Jeff Harmening took over the CEO job at General Mills last week, and the top of his agenda should be figuring out whether it’s time to get out of the U.S. yogurt business.

This may seem like a far-fetched idea for a key line of business for any new leader of a company like General Mills, which thinks of itself as a growth company. But the people in charge of growth companies that can’t seem to grow need to question the wisdom of doing more of what clearly hasn’t been working.

General Mills sales haven’t been growing, and one big reason is slipping sales in the U.S. yogurt business. There just have to be opportunities at least as good in other lines of business for the capital and the management attention that have been invested here in yogurt.

Admittedly, yogurt is not a small part of General Mills, with $1.3 billion of “yogurt and other” sales last full fiscal year in the U.S., or about 13 percent of sales for the company’s largest business segment.

Mills got its start by getting the U.S. license for the Yoplait brand back in the late 1970s, and since then General Mills’ marketing was one of the ways more and more Americans found out yogurt can be a good thing to eat. U.S. consumption of yogurt per person grew 645 percent from 1980 through 2013, as tracked by the U.S. Department of Agriculture.

General Mills decided to buy 51 percent interest in the Yoplait business back in the summer of 2011. At the time, Mills was second only to the French company Danone in U.S. retail market share, according to Euromonitor, a global market research ­provider.

Five years later, Danone’s market share, including the Dannon and Stonyfield Farm brands, had held up pretty well, still with roughly a third of the market. But General Mills’ market share had fallen by nearly 25 percent. It’s now third, also trailing the Greek yogurt innovator Chobani LLC.

Yogurt sales in the last reported quarter for General Mills in the U.S. retail business declined 20 percent, a steeper rate of decline than in the first part of the current fiscal year. This is far worse than the performance of the last full fiscal year, when sales of yogurt in the U.S. slid 7 percent.

The company pointed to a presentation that Harmening gave earlier this year at a Florida investor meeting for some background on its thinking, including its long-term commitment to yogurt. Yogurt is a $78 billion worldwide consumer market that is growing, as one slide in the presentation explained, obviously too attractive a market to quit. In the United States, it plans to both refresh its current products as well as go after premium segments like natural and organic.

The biggest problem here in the United States appears to be what’s printed on the label and not what’s inside. When Greek yogurt sales were surging, introducing Yoplait Greek sounded to many consumers a little like some big food company executive wanted to try to cash in on the Greek yogurt trend. Better to buy a real Greek yogurt, shoppers reasoned, rather than something with Yoplait in the name.

That’s how, in a matter of years, Chobani pulled ahead of Yoplait. It’s an amazing case study, a little like watching a start-up like Tesla overtake Ford in just a few years.

In fairness to General Mills, nothing obvious comes to mind as a quick way to improve the company’s position in the U.S. What the company is trying, including better-tasting products, seems to be a good idea at just about any food company.

On the other hand, a “product renovation” is just the kind of incremental move the consulting firm Bain & Co. discussed in a study of corporate divestitures some years ago, finding that big companies often put off making the tough call on whether to get out or invest a lot more, choosing to tinker instead.

Bain had observed, looking back two decades, that companies really good at knowing when and how to sell a business delivered far better returns to investors than the average company. And the just average companies used a default strategy of buy-and-hold in their portfolio of businesses.

The choices for any business unit that starts to go into the tank boil down to only three, according to the Bain analysis. One is to sell, another they called “milk” for cash flow and the other was “transform,” which is how the consultants talked about fundamentally changing the business, usually by committing a lot of additional time and capital.

Most companies look at these three choices and can’t ever seem to decide, maybe because executives can’t bring themselves to give up on a business that’s long been part of their success. But by putting off the decision they’ve effectively made a decision to milk it, and it continues to decline.

General Mills appears to have passed on at least one opportunity to shake up its yogurt business, the acquisition of Danone’s organic brand Stonyfield Farm. Danone first invested in this early organic innovator years ago, but this year it was forced to sell as part of a divestiture required to let a far bigger deal go through.

This is the kind of acquisition opportunity that does not come along very often. A far from definitive list gleaned from analyst comments earlier in the year has General Mills among several likely Stonyfield Farm bidders, and more candidates have appeared in news reports since then.

Here’s a thought: Maybe the losing bidders on the Stonyfield deal would jump if offered a chance to get into the U.S. market with Yoplait.