.

$$ Donald, Carper, and Fists are pitchers, and the general manager wants to sign at least

one pitcher.

$$ He wants to limit the contracts' dollars for next season to $$34$ million, and the overall

value of the contracts to not exceed $$55$ million to remain under the league salary cap

$($and not be taxed for exceeding the cap$).$

$$ He does not want the average contract length to exceed $2\mathrm{.}5$ years, and he wants the

average age of the players he signs to be $31$ years or less.

Formulate and solve an integer programming model to determine which players the general manager

should sign to contracts to maximize the overall WAR of the players.