Market Basket will open its new Littleton store, which has been under construction off Route 119, by the end of the year. "I am writing to provide with the assurance that we are, after some delay, proceeding rather quickly and have directed all available resources to open the new Market Supermarket in Littleton by the end of December 2014," Arthur T. Demoulas, CEO of Demoulas Super Markets Inc., wrote in his Sept.