"We are encouraged by the progress we've made to realign crew, their schedules and our fleet," Southwest said in a statement Thursday.
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"With another holiday weekend full of important connections for our valued customers and employees, we are eager to return to a state of normalcy. We know even our deepest apologies -- to our customers, to our employees and to all affected through this disruption -- only go so far."
The airline statement also said, "We plan to return to normal operations with minimal disruptions on Friday."
Southwest officials said Thursday that the airline was operating roughly one-third of its schedule. In all, 2,364 Southwest flights were scrubbed Thursday -- about 58% of the carrier's schedule. Another 206 were delayed.
By Friday morning, there had been no flight cancellations out of LAX, or any other major Southern California airport - with the exception of two flights at Ontario International Airport.
Southwest CEO: 'Just no way to apologize enough' for disastrous week of flight cancelations
Southwest CEO: 'Just no way to apologize enough'
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It was a stark comparison to the past week, which saw nearly 16,000 canceled flights, according to Flightaware.
The airline blamed the chaos on last week's brutal winter storm, but company and labor officials are also blaming outdated technology systems that left scheduling systems jammed and crews scattered.
Southwest has set up a page at Southwest.com/traveldisruption for customers to submit refund and reimbursement requests for meals, hotel and alternate transportation, as well as to connect customers to their baggage.
"We have much work ahead of us, including investing in new solutions to manage wide-scale disruptions," the airline said. "We aim to serve our customers and employees with our legendary levels of Southwest hospitality and reliability again very soon."
Thousands of passengers and their luggage also remained in limbo in Southern California and across the nation as Southwest works to recover from the past week.
City News Service contributed to this report.