As the Cincinnati Reds have delved deeper into the analytics side of baseball, they hired some real-life math majors. That will come in handy this offseason because what the club faces is essentially a math problem, i.e., how to get the most out the player payroll.

Payroll is a factor every offseason, but it paramount this offseason because the Reds plan to be players on the free-agent market.

“We certainly feel like the window is open,” Dick Williams, the Reds’ president of baseball operations, told Bobby Nightengale of The Enquirer. “When the window is open, it’s time to make a different kind of trade and a different kind of free-agent deal than the ones you made in the past. You’re not trying to win every trade in terms of long-term capital. Now, you are more short-term focused.”

The Reds say the payroll is going up – they aren’t saying how much.

“We don’t comment on the specifics of payroll,” Williams said, “but directionally, we are going to continue to build and invest in this team and have more financial resources available for us. Our payroll will be bigger this year. There will be good opportunities to invest that payroll.”

Will that increase by enough to fix the bullpen and add an impact bat? Those seem to be the glaring needs for 2020. So let’s take a stab at what kind of money the Reds have to spend.

The payroll last year, according to Baseball Prospectus, was $126.6 million, the highest in club history and up over $25 million from 2018.

The guess here the 2020 payroll will approach $140 million. That would give the Reds significant money to spend on the free-agent market.

How much? Let’s do some math:

The Reds have five players currently under contract who will make $57.5 million: Joey Votto ($25 million), Sonny Gray ($10.1 million), Eugenio Suarez ($9.5 million), Raisel Iglesias ($9 million) and Tucker Barnhart ($3.9 million).

They also hold a $5.5 million option on Freddy Galvis, which they are almost certain to pick up.

That puts them at $63 million.

The next big number is what the arbitration-eligible players will get. The Reds have a slew of them. Anthony DeSclafani, Jose Peraza, Curt Casali, Michael Lorenzen, Derek Dietrich, Trevor Bauer and Kevin Gausman are arbitration-eligible.

The Reds could non-tender any of the above. That’s not going to happen with DeSclafani, Lorenzen, Casali and Bauer. My guess is it doesn’t happen with Peraza either.

It’s hard to see them bringing back Gausman, who made $9.35 million this year. Dietrich would make $3 million or so in arb, but he’s a non-tender candidate based on his second half and emergence of Josh VanMeter.

Everyone in arbitration gets significant raises. But Peraza, DeSclafani, Lorezenzen and Casali aren’t going to break the budget. If they double their salaries from last year, they make a total of $15.4 million – Peraza ($5.4 million), DeSclafani ($4.2 million), Lorenzen ($3.9 million), Casali ($1.9 million).

Bauer will get the biggest chunk of arb money. He made $13 million this year. No player went into arbitration with that high of a salary last year. The highest was Anthony Rendon. He went from $12.3 million to $18.8 million through arb. He was coming off a much better year than Bauer.

My guess is Bauer ends up making $18 million or so.

The arbitration-guesstimate and money committed to signed players adds up $96.4 million for 11 players. The Reds will have 11 or 12 players on the roster, making around the minimum of $550,000. Let’s say that adds up to $7 million.

That gives you a grand total of $103.5 million.

So if the player pool is $140 million, the Reds would have north of $36 million to spend.

That’s a good chunk, especially when you consider the starting rotation is in good shape with Gray, Luis Castillo, Bauer and DeSclafani under control.

The Reds are on record as saying the want to upgrade the offense. The obvious place to do that is the outfield. Right now, on paper, the outfield is Nick Senzel in center, Aristides Aquino in right and Phillip Ervin/Jesse Winker sharing left.

Senzel is also an option to move back to second base.

There are some solid free agents outfielders on the market: Marcell Ozuna (.243 average, 29 homers, 81 RBI, .804 OPS), Corey Dickerson (.309, 12, 59, .906, in 79 games), Nicholas Castellanos (.289, 27, 73, .862) and Starling Marte (.295, 23, 85, .845).

The Reds looked to upgrade catching last year. Former Red Yasmani Grandal (.246, 28, 77, .845) could be available. The Brewers have a mutual option with him.

The most interesting name among relievers: Aroldis Chapman. He can opt-out of his contract with the Yankees after this year. Chapman, 31, doesn’t strike them out like he used to, but he put his best ERA (2.21) since 2016.

The Reds may not get any of the above-mentioned players. But the difference is this offseason is they’re in a position to shop for them.