A manual assembly line is to be designed to make a small toy product. The work elements, their times, and precedence constraints are given in the table below. The workers will work only $1$ shift per day, each shift will last $7$ hours, including $1$ hour for lunch. Local labor laws require that the workers be allowed a $10-$minute break every $1$ work hour. The current work order requires that $342$ toys must be produced per day.