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Sen. Roy Blunt, Missouri Republican, asked why inspectors couldn’t be spared and the cuts fall elsewhere within the Food Safety and Inspection Service — something the American Meat Institute has said should be possible. The institute argued that inspectors should be deemed essential employees, and therefore exempt from furloughs.

But Mr. Werfel said the law doesn’t read that way in this case, and said the furloughs will have to happen.

“It becomes a math issue, ultimately,” he said. “There is no way in which to find other sources of funds because 88 percent of the entire budget are those very people that need to be at those meat plants doing that inspection to keep them open.”

• Tim Devaney contributed to this report.