(FORTUNE Magazine) – In the early 1970s the Egyptian government asked Tom Saaty, a pioneering mathematician with a fistful of awards, to help clarify the Middle East conflict. The Egyptians needed a coherent, analytical way to assess the pros and cons of their less than cozy strategic relationship with the Soviet Union. Saaty, a Wharton professor with a background in arms-control research, tackled the question with game theory, a mathematical process used to analyze complex problems. "The Egyptians drew the conclusion that they should ask the Russians to leave Egypt," says Saaty, now a professor at the University of Pittsburgh's Katz School. The Egyptian government was pleased with his work (and eventually did ask the Russians to leave), but Saaty himself wasn't satisfied with the process. He felt his conclusion was incomplete--that important but intangible information was left out of the final equation because game theory was too rigid. "I couldn't use it to solve a real-life problem," he says.

So the academic started brainstorming in his sweltering Cairo hotel room. He flashed on youthful memories of growing up in Mosul, Iraq, where he lived until he was 14. "My grandmother was a very handsome, very pesky woman," the 72-year-old professor recalls. And she had her own method of figuring out thorny problems, says Saaty. "You'd say, 'Grandma, you don't have a Ph.D. in mathematics.' But she was well informed and would do better than any scientist." She identified sets of options in pairs, then weighed options against each other. Nothing revolutionary here. But Grandma went a step further. "She said, 'I like that better than that--a lot better,'" squeaks Saaty in a granny-esque falsetto, his thinning white hair flopping about. It dawned on Saaty that what he needed--and what Grandma had come up with in a rudimentary way--was a process that captured intensities of human feeling and assigned them numerical values.

So Saaty set about devising such a process, trying to mirror the way real people solve real problems. Governments, corporations, and all manner of institutions are run by people with ideas, agendas, and beliefs, Saaty reasoned, so rather than forcing decision-makers to repress this stuff, why not create a formula that would let them be explicitly opinionated about the problem at hand?

The methodology Saaty developed out of his lengthy ruminations is called the analytic hierarchy process. AHP is still a whippersnapper in the world of mathematics; and game theory, far from being supplanted, is alive and kicking. (The Federal Communications Commission used game theory to auction $7 billion worth of airwaves for wireless communications in 1995.) But for Saaty, an irrepressible man with a fondness for nonlinear math and painfully bad jokes, the proof is in his proof: AHP works, and is used widely for problem solving and decision-making. The Army and Air Force have used his AHP-based software, called Expert Choice. So have the State Department, Xerox, Boeing, GE, IBM, Northrop Grumman, U.S. Steel, and the governments of South Africa, Canada, and Indonesia. Soon corporate America will be hearing a lot more about AHP, because a determined and entrepreneurial disciple of the good professor has tailored the software for the FORTUNE 500. Aly Abulleil, a gregarious 52-year-old Egyptian emigre, is marketing his program, called Aliah (which means "high" or "peak" in Arabic), as a soup-to-nuts tool for business development.

There are early converts. "I was quite impressed," says Harvey Paskin, director of strategic planning at Northrop Grumman's electronic sensors and systems division. "The software is exotic. But the real value is in the interaction of the folks and the result you get." Frank Clark, managing director of long-range planning at U.S. Steel Group, says the software "gives you a discipline to professionally pick someone's gut." Steve Gessner, a program manager at IBM who has used the tool with product designers, says, "He has taken very complex mathematical algorithms and put a very elegant front end on them."

So what exactly is this analytic hierarchy process, and how does it work?

"When we do decision-making, we make paired comparisons with absolute numbers, like if one apple is five times larger than another," Saaty explains. We have other apple-judging criteria--color, flavor, aroma, texture. We can compare our apples by each criterion in the same manner until we exhaust all the matchups: A is five times bigger than B, we determine; A is three times redder than B; A is only half as flavorful as B; and so on. We can then take those paired comparisons and decide which of them are more or less important for our purposes: Is size more important than color? Is size more important than flavor? Is texture more important than size? and so on. Easy as pi.

Right about here is where most decision-makers begin to reach the limits of their processing power. As a decision increases in complexity, the number of possible pairs increases exponentially, and the mind loses track. AHP software keeps the sets of comparisons in order and lets you do what Saaty's grandmother did: choose which ingredients are more important and by how much. The AHP user can move all the variables around and crunch together the preferences of many people. "It helps structure complexity," says Ernest Forman, the George Washington University professor who crunched Saaty's algorithms into computer code 15 years ago on an old Apple II computer.

