The CEO testified Tuesday before a Senate committee about the Max, which has been banned from flying since March.
The hearing, the highest-profile congressional scrutiny of commercial aviation safety in years, heaps pressure on a newly rejigged Boeing senior management team fighting to fix trust with airline customers and passengers shaken by an eight-month safety ban on its 737 MAX following the crashes.
In an emotional exchange, Muilenburg described a private meeting Tuesday with the victims' loved ones - many of whom have attended the hearings displaying large pictures of their smiling children, siblings and spouses. Are you working for free from now on until you can cure this problem?
When discussing conversations with family members during the hearing, Muilenburg repeatedly shifted in his chair and slowed his speech. He turned to her and mentioned: "I'm sorry". Representative John Garamendi of California, for instance accused the company of "pushing profits over quality and safety".
Sen. Maria Cantwell (D-WA) said being the leader in aviation manufacturing requires responsibility.
Boeing's shares were down almost 2 percent in afternoon trading.
U.S. Rep. Peter DeFazio, who heads the House transportation and infrastructure committee, kicked off the hearing by criticizing Boeing for "a lack of candour" over the jetliner's development, and asked why Boeing approved a safety-critical flight control system known as MCAS - linked to two deadly crashes - vulnerable to a single point of failure.
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One document showed an MCAS failure alert was not added to the plane's control panel and another warned if a pilot took longer than 10 seconds to activate the anti-ditch software it could lead to a "catastrophic" failure even though the standard reaction time was four seconds.
He said the company erred when it made a cockpit alert to inform pilots of disagreements between the 737 Max's two angle-of-attack sensors available on only some of the jets instead of all, saying "we got that wrong up front". Tuesday's listening to represented Boeing's broadest acceptance of duty that it made errors, however Muilenburg stopped in need of what some lawmakers and relations had sought.
"We are deeply, deeply sorry, we'll never forget and I want you to know that and we're committed to making the improvements we need to make", he said in his opening statement to the House Transportation Committee.
The 737 MAX, when it returns, will have a number of new safeguards.
In one document from 2015 a Boeing employee questioned the decision to permit MCAS to be triggered by only one of the two sensors mounted on the jet's nose.
"We can and must do better", Muilenburg acknowledged. "We own that. We are responsible for our planes", Muilenburg said. "We've spent over 100,000 engineering and test hours, we've flown more than 800 test flights", Muilenburg said.
Last week, Indonesia investigators found Boeing failed to grasp risks in the design of MCAS.
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"That could be the case", Muilenburg said.
"We need answers. We need reforms on how commercial aircraft are certified", and how manufacturers like Boeing "are watched" by regulators, he added.
Muilenburg said that he had met with some family members of the victims before the Wednesday's hearing.
Boeing's fleet of 737 Max planes has been grounded while officials examined the software issues that contributed to the Lion Air and Ethiopian Air crashes, implemented software upgrades, and retrained pilots. Boeing did not comment Tuesday on the allegation.
Muilenburg told reporters on Wednesday he believes the allegation was in response to concerns about a change in the increase of the production rate.
DeFazio didn't detail the nature of the safety concerns raised by the Boeing manager or how the company responded. Toward the end of the hearing, Representative Jesus Garcia of IL called for Muilenburg's resignation, saying Boeing was either negligent, incompetent or corrupt, starting at the top.
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