At a soup kitchen in Harlem, Toyota’s engineers cut down the wait time for dinner to 18 minutes from as long as 90. At a food pantry on Staten Island, they reduced the time people spent filling their bags to 6 minutes from 11. And at a warehouse in Bushwick, Brooklyn, where volunteers were packing boxes of supplies for victims of Hurricane Sandy, a dose of kaizen cut the time it took to pack one box to 11 seconds from 3 minutes. Toyota has “revolutionized the way we serve our community,” said Margarette Purvis, the chief executive and president of the Food Bank. But Toyota’s initial offer to the charity in 2011 was met with apprehension.

“They make cars; I run a kitchen,” said Daryl Foriest, director of distribution at the Food Bank’s pantry and soup kitchen in Harlem. “This won’t work.”

When Toyota insisted it would, Mr. Foriest presented the company with a challenge. “The line of people waiting to eat is too long,” Mr. Foriest said. “Make the line shorter.”

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