"When confronted with an extremely complex situation with a lot of what-if scenarios, we kind of shut down," explains Jaime Gomez, strategic-planning director at a Corning division that uses AHP. "We just don't know what to do. It's like playing 20 or 30 chess games simultaneously. What the tool does is isolate one particular scenario of that complexity at a time."

In some applications AHP is like a very big and sophisticated sorting machine. Ave Lethbridge, a manager in the human resources department at Toronto Hydro, used Saaty's Expert Choice AHP software to rank 800 job applicants, "from senior VPs right on down," for a major change initiative at the Canadian electric utility. There were too many criteria and too many applicants to evaluate candidates the old-fashioned way, recalls Lethbridge. "You can't do it in your head" and still do justice to the process. Using Expert Choice, department managers at Toronto Hydro established the criteria they were after and let the program sort through the applicants.

Gomez used Aliah's version of AHP to overhaul Optical Corp. of America, a maker of high-tech optical devices for the aerospace and defense industries that Corning had acquired. Corning bought OCA to strengthen its position in the field of photonics. But when Corning brought in its people to plot strategy with OCA's managers, they found they couldn't communicate. Corning execs talked revenue, bottom line, and market share; OCAers answered in engineer-speak, says Gomez.

So Gomez and his boss, Dan Acton, rounded up a group of OCA managers, hunkered down in a room with an Aliah-fueled computer, and brought in Abulleil as a facilitator to tackle three tasks: assess the direction of OCA's market, separate winning OCA products from dogs, and devise a plan to make money. And they did just that, says Gomez. Corning could have slashed staff and budgets on its own, but that would have alienated OCA's managers, who oversaw production of its successful, ultra-high-tech optics. And Corning could have called in consultants, says Gomez. "But what does a consulting company do when they come into an office?" asks Gomez rhetorically. "For the first three days they charge you an arm and a leg to educate themselves." Between OCA and Corning, they had the knowledge needed to retool the division, asserts Gomez. "If you just train people in the uses of the tool, that's a better way of doing it, in my opinion--where you don't waste so much time and money."

Using software for this kind of thing creates difficulties, and not just the obvious garbage-in/garbage-out issue. "The downside," says one manager who has used AHP, "is you can spend a lot of time just getting through the tool itself." Collecting pairwise comparisons from participants can be so painstaking that it makes planners long for the old whiteboard-and-coffee method.

Abulleil refers often to management pioneers--Peter Drucker, Igor Ansoff, Jack Welch, and especially Henry Mintzberg, author of The Rise and Fall of Strategic Planning. "Mintzberg says strategic planning is an oxymoron, that you cannot produce a process that will generate creative strategies," paraphrases Abulleil. "I say to you that I can have a process that leads into creativity by using the stress caused by the pairwise comparison." But the school of thought that holds that strategy comes from the gut resists the lure of a computer program. "The idea of doing strategic planning through software I find ludicrous," says Mintzberg himself (though he has not examined AHP software). Strategy, he says, "is a process of synthesis...a process of debate, discussion, exchange."

Nevertheless, Abulleil has grand ambitions. "We are going to become the de facto standard of strategic planning," says Abulleil. "I want people to say 'This is Aliah strategy' like they say 'This is Xerox, or Kleenex.'" A big dream for his tiny private firm, called Aliah Decision Models, which is on the same floor of the same Pittsburgh building as Saaty's Expert Choice and has revenues of $1.6 million and only 23 full-time employees. One of them is Abulleil's son Tarek, who maxed out his credit cards to help fund the firm in its early years. ("He gave up Carnegie Mellon to clean toilets and answer the phone," quips Abulleil. Actually, he manages accounting.)

So these guys are not in SAP's league. But last March, Aliah formed an alliance with Arthur Andersen's Pittsburgh practice. And this March, says Abulleil, IBM--which has been working with Aliah for years--agreed to use the software. (IBM refused to comment.)

Saaty, AHP's No. 1 booster, is willing to admit that AHP may not be the best tool for all decisions. "I don't recommend using AHP to go out to a restaurant. It would spoil my pleasure. I want caprice." Certain things are still best left to the gut.